2 require "../eruby_util.rb"
26 ``Okay. Your duties are as follows: Get Breen. I don't care
27 how you get him, but get him soon. That faker! He posed for twenty
28 years as a scientist without ever being apprehended. Well, I'm going
29 to do some apprehending that'll make all previous apprehending look
30 like no apprehension at all. You with me?''
33 Battle, very much confused. ``What's that thing you have?''
35 ``Piggy-back heat-ray. You transpose the air in its path into an
36 unstable isotope which tends to carry all energy as heat. Then you
37 shoot your juice light, or whatever along the isotopic path and you
38 burn whatever's on the receiving end. You want a few?''
40 ``No,'' said Battle. ``I have my gats. What else have you
41 got for offense and defense?'' Underbottam opened a cabinet and
42 proudly waved an arm. ``Everything,'' he said.
44 ``Disintegraters, heat-rays, bombs of every type. And impenetrable
45 shields of energy, massive and portable. What more do I need?''
47 From THE REVERSIBLE REVOLUTIONS by Cecil Corwin, Cosmic Stories, March
48 1941. Art by Morey, Bok, Kyle, Hunt, Forte. Copyright expired.
50 \normalsize\normalfont
52 <% begin_sec("Fields of Force",0) %>\index{force!fields of}\index{fields of force}
54 Cutting-edge science readily infiltrates popular culture,
55 though sometimes in garbled form. The Newtonian imagination
56 populated the universe mostly with that nice solid stuff
57 called matter, which was made of little hard balls called
58 atoms. In the early twentieth century, consumers of pulp
59 fiction and popularized science began to hear of a new image
60 of the universe, full of x-rays, N-rays, and Hertzian waves.
61 What they were beginning to soak up through their skins was
62 a drastic revision of Newton's concept of a universe made of
63 chunks of matter which happened to interact via forces. In
64 the newly emerging picture, the universe was \emph{made} of
65 force, or, to be more technically accurate, of ripples in
66 universal fields of force. Unlike the average reader of
67 Cosmic Stories in 1941, you now possess enough technical
68 background to understand what a ``force field'' really is.
70 <% begin_sec("Why fields?") %>
72 <% begin_sec("Time delays in forces exerted at a distance") %>
74 What convinced physicists that they needed this new concept
75 of a field of force? Although we have been dealing mostly
76 with electrical forces, let's start with a magnetic example.
77 (In fact the main reason I've delayed a detailed discussion
78 of magnetism for so long is that mathematical calculations
79 of magnetic effects are handled much more easily with the
80 concept of a field of force.) First a little background
81 leading up to our example. A bar magnet, \figref{baratoms}, has an axis
82 about which many of the electrons' orbits are oriented. The
83 earth itself is also a magnet, although not a bar-shaped
84 one. The interaction between the earth-magnet and the bar
85 magnet, \figref{barhang}, makes them want to line up their axes in
86 opposing directions (in other words such that their
87 electrons rotate in parallel planes, but with one set
88 rotating clockwise and the other counterclockwise as seen
89 looking along the axes). On a smaller scale, any two bar
90 magnets placed near each other will try to align themselves head-to-tail, \figref{barns}.
97 %q{A bar magnet's atoms are (partially) aligned.}
105 %q{A bar magnet interacts with our magnetic planet.}
113 %q{Magnets aligned north-south.}
121 %q{The second magnet is reversed.}
129 %q{Both magnets are reversed.}
134 Now we get to the relevant example. It is clear that two
135 people separated by a paper-thin wall could use a pair of
136 bar magnets to signal to each other. Each person would feel
137 her own magnet trying to twist around in response to any
138 rotation performed by the other person's magnet. The
139 practical range of communication would be very short for
140 this setup, but a sensitive electrical apparatus could pick
141 up magnetic signals from much farther away. In fact, this is
142 not so different from what a radio does: the electrons
143 racing up and down the transmitting antenna create forces on
144 the electrons in the distant receiving antenna. (Both
145 magnetic and electric forces are involved in real radio
146 signals, but we don't need to worry about that yet.)
148 A question now naturally arises as to whether there is any
149 time delay in this kind of communication via magnetic (and
150 electric) forces. Newton would have thought not, since he
151 conceived of physics in terms of instantaneous action at a
152 distance. We now know, however, that there is such a time
153 delay. If you make a long-distance phone call that is routed
154 through a communications satellite, you should easily be
155 able to detect a delay of about half a second over the
156 signal's round trip of 50,000 miles. Modern measurements
157 have shown that electric, magnetic, and gravitational forces
158 all travel at the speed of light, $3\times10^8$ m/s. (In
159 fact, we will soon discuss how light itself is made of
160 electricity and magnetism.)
162 If it takes some time for forces to be transmitted through
163 space, then apparently there is some \emph{thing} that
164 travels \emph{through} space. The fact that the phenomenon
165 travels outward at the same speed in all directions strongly
166 evokes wave metaphors such as ripples on a pond.
169 <% begin_sec("More evidence that fields of force are real: they carry energy.") %>\label{fieldenergy}
171 The smoking-gun argument for this strange notion of
172 traveling force ripples comes from the fact that they carry energy.
174 First suppose that the person holding the bar magnet on the
175 right decides to reverse hers, resulting in configuration
176 \figref{barnn}. She had to do mechanical work to twist it, and if she
177 releases the magnet, energy will be released as it flips
178 back to \figref{barns}. She has apparently stored energy by going from
179 \figref{barns} to \figref{barnn}. So far everything is easily explained without
180 the concept of a field of force.
182 But now imagine that the two people start in position \figref{barns}
183 and then simultaneously flip their magnets extremely quickly
184 to position \figref{barsn}, keeping them lined up with each other the
185 whole time. Imagine, for the sake of argument, that they can
186 do this so quickly that each magnet is reversed while the
187 force signal from the other is still in transit. (For a more
188 realistic example, we'd have to have two radio antennas, not
189 two magnets, but the magnets are easier to visualize.)
190 During the flipping, each magnet is still feeling the forces
191 arising from the way the other magnet \emph{used} to be
192 oriented. Even though the two magnets stay aligned during
193 the flip, the time delay causes each person to feel
194 resistance as she twists her magnet around. How can this be?
195 Both of them are apparently doing mechanical work, so they
196 must be storing magnetic energy somehow. But in the
197 traditional Newtonian conception of matter interacting via
198 instantaneous forces at a distance, interaction energy arises
199 from the relative positions of objects that are interacting
200 via forces. If the magnets never changed their orientations
201 relative to each other, how can any magnetic energy have been stored?
203 The only possible answer is that the energy must have gone
204 into the magnetic force ripples crisscrossing the space
205 between the magnets. Fields of force apparently carry energy
206 across space, which is strong evidence that they are real things.
208 This is perhaps not as radical an idea to us as it was to
209 our ancestors. We are used to the idea that a radio
210 transmitting antenna consumes a great deal of power, and
211 somehow spews it out into the universe. A person working
212 around such an antenna needs to be careful not to get too
213 close to it, since all that energy can easily cook flesh (a
214 painful phenomenon known as an ``RF burn'').
218 <% begin_sec("The gravitational field") %>\index{gravitational field}\index{field!gravitational}
225 The wind patterns in a certain area of the ocean could be
226 charted in a ``sea of arrows'' representation like this. Each
227 arrow represents both the wind's strength and its direction at a
233 Given that fields of force are real, how do we define,
234 measure, and calculate them? A fruitful metaphor will be the
235 wind patterns experienced by a sailing ship. Wherever the
236 ship goes, it will feel a certain amount of force from the
237 wind, and that force will be in a certain direction. The
238 weather is ever-changing, of course, but for now let's just
239 imagine steady wind patterns. Definitions in physics are
240 operational, i.e., they describe how to measure the thing
241 being defined. The ship's captain can measure the wind's
242 ``field of force'' by going to the location of interest and
243 determining both the direction of the wind and the strength
244 with which it is blowing. Charting all these measurements on
245 a map leads to a depiction of the field of wind force like
246 the one shown in the figure. This is known as the ``sea of
247 arrows'' method of visualizing a field.
249 Now let's see how these concepts are applied to the
250 fundamental force fields of the universe. We'll start with
251 the gravitational field, which is the easiest to understand.
252 As with the wind patterns, we'll start by imagining gravity
253 as a static field, even though the existence of the tides
254 proves that there are continual changes in the gravity field
255 in our region of space.
256 When the gravitational field was introduced in chapter
257 \ref{ch:2}, I avoided discussing its direction explicitly, but
258 defining it is easy enough: we simply go to the
259 location of interest and measure the direction of the
260 gravitational force on an object, such as a weight tied to
261 the end of a string.\index{gravitational field}\index{field!gravitational}
263 In chapter \ref{ch:2}, I defined the gravitational field in
264 terms of the energy required to raise a unit mass through a unit distance.
265 However, I'm going to give a different definition now, using an
266 approach that will be more easily adapted to electric and magnetic fields.
267 This approach is based on force rather than energy.
268 We couldn't carry out the energy-based definition without dividing by
269 the mass of the object involved, and the same is true for the force-based definition.
270 For example, gravitational forces are weaker on the moon than on
271 the earth, but we cannot specify the strength of gravity
272 simply by giving a certain number of newtons. The number of
273 newtons of gravitational force depends not just on the
274 strength of the local gravitational field but also on the
275 mass of the object on which we're testing gravity, our
276 ``test mass.'' A boulder on the moon feels a stronger
277 gravitational force than a pebble on the earth. We can get
278 around this problem by defining the strength of the
279 gravitational field as the force acting on an object,
280 \emph{divided by the object's mass:}
282 The gravitational field vector, $\vc{g}$, at any location in space is
283 found by placing a test mass $m_t$ at that point. The field vector
284 is then given by $\vc{g}=\vc{F}/m_t$, where $\vc{F}$ is the gravitational
285 force on the test mass.
287 We now have three ways of representing a gravitational field. The
288 magnitude of the gravitational field near the surface of
289 the earth, for instance, could be written as 9.8 N/kg, 9.8 $\zu{J}/\zu{kg}\cdot\zu{m}$,
290 or 9.8 $\zu{m}/\zu{s}^2$. If we already had two names for it, why invent
291 a third? The main reason is that
292 it prepares us with the right approach for defining other fields.
294 The most subtle point about all this is that the gravitational
295 field tells us about what forces \emph{would} be exerted on
296 a test mass by the earth, sun, moon, and the rest of the
297 universe, \emph{if} we inserted a test mass at the point in
298 question. The field still exists at all the places where
299 we didn't measure it.
301 \begin{eg}{Gravitational field of the earth}\label{eg:gravfieldofsphere}
303 What is the magnitude of the earth's gravitational
304 field, in terms of its mass, $M$, and the distance
308 Substituting $|\vc{F}|= GMm_{t}/ r^2$ into the definition of the
309 gravitational field, we find $|\vc{g}|= GM/ r^2$. This expression
310 could be used for the field of any spherically symmetric mass distribution, since
311 the equation we assumed for the gravitational force would apply in any such
315 <% begin_sec("Sources and sinks") %>\index{sinks in fields}\index{sources of fields}
323 The gravitational field surrounding a clump of mass such as
334 The gravitational fields of the earth and moon superpose.
335 Note how the fields cancel at one point, and how there is no boundary
336 between the interpenetrating fields surrounding the two bodies.
342 If we make a \index{sea-of-arrows representation}sea-of-arrows
343 picture of the gravitational fields surrounding the earth,
344 \figref{sink}, the result is evocative of water going down a drain.
345 For this reason, anything that creates an inward-pointing
346 field around itself is called a sink. The earth is a
347 gravitational sink. The term ``source'' can refer specifically
348 to things that make outward fields, or it can be used as a
349 more general term for both ``outies'' and ``innies.''
350 However confusing the terminology, we know that gravitational
351 fields are only attractive, so we will never find a region
352 of space with an outward-pointing field pattern.
354 Knowledge of the field is interchangeable with knowledge of
355 its sources (at least in the case of a static, unchanging
356 field). If aliens saw the earth's gravitational field
357 pattern they could immediately infer the existence of the
358 planet, and conversely if they knew the mass of the earth
359 they could predict its influence on the surrounding
363 <% begin_sec("Superposition of \index{superposition of fields}\index{fields!superposition of}fields") %>
365 A very important fact about all fields of force is that when
366 there is more than one source (or sink), the fields add
367 according to the rules of vector addition. The gravitational
368 field certainly will have this property, since it is defined
369 in terms of the force on a test mass, and forces add like
370 vectors. Superposition is an important characteristics of
371 waves, so the superposition property of fields is consistent
372 with the idea that disturbances can propagate outward
377 \begin{eg}{Reduction in gravity on Io due to Jupiter's gravity}
379 The average gravitational field on Jupiter's moon
380 Io is 1.81 N/kg. By how much is this reduced when Jupiter is
381 directly overhead? Io's orbit has a radius of $ 4.22\times10^8$ m,
382 and Jupiter's mass is $ 1.899\times10^{27}$ kg.
385 By the shell theorem, we can treat the Jupiter as if its
386 mass was all concentrated at its center, and likewise for
387 Io. If we visit Io and land at the point where Jupiter is
388 overhead, we are on the same line as these two centers, so
389 the whole problem can be treated one-dimensionally, and
390 vector addition is just like scalar addition. Let's use
391 positive numbers for downward fields (toward the center of
392 Io) and negative for upward ones. Plugging the appropriate
393 data into the expression derived in example \ref{eg:gravfieldofsphere},
394 we find that the Jupiter's contribution to the field is
395 $- 0.71$ N/kg. Superposition says that we can find the actual
396 gravitational field by adding up the fields created by Io
397 and Jupiter: $1.81-0.71$ N/kg = 1.1 N/kg.
398 You might think that this reduction would create some
399 spectacular effects, and make Io an exciting tourist
400 destination. Actually you would not detect any difference if
401 you flew from one side of Io to the other. This is because
402 your body and Io both experience Jupiter's gravity, so you
403 follow the same orbital curve through the space around Jupiter.
412 The part of the LIGO gravity wave detector at
413 Hanford Nuclear Reservation, near Richland, Washington. The other half of the
414 detector is in Louisiana.
425 <% begin_sec("Gravitational waves") %>
426 \index{waves!gravitational}\index{gravitational waves}
428 A source that sits still will create a static field pattern,
429 like a steel ball sitting peacefully on a sheet of rubber. A
430 moving source will create a spreading wave pattern in the
431 field, like a bug thrashing on the surface of a pond.
432 Although we have started with the gravitational field as the
433 simplest example of a static field, stars and planets do
434 more stately gliding than thrashing, so gravitational waves
435 are not easy to detect. Newton's theory of gravity does not
436 describe gravitational waves, but they are predicted by
437 Einstein's general theory of relativity. J.H. Taylor and
438 R.A. Hulse \index{Taylor, J.H.}\index{Hulse, R.A.}
439 were awarded the Nobel Prize in 1993 for giving
440 indirect evidence that Einstein's waves actually exist. They
441 discovered a pair of exotic, ultra-dense stars called
442 neutron stars orbiting one another very closely, and showed
443 that they were losing orbital energy at the rate predicted
444 by Einstein's theory.
446 A Caltech-MIT collaboration has built
447 a pair of gravitational wave detectors
448 called LIGO\index{LIGO} to search for more direct evidence of gravitational
449 waves. Since they are essentially the most sensitive
450 vibration detectors ever made, they are located in quiet
451 rural areas, and signals will be compared between them to
452 make sure that they were not due to passing trucks. The
453 project began operating at full sensitivity in 2005, and is now able to
454 detect a vibration that causes a change of $10^{-18}$ m in the distance between
455 the mirrors at the ends of the 4-km vacuum tunnels. This is a thousand times
456 less than the size of an atomic nucleus!
457 There is only enough funding to keep the detectors
458 operating for a few more years, so
459 the physicists can only hope that during that time,
460 somewhere in the universe, a sufficiently violent cataclysm
461 will occur to make a detectable gravitational wave. (More
462 accurately, they want the wave to arrive in our solar system
463 during that time, although it will have been produced
464 millions of years before.)
468 <% begin_sec("The electric field",3) %>\index{electric field}\index{field!electric}
470 <% begin_sec("Definition") %>
472 The definition of the electric field is directly analogous
473 to, and has the same motivation as, the definition of the
476 The electric field vector, $\vc{E}$, at any location in space is
477 found by placing a test charge $q_t$ at that point. The electric
478 field vector is then given by $\vc{E}=\vc{F}/q_t$, where
479 $\vc{F}$ is the electric force on the test charge.
481 Charges are what create electric fields. Unlike gravity,
482 which is always attractive, electricity displays both
483 attraction and repulsion. A positive charge is a source of
484 electric fields, and a negative one is a sink.
486 The most difficult point about the definition of the
487 electric field is that the force on a negative charge is in
488 the opposite direction compared to the field. This follows
489 from the definition, since dividing a vector by a negative
490 number reverses its direction. It's as though we had some
491 objects that fell upward instead of down.
493 <% self_check('pointchargefield',<<-'SELF_CHECK'
494 Find an equation for the magnitude of the field of a
495 single point charge $Q$.
499 \begin{eg}{Superposition of electric fields }\label{eg:square}
501 Charges $q$ and $- q$ are at a distance $b$ from
502 each other, as shown in the figure. What is the electric
503 field at the point P, which lies at a third corner of the square?
506 The field at P is the vector sum of the fields
507 that would have been created by the two charges independently.
508 Let positive $x$ be to the right and let positive $y$ be up.
510 Negative charges have fields that point at them, so the
511 charge $-q$ makes a field that points to the right, i.e., has
512 a positive $x$ component. Using the answer to the self-check, we have
514 E_{-q,x} &= \frac{ kq}{ b^2} \\
515 E_{-q,y} &= 0 \qquad .
517 Note that if we had blindly ignored the absolute value signs
518 and plugged in $- q$ to the equation, we would have
519 incorrectly concluded that the field went to the left.
521 By the Pythagorean theorem, the positive charge is at a
522 distance $\sqrt{2} b$ from P, so the magnitude of its contribution to
523 the field is $E= kq/2 b^2$. Positive charges have
524 fields that point away from them, so the field vector is at
525 an angle of 135\degunit counterclockwise from the $x$ axis. \\
527 E_{q,x} &= \frac{ kq}{2 b^2} \zu{cos}\ 135\degunit \\
528 &= -\frac{ kq}{2^\zu{3/2} b^2} \\
529 E_{q,y} &= \frac{ kq}{2 b^2} \zu{sin}\ 135\degunit \\
530 &= \frac{ kq}{2^\zu{3/2} b^2}
535 E_\zu{x} &= \left(1-2^{-\zu{3/2}}\right)\frac{ kq}{ b^2} \\
536 E_{y} &= \frac{ kq}{2^\zu{3/2} b^2}
541 <% begin_sec("Dipoles") %>\index{dipole!electric}\index{electric dipole}
547 %q{Example \ref{eg:square}.}
556 A dipole field. Electric fields diverge from a positive
557 charge and converge on a negative charge.
566 %q{A water molecule is a dipole.}
571 The simplest set of sources that can occur with electricity
572 but not with gravity is the \index{dipole!electric}\emph{dipole},
573 consisting of a positive charge and a negative charge
574 with equal magnitudes. More generally, an electric dipole can
575 be any object with an imbalance of positive charge on one
576 side and negative on the other. A water molecule, \figref{watermolecule}, is a
577 dipole because the electrons tend to shift away from the
578 hydrogen atoms and onto the oxygen atom.
585 1. A uniform electric field created by some charges
586 ``off-stage.'' 2. A dipole is placed in the field. 3. The dipole aligns with the field.
595 Your microwave oven acts on water molecules with electric
596 fields. Let us imagine what happens if we start with a
597 uniform electric field, \figref{dipoleinfield}/1, made by some external charges,
598 and then insert a dipole, \figref{dipoleinfield}/2, consisting of two charges
599 connected by a rigid rod. The dipole disturbs the field
600 pattern, but more important for our present purposes is that
601 it experiences a torque. In this example, the positive
602 charge feels an upward force, but the negative charge is
603 pulled down. The result is that the dipole wants to align
604 itself with the field, \figref{dipoleinfield}/3. The microwave oven heats food
605 with electrical (and magnetic) waves. The alternation of the
606 torque causes the molecules to wiggle and increase the
607 amount of random motion. The slightly vague definition of a
608 dipole given above can be improved by saying that a dipole
609 is any object that experiences a torque in an electric field.
611 What determines the torque on a dipole placed in an
612 externally created field? Torque depends on the force, the
613 distance from the axis at which the force is applied, and
614 the angle between the force and the line from the axis to
615 the point of application. Let a dipole consisting of charges
616 $+q$ and $-q$ separated by a distance $\ell$ be placed in an
617 external field of magnitude $|\vc{E}|$, at an angle $\theta$ with
618 respect to the field. The total torque on the dipole is
620 \tau &= \frac{\ell}{2}q|\vc{E}|\sin \theta+\frac{\ell}{2}q|\vc{E}|\sin \theta \\
621 &= \ell q|\vc{E}|\sin \theta \qquad .
623 (Note that even though the two forces are in opposite
624 directions, the torques do not cancel, because they are both
625 trying to twist the dipole in the same direction.) The
626 quantity is called the dipole \index{moment!dipole}\index{dipole
627 moment}moment, notated $D$. (More complex dipoles can also
628 be assigned a dipole moment --- they are defined as having
629 the same dipole moment as the two-charge dipole that would
630 experience the same torque.)
632 Employing a little more mathematical elegance, we can define
633 a dipole moment \emph{vector},
635 \vc{D} = \sum q_i \vc{r}_i \qquad ,
637 where $\vc{r}_i$ is the position vector of the charge labeled by the
638 index $i$. We can then write the torque in terms of a vector
639 cross product (page \pageref{vectorcrossproductdef}),
641 \btau = \vc{D}\times\vc{E} \qquad .
644 No matter how we notate it, the definition of the dipole moment requires
645 that we choose point from which we measure all the position vectors
646 of the charges. However, in the commonly encountered special case where
647 the total charge of the object is zero,
648 the dipole moment is the same regardless of this choice.
652 \begin{eg}{Dipole moment of a molecule of NaCl gas}
654 In a molecule of NaCl gas, the center-to-center
655 distance between the two atoms is about 0.6 nm. Assuming
656 that the chlorine completely steals one of the sodium's
657 electrons, compute the magnitude of this molecule's dipole moment.
660 The total charge is zero, so it doesn't matter where we choose the
661 origin of our coordinate system. For convenience, let's choose
662 it to be at one of the atoms, so that the charge on that atom
663 doesn't contribute to the dipole moment.
664 The magnitude of the dipole moment is then
666 D &= (6\times10^{-10}\ \zu{m})( e) \\
667 &= (6\times10^{-10}\ \zu{m})( 1.6\times10^{-19}\ \zu{C}) \\
668 &= 1\times10^{-28}\ \zu{C}\cdot\zu{m}
674 <% fig('eg-dipole-vector','Example \ref{eg:dipole-vector}.')
677 \begin{eg}{Dipole moments as vectors}\label{eg:dipole-vector}
678 \egquestion The horizontal and vertical spacing between the charges in the figure is $b$. Find the dipole
681 \eganswer Let the origin of the coordinate system be at the leftmost charge.
683 \vc{D} &= \sum q_i \vc{r}_i \\
684 &= (q)(\vc{0})+(-q)(b\hat{\vc{x}})+(q)(b\hat{\vc{x}}+b\hat{\vc{y}})+(-q)(2b\hat{\vc{x}}) \\
685 &= -2bq\hat{\vc{x}}+bq\hat{\vc{y}}
689 <% begin_sec("Alternative definition of the electric field") %>\label{subsec:efielddipoledef}
691 The behavior of a dipole in an externally created field
692 leads us to an alternative definition of the electric field:
694 The electric field vector, $E$, at any location in space is
695 defined by observing the torque exerted on a test dipole
696 $D_t$ placed there. The direction of the field is the
697 direction in which the field tends to align a dipole (from
698 $-$ to +), and the field's magnitude is $|\vc{E}|=\tau/D_t\sin\theta$.
699 In other words, the field vector is the vector that satisfies the
700 equation $\btau = \vc{D}_t\times\vc{E}$ for any test dipole $\vc{D}_t$
701 placed at that point in space.
703 The main reason for introducing a second definition for the
704 same concept is that the magnetic field is most easily
705 defined using a similar approach.
710 In the definition of the electric field, does the test
711 charge need to be 1 coulomb? Does it need to be positive?
715 Does a charged particle such as an electron or proton
716 feel a force from its own electric field?
720 Is there an electric field surrounding a wall socket that
721 has nothing plugged into it, or a battery that is just sitting on a table?
725 In a flashlight powered by a battery, which way do the
726 electric fields point? What would the fields be like inside
727 the wires? Inside the filament of the bulb?
731 Criticize the following statement: ``An electric field
732 can be represented by a sea of arrows showing how current is flowing.''
736 The field of a point charge, $|\vc{E}|=kQ/r^2$, was
737 derived in a self-check. How would the field pattern
738 of a uniformly charged sphere compare with the field of a point charge?
742 The interior of a perfect electrical conductor in
743 equilibrium must have zero electric field, since otherwise
744 the free charges within it would be drifting in response to
745 the field, and it would not be in equilibrium. What about
746 the field right at the surface of a perfect conductor?
747 Consider the possibility of a field perpendicular to the
748 surface or parallel to it.
755 %q{Discussion question \ref{dq:dipole-moments}.}
760 \begin{dq}\label{dq:dipole-moments}
761 Compare the dipole moments of the molecules and molecular
762 ions shown in the figure.
766 Small pieces of paper that have not been electrically
767 prepared in any way can be picked up with a charged object
768 such as a charged piece of tape. In our new terminology, we
769 could describe the tape's charge as inducing a dipole moment
770 in the paper. Can a similar technique be used to induce not
771 just a dipole moment but a charge?
774 % ------------------------------------------------------------------------------------
778 <% begin_sec("Voltage Related to Field") %>%
779 \index{electric field!related to voltage}\index{voltage!related to electric field}
780 <% begin_sec("One dimension") %>
781 Voltage is electrical energy per unit charge, and electric
782 field is force per unit charge. For a particle moving in one
783 dimension, along the $x$ axis, we can therefore relate
784 voltage and field if we start from the relationship between
785 interaction energy and force,
787 \der U = -F_x\der x \qquad ,
789 and divide by charge,
791 \frac{\der U}{q} = -\frac{F_x}{q}\der x \qquad ,
795 \der V = -E_x \der x \qquad ,
799 \frac{\der V}{\der x} = -E_x \qquad .
801 The interpretation is that a strong
802 electric field occurs in a region of space where the voltage is
803 rapidly changing. By analogy, a steep hillside is a place on
804 the map where the altitude is rapidly changing.
806 \begin{eg}{Field generated by an electric eel}
808 Suppose an electric eel is 1 m long, and
809 generates a voltage difference of 1000 volts between its
810 head and tail. What is the electric field in the water around it?
813 We are only calculating the amount of field, not
814 its direction, so we ignore positive and negative signs.
815 Subject to the possibly inaccurate assumption of a constant
816 field parallel to the eel's body, we have
818 |\vc{E}| &= \frac{\der V}{\der x} \\
819 &\approx \frac{\Delta V}{\Delta x} \qquad \text{[assumption of constant field]} \\
820 &= 1000\ \zu{V/m} \qquad .
825 \enlargethispage{-6\baselineskip}
827 \begin{eg}{Relating the units of electric field and voltage}
828 From our original definition of the electric field, we
829 expect it to have units of newtons per coulomb, N/C. The
830 example above, however, came out in volts per meter, V/m.
831 Are these inconsistent? Let's reassure ourselves that this
832 all works. In this kind of situation, the best strategy is
833 usually to simplify the more complex units so that they
834 involve only mks units and coulombs. Since voltage is
835 defined as electrical energy per unit charge, it has units of J/C:
837 \frac{\zu{V}}{\zu{m}} &= \frac{\zu{J/C}}{\zu{m}} \\
838 &= \frac{\zu{J}}{\zu{C}\cdot\zu{m}} \qquad .
840 To connect joules to newtons, we recall that work equals
841 force times distance, so $\zu{J}=\zu{N}\cdot\zu{m}$, so
843 \frac{\zu{V}}{\zu{m}} &= \frac{\zu{N}\cdot\zu{m}}{\zu{C}\cdot\zu{m}} \\
844 &= \frac{\zu{N}}{\zu{C}}
846 As with other such difficulties with electrical units, one
847 quickly begins to recognize frequently occurring combinations.
849 \begin{eg}{Voltage associated with a point charge}
851 What is the voltage associated with a point charge?
853 \eganswer As derived previously in self-check \ref{sc:pointchargefield} on page \pageref{sc:pointchargefield}, the field is
855 |\vc{E}| = \frac{ kQ}{ r^2}
857 The difference in voltage between two points on the same radius line is
859 \Delta V &= -\int \der V \\
860 &= -\int E_{x} \der x
862 In the general discussion above, $x$ was just a generic name
863 for distance traveled along the line from one point to the
864 other, so in this case $x$ really means $r$.
866 \Delta V &= -\int_{ r_1}^{ r_2} E_{r} \der r \\
867 &= -\int_{ r_1}^{ r_2} \frac{ kQ}{ r^2} \der r \\
868 &= \left.\frac{ kQ}{ r}\right]_{ r_1}^{ r_2} \\
869 &= \frac{ kQ}{ r_2}-\frac{ kQ}{ r_1} \qquad .
871 The standard convention is to use $r_1=\infty$ as a reference
872 point, so that the voltage at any distance $r$ from the charge is
874 V = \frac{ kQ}{ r} \qquad .
876 The interpretation is that if you bring a positive test
877 charge closer to a positive charge, its electrical energy is
878 increased; if it was released, it would spring away,
879 releasing this as kinetic energy.
882 <% self_check('pointchargevtoe',<<-'SELF_CHECK'
883 Show that you can recover the expression for the field of a
884 point charge by evaluating the derivative $E_{x}=-\der V/\der x$.
889 <% begin_sec("Two or three dimensions",4,'evthreed') %>
890 The topographical map in figure \figref{topo} suggests a good
891 way to visualize the relationship between field and voltage
892 in two dimensions. Each contour on the map is a line of
893 constant height; some of these are labeled with their
894 elevations in units of feet. Height is related to gravitational
895 energy, so in a gravitational analogy, we can
896 think of height as representing voltage. Where the contour
897 lines are far apart, as in the town, the slope is gentle.
898 Lines close together indicate a steep slope.
904 A topographical map of Shelburne Falls,
905 Mass. \photocredit{USGS}
912 'voltageofpointcharge',
914 The constant-voltage curves surrounding a point charge.
915 Near the charge, the curves are so closely spaced that they blend
916 together on this drawing due to the finite width with which they were
917 drawn. Some electric fields are shown as arrows.
923 If we walk along a straight line, say straight east from the
924 town, then height (voltage) is a function of the east-west
925 coordinate $x$. Using the usual mathematical definition of
926 the slope, and writing $V$ for the height in order to remind
927 us of the electrical analogy, the slope along such a line is
928 $\der V/\der x$ (the rise over the run).
930 What if everything isn't confined to a straight line? Water
931 flows downhill. Notice how the streams on the map cut
932 perpendicularly through the lines of constant height.
934 It is possible to map voltages in the same way, as shown in
935 figure \figref{voltageofpointcharge}. The electric field is strongest where the
936 constant-voltage curves are closest together, and the
937 electric field vectors always point perpendicular to the
938 constant-voltage curves.
940 The one-dimensional relationship $E=-\der V/\der x$ generalizes
941 to three dimensions as follows:
943 E_x &= -\frac{\der V}{\der x} \\
944 E_y &= -\frac{\der V}{\der y} \\
945 E_z &= -\frac{\der V}{\der z}
947 This can be notated as a gradient (page \pageref{gradandlineintegral}),
949 \vc{E} = \nabla V \qquad ,
951 and if we know the field and want to find the voltage, we can
954 \Delta V = \int_C \vc{E}\cdot\der\vc{r} \qquad ,
956 where the quantity inside the integral is a vector dot product.
958 <% self_check('topointerp',<<-'SELF_CHECK'
959 Imagine that figure \figref{topo} represents voltage rather
960 than height. (a) Consider the stream the starts near the
961 center of the map. Determine the positive and negative signs
962 of $\der V/\der x$ and $\der V/\der y$, and relate these to the direction of
963 the force that is pushing the current forward against the
964 resistance of friction. (b) If you wanted to find a lot of
965 electric charge on this map, where would you look?
969 \enlargethispage{-2\baselineskip}
971 Figure \figref{twodvmaps} %on page \pageref{fig:twodvmaps} % convinced it to put on same page
972 shows some examples of ways
973 to visualize field and voltage patterns.
979 Two-dimensional field and voltage patterns.
980 Top: A uniformly charged rod. Bottom: A dipole.
981 In each case, the diagram on the left shows the field vectors and constant-voltage curves, while the one on the right shows
982 the voltage (up-down coordinate) as a function of x and y.
983 Interpreting the field diagrams: Each arrow represents the field at the point where its tail has been positioned. For clarity,
984 some of the arrows in regions of very strong field strength are not shown --- they would be too long to show.
985 Interpreting the constant-voltage curves: In regions of very strong fields, the curves are not shown because they would
986 merge together to make solid black regions.
987 Interpreting the perspective plots: Keep in mind that even though we're visualizing things in three dimensions, these are
988 really two-dimensional voltage patterns being represented. The third (up-down) dimension represents voltage, not position.
998 <% begin_sec("Fields by Superposition",4) %>
999 <% begin_sec("Electric field of a continuous charge distribution") %>
1001 Charge really comes in discrete chunks, but often it is
1002 mathematically convenient to treat a set of charges as if
1003 they were like a continuous fluid spread throughout a region
1004 of space. For example, a charged metal ball will have charge
1005 spread nearly uniformly all over its surface, and for
1006 most purposes it will make sense to ignore the fact that
1007 this uniformity is broken at the atomic level. The electric
1008 field made by such a continuous charge distribution is the
1009 sum of the fields created by every part of it. If we let the
1010 ``parts'' become infinitesimally small, we have a sum of an
1011 infinitely many infinitesimal numbers: an
1012 integral. If it was a discrete sum, as in example \ref{eg:square} on
1013 page \pageref{eg:square}, we would have a total
1014 electric field in the $x$ direction that was the sum of all
1015 the $x$ components of the individual fields, and similarly
1016 we'd have sums for the $y$ and $z$ components. In the
1017 continuous case, we have three integrals. Let's keep it simple
1018 by starting with a one-dimensional example.
1024 %q{Example \ref{eg:chargedrod}.}
1028 \begin{eg}{Field of a uniformly charged rod}\label{eg:chargedrod}
1030 A rod of length $L$ has charge $Q$ spread
1031 uniformly along it. Find the electric field at a point a
1032 distance $d$ from the center of the rod, along the rod's axis.
1035 This is a one-dimensional
1036 situation, so we really only need to do a single integral
1037 representing the total field along the axis. We imagine
1038 breaking the rod down into short pieces of length $\der z$, each
1039 with charge $\der q$. Since charge is uniformly spread along the
1040 rod, we have $\der q=\lambda\der z$, where $\lambda= Q/ L$ (Greek lambda) is the
1041 charge per unit length, in units of coulombs per meter. Since the pieces are infinitesimally
1042 short, we can treat them as point charges and use the
1043 expression $k\der q/ r^2$ for their contributions to the field,
1044 where $r= d- z$ is the distance from the charge at $z$ to the
1045 point in which we are interested.
1047 E_{z} &= \int \frac{ k\der q }{ r^2} \\
1048 &= \int_{- L/2}^{+ L/2} \frac{ k\lambda\der z }{ r^2} \\
1049 &= k \lambda \int_{- L/2}^{+ L/2} \frac{\der z}{( d- z)^2}
1051 The integral can be looked up in a table, or reduced to an
1052 elementary form by substituting a new variable for
1053 $d- z$. The result is
1055 E_{z} &= k\lambda\left(\frac{1}{ d- z}\right)_{- L/2}^{+ L/2} \\
1056 &= \frac{ kQ}{ L} \left(\frac{1}{ d- L/2}-\frac{1}{ d+ L/2}\right) \qquad .
1058 For large values of $d$, this expression gets
1059 smaller for two reasons: (1) the denominators of the
1060 fractions become large, and (2) the two fractions become
1061 nearly the same, and tend to cancel out. This makes sense,
1062 since the field should get weaker as we get farther away from
1063 the charge. In fact, the field at large distances must approach
1064 $ kQ/ d^2$ (homework problem \ref{hw:distantrsquared}).
1066 It's also interesting to note that the field becomes
1067 infinite at the ends of the rod, but is not infinite on the
1068 interior of the rod. Can you explain physically why this happens?
1071 Example \ref{eg:chargedrod} was one-dimensional. In the general three-dimensional
1072 case, we might have to integrate all three components of the field. However,
1073 there is a trick that lets us avoid this much complication. The voltage is a scalar,
1074 so we can find the voltage by doing just a single integral, then use the voltage
1081 %q{Example \ref{eg:chargedrodside}.}
1085 \begin{eg}{Voltage, then field}\label{eg:chargedrodside}
1087 A rod of length $L$ is uniformly charged with charge $Q$.
1088 Find the field at a point lying in the midplane of the rod
1092 By symmetry, the field has only a radial component, $E_R$, pointing
1093 directly away from the rod (or toward it for $Q<0$). The brute-force approach,
1094 then, would be to evaluate the integral $E=\int |\der\vc{E}|\zu{cos}\ \theta$,
1095 where $\der\vc{E}$ is the contribution to the field from a charge $\der q$ at some
1096 point along the rod, and $\theta$ is the angle $\der\vc{E}$ makes with the radial line.
1098 It's easier, however, to find the voltage first, and then find the field from the
1099 voltage. Since the voltage is a scalar,
1100 we simply integrate the contribution $\der V$ from each charge $\der q$, without
1101 even worrying about angles and directions. Let $z$ be the coordinate that measures distance
1102 up and down along the rod, with $z=0$ at the center of the rod. Then the distance
1103 between a point $z$ on the rod and the point of interest is
1104 $r=\sqrt{ z^2+ R^2}$, and we have
1106 V &= \int \frac{ k\der q}{ r} \\
1107 &= k\lambda \int_{- L/2}^{+ L/2}\frac{\der z}{ r} \\
1108 &= k\lambda \int_{- L/2}^{+ L/2}\frac{\der z}{\sqrt{ z^2+ R^2}} \\
1110 The integral can be looked up in a table, or evaluated using computer software:
1112 V &= \left. k\lambda\: \zu{ln}\left( z+\sqrt{ z^2+ R^2}\right)\right|_{- L/2}^{+ L/2} \\
1113 &= k\lambda\: \zu{ln}\left(%
1114 \frac{ L/2+\sqrt{ L^2/4+ R^2}}%
1115 {- L/2+\sqrt{ L^2/4+ R^2}}%
1118 The expression inside the parentheses can be simplified a little.
1119 Leaving out some tedious algebra, the result is
1121 V = 2 k\lambda\: \zu{ln}\left(%
1123 +\sqrt{1+\frac{ L^2}{4 R^2}}%
1127 This can readily be differentiated to find the field:
1129 E_{R} &= -\frac{\der V}{\der R} \\
1133 +(1/2)(1+ L^2/4 R^2)^{-1/2}(- L^2/2 R^3)
1135 L/2 R+(1+ L^2/4 R^2)^{1/2}
1136 } \qquad , \qquad \\
1137 \intertext{or, after some simplification,}
1138 E_{R} &= \frac{ k\lambda L}{ R^2\sqrt{1+ L^2/4 R^2}}
1141 For large values of $R$, the square root approaches one, and we have
1142 simply $E_{R}\approx k\lambda L/ R^2= k Q/ R^2$.
1143 In other words, the field very far away is the same regardless of whether the charge
1144 is a point charge or some other shape like a rod. This is intuitively appealing, and
1145 doing this kind of check also helps to reassure one that the final result is correct.
1147 The preceding example, although it involved some messy algebra, required only straightforward
1148 calculus, and no vector operations at all, because we only had to integrate a scalar
1149 function to find the voltage. The next example is one in which
1150 we can integrate either the field or the voltage without too much complication.
1156 %q{Example \ref{eg:ring}.}
1160 \begin{eg}{On-axis field of a ring of charge}\label{eg:ring}
1162 Find the voltage and field along the axis of a uniformly charged ring.
1165 Integrating the voltage is straightforward.
1167 V &= \int \frac{ k\der q}{ r} \\
1168 &= k \int \frac{\der q}{\sqrt{ b^2+ z^2}} \\
1169 &= \frac{ k}{\sqrt{ b^2+ z^2}} \int \der q \\
1170 &= \frac{ kQ}{\sqrt{ b^2+ z^2}} \qquad ,
1172 where $Q$ is the total charge of the ring.
1173 This result could have been derived without calculus, since the distance
1174 $r$ is the same for every point around the ring, i.e., the integrand is a constant.
1175 It would also be straightforward to find the field by differentiating this expression
1176 with respect to $z$ (homework problem \ref{hw:ringve}).
1178 Instead, let's see how to find the field by direct integration. By symmetry, the
1179 field at the point of interest can have only a component along the axis of
1180 symmetry, the $z$ axis:
1185 To find the field in the $z$ direction, we integrate the $z$ components
1186 contributed to the field by each infinitesimal part of the ring.
1188 E_{z} &= \int \der E_z \\
1189 &= \int |\der\vc{E}|\:\zu{cos}\:\theta \qquad ,
1191 where $\theta$ is the angle shown in the figure.
1193 E_{z} &= \int \frac{ k\der q}{ r^2}\:\zu{cos}\:\theta \\
1194 &= k \int \frac{\der q}{ b^2+ z^2}\:\zu{cos}\:\theta
1196 Everything inside the integral is a constant, so we have
1198 E_{z} &= \frac{ k}{ b^2+ z^2}\:\zu{cos}\:\theta \int \der q \\
1199 &= \frac{ kQ}{ b^2+ z^2}\:\zu{cos}\:\theta \\
1200 &= \frac{ kQ}{ b^2+ z^2}\:\frac{ z}{ r} \\
1201 &= \frac{ kQz}{\left( b^2+ z^2\right)^\zu{3/2}}
1205 In all the examples presented so far, the charge has been confined to a one-dimensional
1206 line or curve. Although it is possible, for example, to put charge on a piece of wire,
1207 it is more common to encounter practical devices in which the charge is distributed
1208 over a two-dimensional surface, as in the flat metal plates used in Thomson's experiments.
1209 Mathematically, we can approach this type of calculation with the divide-and-conquer
1210 technique: slice the surface into lines or curves whose fields we know how to calculate,
1211 and then add up the contributions to the field from all these slices. In the limit where the
1212 slices are imagined to be infinitesimally thin, we have an integral.
1214 \begin{eg}{Field of a uniformly charged disk}\label{eg:diskofcharge}
1216 A circular disk is uniformly charged. (The disk must be an insulator; if it was a conductor,
1217 then the repulsion of all the charge would cause it to collect more densely near the
1218 edge.) Find the field at a point on the axis, at a distance $z$ from the plane of the disk.
1224 %q{Example \ref{eg:diskofcharge}: geometry.}
1231 %q{Example \ref{eg:diskofcharge}: the field on both sides (for $\sigma>0$).}
1238 %q{A capacitor consisting of two disks with opposite charges.}
1244 We're given that every part of the disk has the same charge per unit area, so rather
1245 than working with $Q$, the total charge, it will be easier to use the charge
1246 per unit area, conventionally notated $\sigma$ (Greek sigma), $\sigma= Q/\pi b^2$.
1248 Since we already know the field due to a ring of charge, we can solve the problem
1249 by slicing the disk into rings, with each ring extending from $r$ to $r+\der r$.
1250 The area of such a ring equals its circumference multiplied by its width,
1251 i.e., $2\pi r\der r$, so its charge is $\der q=2\pi\sigma r\der r$,
1252 and from the result of example \ref{eg:ring}, its contribution to the field is
1254 \der E_{z} &= \frac{ kz\der q}{\left( r^2+ z^2\right)^\zu{3/2}} \\
1255 &= \frac{2\pi\sigma kzr\der r}{\left( r^2+ z^2\right)^\zu{3/2}} \\
1259 E_{z} &= \int \der E_{z} \\
1262 \frac{ r\der r}{\left( r^2+ z^2\right)^\zu{3/2}} \\
1263 &= 2\pi\sigma kz \left. %
1264 \frac{-1}{\sqrt{ r^2+ z^2}}
1265 \right|_{ r=0}^{ r=\zu{b}} \\
1266 &= 2\pi\sigma k\left(1-\frac{ z}{\sqrt{ b^2+ z^2}}\right)
1270 The result of example \ref{eg:diskofcharge} has some interesting properties.
1271 First, we note that it was derived on the unspoken assumption
1272 of $z>0$. By symmetry, the field on the other side of the disk must be equally
1273 strong, but in the opposite direction, as shown in
1274 figures \figref{diskfield} and \figref{diskfieldgraph}. Thus there is a discontinuity in the field
1275 at $z=0$. In reality, the disk will have some finite thickness, and the switching
1276 over of the field will be rapid, but not discontinuous.
1278 At large values of $z$, i.e., $z\gg b$, the field rapidly approaches
1279 the $1/r^2$ variation that we expect when we are so far from the disk that the disk's
1280 size and shape cannot matter (homework problem \ref{hw:distantrsquared}).
1286 %q{Example \ref{eg:diskofcharge}: variation of the field ($\sigma>0$).},
1293 A practical application is the case of a capacitor, \figref{diskcap}, having
1294 two parallel circular plates very close together. In normal operation, the charges
1295 on the plates are opposite, so one plate has fields pointing into it and the other
1296 one has fields pointing out. In a real capacitor, the plates are a metal conductor,
1297 not an insulator, so the charge will tend to arrange itself more densely near the edges,
1298 rather than spreading itself uniformly on each plate. Furthermore, we have only calculated
1299 the \emph{on-axis} field in example \ref{eg:diskofcharge}; in the off-axis region,
1300 each disk's contribution
1301 to the field will be weaker, and it will also point away from the axis a little. But if we are
1302 willing to ignore these complications for the sake of a rough analysis, then the fields
1303 superimpose as shown in figure \figref{diskcap}: the fields cancel the outside of the
1304 capacitor, but between the plates its value is double that contributed by a single
1305 plate. This cancellation on the outside is a very useful property for a practical
1306 capacitor. For instance, if you look at the printed circuit board in a typical piece
1307 of consumer electronics, there are many capacitors, often placed fairly close together.
1308 If their exterior fields didn't cancel out nicely, then each capacitor would interact
1309 with its neighbors in a complicated way, and the behavior of the circuit would depend
1310 on the exact physical layout, since the interaction would be stronger or weaker depending
1311 on distance. In reality, a capacitor does create weak external electric fields, but their
1312 effects are often negligible, and we can then use the \emph{lumped-circuit approximation},
1313 \index{lumped-circuit approximation!for capacitors}\label{lumped-circuit-approx}
1314 which states that each component's behavior depends only on the currents that flow in
1315 and out of it, not on the interaction of its fields with the other components.
1320 <% begin_sec("The field near a charged surface",nil,'surfacefield') %>
1321 From a theoretical point of view, there is something even
1322 more intriguing about example \ref{eg:diskofcharge}:
1323 the magnitude of the field for small values
1324 of $z$ ($z\ll b$) is $E=2\pi k\sigma$, which doesn't depend on $b$ at all for a fixed value
1325 of $\sigma$. If we made a disk with twice the radius, and covered
1326 it with the same number of coulombs per square meter
1327 (resulting in a total charge four times as great), the field close to the disk would
1328 be unchanged! That is, a flea living near the center of the disk, \figref{fleathinking}, would have no way
1329 of determining the size of her flat ``planet'' by measuring the local field
1330 and charge density. (Only by leaping off the surface into outer space would she
1331 be able to measure fields that were dependent on $b$. If she traveled very
1332 far, to $z\gg b$, she would be in the region where the field is well approximated by
1333 $|\vc{E}|\approx kQ/z^2=k\pi b^2\sigma/z^2$, which she could solve for $b$.)
1340 Close to the surface, the relationship between $E$ and $\sigma$ is a fixed one, regardless
1341 of the geometry. The flea can't determine the size or shape of her world by
1342 comparing $E$ and $\sigma$.
1347 What is the reason for this surprisingly simple behavior of the field? Is it a
1348 piece of mathematical trivia, true only in this particular case? What if the shape
1349 was a square rather than a circle? In other words, the flea gets no information about
1350 the \emph{size} of the disk from measuring $E$, since $E=2\pi k\sigma$, independent
1351 of $b$, but what if she didn't know the \emph{shape}, either? If the result for a square
1352 had some other geometrical factor in front instead of $2\pi$, then she could tell which
1353 shape it was by measuring $E$. The surprising mathematical fact, however, is that the
1354 result for a square, indeed for any shape whatsoever, is $E=2\pi\sigma k$. It doesn't
1355 even matter whether the surface is flat or warped, or whether the density of charge
1356 is different at parts of the surface which are far away compared to the flea's distance
1364 Fields contributed by nearby parts of the surface, P, Q, and R, contribute
1365 to $E_\perp$. Fields due to distant charges, S, and T, have very small contributions to
1366 $E_\perp$ because of their shallow angles.
1371 This universal $E_\perp=2\pi k\sigma$ field perpendicular to
1372 a charged surface can be proved mathematically based
1373 on Gauss's law\footnote{rhymes with ``mouse''} (section \ref{sec:gauss}), but we can understand
1374 what's happening on qualitative grounds. Suppose on night, while the flea is asleep,
1375 someone adds more surface area, also positively charged, around the outside edge of
1376 her disk-shaped world, doubling its radius. The added charge, however, has very little
1377 effect on the field in her environment, as long as she stays at low altitudes above
1378 the surface. As shown in figure \figref{fleavectors}, the new charge
1379 to her west contributes a field, T, that is almost purely ``horizontal'' (i.e., parallel
1380 to the surface) and to the east. It has
1381 a negligible upward component, since the angle is so shallow. This new eastward
1382 contribution to the field
1383 is exactly canceled out by the westward field, S, created by the new charge to her east.
1384 There is likewise almost perfect cancellation between any other pair of opposite
1387 A similar argument can be made as to the shape-independence of the result, as long as
1388 the shape is symmetric.
1389 For example, suppose that the next night, the tricky real estate developers
1390 decide to add corners to the disk and transform it into a square. Each corner's
1391 contribution to the field measured at the center is canceled by the field due to the
1392 corner diagonally across from it.
1394 What if the flea goes on a trip away from the center of the disk? The perfect cancellation
1395 of the ``horizontal'' fields contributed by distant charges will no longer occur, but the
1396 ``vertical'' field (i.e., the field perpendicular to the surface)
1397 will still be $E_\perp=2\pi k\sigma$, where $\sigma$ is the local charge density,
1398 since the distant charges can't contribute to the vertical field. The same result applies
1399 if the shape of the surface is asymmetric, and doesn't even have any well-defined
1400 geometric center: the component perpendicular to the surface is $E_\perp=2\pi k\sigma$,
1401 but we may have $E_\parallel\neq0$. All of the above arguments can be made more rigorous
1402 by discussing mathematical limits rather than using words like ``very small.'' There is not
1403 much point in giving a rigorous proof here, however, since we will be able to demonstrate
1404 this fact as a corollary of Gauss' Law in section \ref{sec:gauss}. The result
1407 At a point lying a distance $z$ from a charged surface, the component of the electric field
1408 perpendicular to the surface obeys
1410 \lim_{z\rightarrow 0} E_\perp = 2\pi k\sigma \qquad ,
1412 where $\sigma$ is the charge per unit area. This is true regardless of the shape or
1413 size of the surface.
1419 %q{Example \ref{eg:pointlinesurface}.}
1423 \begin{eg}{The field near a point, line, or surface charge}\label{eg:pointlinesurface}
1425 Compare the variation of the electric field with distance, $d$, for small
1426 values of $d$ in the case of a point charge, an infinite line of charge,
1427 and an infinite charged surface.
1430 For a point charge, we have already found $E\propto d^{-2}$ for the
1431 magnitude of the field, where we are now using $d$ for the quantity we would
1432 ordinarily notate as $r$. This is true for all values of $d$, not just for small
1433 $d$ --- it has to be that way, because the point charge has no size, so if $E$
1434 behaved differently for small and large $d$, there would be no way to decide what
1435 $d$ had to be small or large relative to.
1437 For a line of charge, the result of example \ref{eg:chargedrodside} is
1439 E = \frac{ k\lambda L}%
1440 { d^2\sqrt{1+ L^2/4 d^2}} \qquad .
1442 In the limit of $d\ll L$, the quantity inside the square root is dominated
1443 by the second term, and we have $E\propto d^{-1}$.
1445 %\enlargethispage{-2\baselineskip}
1447 Finally, in the case of a charged surface, the result is simply
1448 $E=2\pi\sigma k$, or $E\propto d^{0}$.
1450 Notice the lovely simplicity of the pattern, as shown in figure
1451 \figref{pointlinesurface}. A point is zero-dimensional: it has no length,
1452 width, or breadth. A line
1453 is one-dimensional, and a surface is two-dimensional.
1454 As the dimensionality of the charged object changes from 0 to 1, and then to 2,
1455 the exponent in the near-field expression goes from 2 to 1 to 0.
1462 <% begin_sec("Energy in Fields",4,'fieldenergy') %>
1463 <% begin_sec("Electric field energy") %>
1464 Fields possess energy, as argued on page \pageref{fieldenergy}, but how much energy?
1465 The answer can be found using the following elegant approach. We assume that the electric energy
1466 contained in an infinitesimal volume of space $\der v$ is given by
1467 $\der U_e=f(\vc{E})\der v$, where $f$ is some function, which we wish to determine,
1468 of the field \vc{E}.
1469 It might seem that we would have no easy way to determine the function $f$, but many of
1470 the functions we could cook up would violate the symmetry of space. For instance,
1471 we could imagine $f(\vc{E})=aE_y$, where $a$ is some constant with the appropriate units.
1472 However, this would violate the symmetry of space, because it would give the $y$ axis
1473 a different status from $x$ and $z$. As discussed on page \pageref{subsec:dotproduct},
1474 if we wish to calculate a scalar based on some vectors, the dot product is the only way
1475 to do it that has the correct symmetry properties. If all we have is one vector, \vc{E}, then
1476 the only scalar we can form is $\vc{E}\cdot\vc{E}$, which is the square of the magnitude
1477 of the electric field vector.
1483 %q{Two oppositely charged capacitor plates are pulled apart.}
1487 In principle, the energy function we are seeking could
1488 be proportional to $\vc{E}\cdot\vc{E}$, or to any function computed from it, such as
1489 $\sqrt{\vc{E}\cdot\vc{E}}$ or $(\vc{E}\cdot\vc{E})^7$. On physical grounds, however,
1490 the only possibility that works is $\vc{E}\cdot\vc{E}$. Suppose, for instance, that
1491 we pull apart two oppositely charged capacitor plates, as shown in figure
1492 \figref{pullplates}. We are doing work by pulling them apart against the force
1493 of their electrical attraction, and this quantity of mechanical work equals the
1494 increase in electrical energy, $U_e$.
1495 Using our previous approach to energy, we would have thought of
1496 $U_e$ as a quantity which depended on the distance of the positive and negative charges
1497 from each other, but now we're going to imagine $U_e$ as being stored within the electric field
1498 that exists in the space between and around the charges. When the plates are
1499 touching, their fields cancel everywhere, and there is zero electrical energy.
1500 When they are separated, there is still approximately zero field on the outside, but the field
1501 between the plates is nonzero, and holds some energy.
1502 Now suppose we carry out the whole process, but with the plates carrying double
1503 their previous charges. Since Coulomb's law involves the product $q_1q_2$ of two
1504 charges, we have quadrupled the force between any given pair of charged particles,
1505 and the total attractive force is therefore also four times greater than before.
1506 This means that the work done in separating the plates is four times greater,
1507 and so is the energy $U_e$ stored in the field.
1508 The field, however, has merely been doubled at any given location:
1509 the electric field $\vc{E}_+$ due to the positively charged plate
1510 is doubled, and similarly for the contribution $\vc{E}_-$ from the negative one,
1511 so the total electric field $\vc{E}_++\vc{E}_-$ is also doubled. Thus
1512 doubling the field results in an electrical energy which is four times greater,
1513 i.e., the energy density must be proportional to the square of the field,
1514 $\der U_e\propto(\vc{E}\cdot\vc{E})\der v$. For ease of notation, we write this as
1515 $\der U_e\propto E^2\der v$, or $\der U_e=aE^2\der v$, where $a$ is a constant
1517 Note that we never really made use of any of the details of the geometry of
1518 figure \figref{pullplates}, so the reasoning is of general validity.
1519 In other words, not only is $\der U_e=aE^2\der v$ the function that works in this
1520 particular case, but there is every reason to believe that it would work
1521 in other cases as well.
1523 It now remains only to find $a$. Since the constant
1524 must be the same in all situations, we only need to find one example in which we
1525 can compute the field and the energy, and then we can determine $a$.
1526 The situation shown in figure \figref{pullplates} is just about the easiest example
1527 to analyze. We let the square capacitor plates be uniformly covered with charge densities
1528 $+\sigma$ and $-\sigma$, and we write $b$ for the lengths of their sides.
1529 Let $h$ be the gap between the plates after they have been separated.
1530 We choose $h\ll b$, so that the field experienced by the negative plate due
1531 to the positive plate is $E_+=2\pi k\sigma$. The charge of the negative plate is
1532 $-\sigma b^2$, so the magnitude of the force attracting it back toward the positive plate
1533 is $(\text{force})=(\text{charge})(\text{field})=2\pi k\sigma^2 b^2$. The amount of work
1534 done in separating the plates is
1535 $(\text{work})=(\text{force})(\text{distance})=2\pi k\sigma^2 b^2h$.
1536 This is the amount of energy that has been stored in the field between the two plates,
1537 $U_e=2\pi k\sigma^2 b^2h=2\pi k\sigma^2 v$, where $v$ is the volume of the region
1540 We want to equate this to $U_e=aE^2v$. (We can write $U_e$ and $v$ rather than
1541 $\der U_e$ and $\der v$, since the field is constant in the region between the plates.)
1542 The field between the plates
1543 has contributions from both plates, $E=E_++E_-=4\pi k\sigma$. (We only used half this
1544 value in the computation of the work done on the moving plate, since the moving plate
1545 can't make a force on itself. Mathematically, each plate is in a region where its
1546 own field is reversing directions, so we can think of its own contribution to the field
1547 as being zero within itself.) We then have $aE^2v=a\cdot 16\pi^2k^2\sigma^2 \cdot v$, and setting
1548 this equal to $U_e=2\pi k\sigma^2 v$ from the result of the work computation, we find
1549 $a=1/8\pi k$. Our final result is as follows:
1551 The electric energy possessed by an electric field \vc{E} occupying an infinitesimal
1552 volume of space $\der v$ is given by
1554 \der U_e = \frac{1}{8\pi k}E^2 \der v \qquad ,
1556 where $E^2=\vc{E}\cdot\vc{E}$ is the square of the magnitude of the electric field.
1557 \index{energy density!of electric field}\index{electric field!energy density of}
1559 This is reminiscent of how waves behave: the energy content of a wave is
1560 typically proportional to the square of its amplitude.
1564 <% self_check('reverseex',<<-'SELF_CHECK'
1565 We can think of the quantity $\der U_{e}/\der v$ as the
1566 \emph{energy density} due to the electric field, i.e., the number of joules
1567 per cubic meter needed in order to create that field. (a) How does this quantity
1568 depend on the components of the field vector, $E_x$, $E_y$, and $E_z$? (b) Suppose
1569 we have a field with $E_x\neq0$, $E_y$=0, and $E_z$=0. What would happen to the
1570 energy density if we reversed the sign of $E_x$?
1574 \begin{eg}{A numerical example}\label{eg:numparplate}
1576 A capacitor has plates whose areas are $10^{-4}\ \zu{m}^2$, separated
1577 by a gap of $10^{-5}$ m. A 1.5-volt battery is connected across it.
1578 How much energy is sucked out of the battery and stored in the electric field
1579 between the plates? (A real capacitor typically has an insulating material between
1580 the plates whose molecules interact electrically with the charge in the plates.
1581 For this example, we'll assume that there is just a vacuum in between the plates. The
1582 plates are also typically rolled up rather than flat.)
1585 To connect this with our previous calculations, we need to find the charge
1586 density on the plates in terms of the voltage we were given. Our previous
1587 examples were based on the assumption that the gap between the plates
1588 was small compared to the size of the plates. Is this valid here? Well, if the
1589 plates were square, then the area of $10^{-4}\ \zu{m}^2$ would imply
1590 that their sides were $10^{-2}$ m in length. This is indeed very large
1591 compared to the gap of $10^{-5}$ m, so this assumption appears to be valid
1592 (unless, perhaps, the plates have some very strange, long and skinny shape).
1594 Based on this assumption, the field is relatively uniform in the whole volume
1595 between the plates, so we can use a single symbol, $E$, to represent its magnitude,
1596 and the relation $E=\der V/\der x$ is equivalent to
1597 $E=\Delta V/\Delta x=(\text{1.5 V})/(\text{gap})= 1.5\times10^5\ \zu{V}/\zu{m}$.
1599 Since the field is uniform, we can dispense with the calculus, and
1600 replace $\der U_{e} = (1/8\pi k) E^2 \der v$
1601 with $U_{e} = (1/8\pi k) E^2 v$.
1602 The volume equals the area multiplied by the
1605 U_{e} &= (1/8\pi k) E^2(\text{area})(\text{gap})\\
1606 &= \frac{1}{8\pi\times9\ \times10^9\ %
1607 \zu{N}\unitdot\munit^2/\zu{C}^2}%
1608 ( 1.5\times10^5\ \zu{V}/\zu{m})^2%
1609 (10^{-4}\ \zu{m}^2)(10^{-5}\ \zu{m})\\
1610 &= 1\times10^{-10}\ \zu{J}
1614 <% self_check('numercapunits',<<-'SELF_CHECK'
1615 Show that the units in the preceding example really
1616 do work out to be joules.
1620 \begin{eg}{Why $k$ is on the bottom}
1621 It may also seem strange that the constant $k$
1622 is in the denominator of the equation $\der U_{e} = (1/8\pi k) E^2 \der v$.
1623 The Coulomb constant $k$ tells us how strong electric
1624 forces are, so shouldn't it be on top? No.
1625 Consider, for instance, an alternative universe in which
1626 electric forces are twice as strong as in ours. The numerical value
1627 of $k$ is doubled. Because $k$ is doubled, all the
1628 electric field strengths are doubled as well, which
1629 quadruples the quantity $E^2$. In the expression $E^2/8\pi k$, we've
1630 quadrupled something on top and doubled something on the
1631 bottom, which makes the energy twice as big. That makes perfect sense.
1634 \begin{eg}{Potential energy of a pair of opposite charges}\label{eg:upointcharges}
1635 Imagine taking two opposite charges, \figref{upointcharges}, that were
1636 initially far apart and allowing them to come together under
1637 the influence of their electrical attraction.
1643 %q{Example \ref{eg:upointcharges}.}
1647 According to our old approach, electrical energy is lost
1648 because the electric force did positive work as it brought
1649 the charges together. (This makes sense because as they come
1650 together and accelerate it is their electrical energy that is
1651 being lost and converted to kinetic energy.)
1653 By the new method, we must ask how the energy stored in the
1654 electric field has changed. In the region indicated
1655 approximately by the shading in the figure, the superposing
1656 fields of the two charges undergo partial cancellation
1657 because they are in opposing directions. The energy in the
1658 shaded region is reduced by this effect. In the unshaded
1659 region, the fields reinforce, and the energy is increased.
1661 It would be quite a project to do an actual numerical
1662 calculation of the energy gained and lost in the two regions
1663 (this is a case where the old method of finding energy gives
1664 greater ease of computation), but it is fairly easy to
1665 convince oneself that the energy is less when the charges
1666 are closer. This is because bringing the charges together
1667 shrinks the high-energy unshaded region and enlarges the
1668 low-energy shaded region.
1675 Example \ref{eg:sphericalcap}. Part of the
1676 outside sphere has been drawn as if it is transparent, in order to
1677 show the inside sphere.
1683 \begin{eg}{A spherical capacitor}\label{eg:sphericalcap}\index{capacitor!spherical}
1685 A spherical capacitor, \figref{sphericalcap}, consists of two concentric spheres
1686 of radii $a$ and $b$.
1687 Find the energy required to charge up the capacitor
1688 so that the plates hold charges $+ q$ and $- q$.
1691 On page \pageref{shelltheoremsubsection}, I proved that for
1692 \emph{gravitational} forces, the interaction of a spherical
1693 shell of mass with other masses outside it is the same as
1694 if the shell's mass was concentrated at its center. On the
1695 interior of such a shell, the forces cancel out exactly.
1696 Since gravity and the electric force both vary as $1/ r^2$,
1697 the same proof carries over immediately to electrical forces. The magnitude of the outward
1698 electric field contributed by the charge $+ q$ of the central sphere
1701 |\vc{E}_+| = \left\{
1708 where $r$ is the distance from the center. Similarly, the magnitude of the
1709 \emph{inward} field contributed
1710 by the outside sphere is
1712 |\vc{E}_-| = \left\{
1719 In the region outside the whole capacitor, the two fields are equal in
1720 magnitude, but opposite in direction, so they cancel. We then have for
1726 kq/ r^2, & a< r< b \\
1731 so to calculate the energy, we only need to worry about the
1732 region $a< r< b$. The energy density in this region is
1734 \frac{\der U_{e}}{\der v} &= \frac{1}{8\pi k} E^2 \\
1735 &= \frac{ kq^2}{8\pi} r^{-4} \qquad .
1737 This expression only depends on $r$, so the energy density is constant across
1738 any sphere of radius $r$. We can slice the region $a< r< b$ into
1739 concentric spherical layers, like an onion, and the energy within one such layer,
1740 extending from $r$ to $r+\der r$ is
1742 \der U_{e} &= \frac{\der U_{e}}{\der v} \der v \\
1743 &= \frac{\der U_{e}}{\der v} (\text{area of shell}) (\text{thickness of shell}) \\
1744 &= (\frac{ kq^2}{8\pi} r^{-4}) (4\pi r^2) (\der r) \\
1745 &= \frac{ kq^2}{2} r^{-2}\der r \qquad .
1747 Integrating over all the layers to find the total energy, we have
1749 U_{e} &= \int \der U_{e} \\
1750 &= \int_{a}^{b} \frac{ kq^2}{2} r^{-2}\der r \\
1751 &= \left.-\frac{ kq^2}{2} r^{-1}\right|_{a}^{b} \\
1752 &= \frac{ kq^2}{2}\left(\frac{1}{a}-\frac{1}{b}\right) \\
1759 \begin{dq}\label{eg:pointchargeincap}
1760 The figure shows a positive charge in the gap between two
1761 capacitor plates. Compare the energy of the electric fields in the
1762 two cases. Does this agree with what you would have expected based
1763 on your knowledge of electrical forces?
1766 \begin{dq}\label{eg:sphericalcap}
1767 The figure shows a spherical capacitor. In the text, the energy stored
1768 in its electric field is shown to be
1770 U_{e} = \frac{ kq^2}{2}\left(\frac{1}{a}-\frac{1}{b}\right) \qquad . \\
1772 What happens if the difference between $b$ and $a$ is very small? Does this
1773 make sense in terms of the mechanical work needed in order to separate the
1774 charges? Does it make sense in terms of the energy stored in the electric
1775 field? Should these two energies be added together?
1777 Similarly, discuss the cases of $b\rightarrow\infty$ and $a\rightarrow0$.
1783 %q{Discussion question \ref{eg:pointchargeincap}.}
1790 %q{Discussion question \ref{eg:sphericalcap}.},
1797 Criticize the following statement: ``A solenoid makes a charge in
1798 the space surrounding it, which dissipates when you release the
1803 In example \ref{eg:upointcharges} on page \pageref{eg:upointcharges},
1804 I argued that for the charges shown in the figure, the
1806 energy when the charges are closer together, because the region
1807 of cancellation expanded, while the region of reinforcing fields
1808 shrank. Perhaps a simpler
1809 approach is to consider the two extreme possibilities: the case
1810 where the charges are infinitely far apart, and the one in which
1811 they are at zero distance from each other, i.e., right on top of each
1812 other. Carry out this reasoning for the case of (1) a positive
1813 charge and a negative charge of equal magnitude, (2) two positive
1814 charges of equal magnitude, (3) the gravitational energy of two
1819 <% begin_sec("Gravitational field energy") %>
1820 Example \ref{eg:sphericalcap} depended on the close analogy between electric and gravitational
1821 forces. In fact, every argument, proof, and example discussed so far in this section is equally
1822 valid as a gravitational example, provided we take into account one fact: only positive mass
1823 exists, and the gravitational force between two masses is attractive. This is the opposite of
1824 what happens with electrical forces, which are repulsive in the case of two positive charges.
1825 As a consequence of this, we need to assign a \emph{negative} energy density to the gravitational
1826 field! For a gravitational field, we have
1828 \der U_g = -\frac{1}{8\pi G}g^2 \der v \qquad ,
1830 where $g^2=\vc{g}\cdot\vc{g}$ is the square of the magnitude of the gravitational field.
1831 \index{energy density!of gravitational field}\index{gravitational field!energy density of}
1834 <% begin_sec("Magnetic field energy") %>
1835 So far we've only touched in passing on the topic of magnetic fields, which
1836 will deal with in detail in chapter \ref{ch:em}. Magnetism is an interaction between
1837 moving charge and moving charge, i.e., between currents and currents. Since a current
1838 has a direction in space,\footnote{Current is a scalar, since the definition
1839 $I=\der q/\der t$ is the derivative of a scalar. However, there is a closely related
1840 quantity called the current \emph{density}, \vc{J}, which is a vector, and \vc{J} is
1841 in fact the more fundamentally important quantity.\index{current density}}
1842 while charge doesn't, we can anticipate that the mathematical rule connecting
1843 a magnetic field to its source-currents will have to be completely different from
1844 the one relating the electric field to its source-charges. However, if you look carefully
1845 at the argument leading to the relation $\der U_e/\der v = E^2/8\pi k$, you'll see that
1846 these mathematical details were only necessary to the part of the argument in which
1847 we fixed the constant of proportionality. To establish $\der U_e/\der v \propto E^2$,
1848 we only had to use three simple facts:
1850 \item The field is proportional to the source.
1851 \item Forces are proportional to fields.
1852 \item Field contributed by multiple sources add like vectors.
1854 All three of these statements are true for the magnetic field as well, so without
1855 knowing anything more specific about magnetic fields --- not even what units are
1856 used to measure them! --- we can state with certainty that the energy density in
1857 the magnetic field is proportional to the square of the magnitude of the magnetic field.\label{b-field-energy-propto}
1858 \index{energy density!of magnetic field}\index{magnetic field!energy density of}%
1859 The constant of proportionality is given on p.~\pageref{benergy}.
1863 % -------------------------------------------------------------------------------------
1867 <% begin_sec("LRC Circuits",4) %>
1868 The long road leading from the light bulb to the computer
1869 started with one very important step: the introduction of
1870 feedback into electronic circuits. Although the principle of
1871 feedback has been understood and and applied to mechanical
1872 systems for centuries, and to electrical ones since the
1873 early twentieth century, for most of us the word evokes an
1874 image of Jimi Hendrix (or some more recent guitar hero)
1875 intentionally creating earsplitting screeches, or of the
1876 school principal doing the same inadvertently in the
1877 auditorium. In the guitar example, the musician stands in
1878 front of the amp and turns it up so high that the sound
1879 waves coming from the speaker come back to the guitar string
1880 and make it shake harder. This is an example of \emph{positive}
1881 feedback: the harder the string vibrates, the stronger the
1882 sound waves, and the stronger the sound waves, the harder
1883 the string vibrates. The only limit is the power-handling
1884 ability of the amplifier.
1886 Negative feedback is equally important. Your thermostat, for
1887 example, provides negative feedback by kicking the heater
1888 off when the house gets warm enough, and by firing it up
1889 again when it gets too cold. This causes the house's
1890 temperature to oscillate back and forth within a certain
1891 range. Just as out-of-control exponential freak-outs are a
1892 characteristic behavior of positive-feedback systems,
1893 oscillation is typical in cases of negative feedback. You
1894 have already studied negative feedback extensively in section \ref{sec:resonance}
1895 in the case of a mechanical system, although we didn't call it that.
1897 <% begin_sec("Capacitance and inductance") %>%
1898 \index{capacitor}\index{capacitor!capacitance}\index{inductor}\index{inductor!inductance}
1899 In a mechanical oscillation, energy is exchanged repetitively
1900 between potential and kinetic forms, and may also be
1901 siphoned off in the form of heat dissipated by friction. In
1902 an electrical circuit, resistors are the circuit elements
1903 that dissipate heat. What are the electrical analogs of
1904 storing and releasing the potential and kinetic energy of a
1905 vibrating object? When you think of energy storage in an
1906 electrical circuit, you are likely to imagine a battery, but
1907 even rechargeable batteries can only go through 10 or 100
1908 cycles before they wear out. In addition, batteries are not
1909 able to exchange energy on a short enough time scale for
1910 most applications. The circuit in a musical synthesizer may
1911 be called upon to oscillate thousands of times a second, and
1912 your microwave oven operates at gigahertz frequencies.
1913 Instead of batteries, we generally use
1915 inductors to store energy in oscillating circuits. Capacitors,
1916 which you've already encountered, store energy in electric fields.
1917 An inductor does the same with magnetic fields.
1919 <% begin_sec("Capacitors") %>
1920 A capacitor's energy exists in its surrounding electric
1921 fields. It is proportional to the square of the field
1922 strength, which is proportional to the charges on the
1923 plates. If we assume the plates carry charges that are the
1924 same in magnitude, $+q$ and $-q$, then the energy stored in
1925 the capacitor must be proportional to $q^2$. For historical
1926 reasons, we write the constant of proportionality as $1/2C$,
1928 U_C = \frac{1}{2C}q^2 \qquad .
1930 The constant $C$ is a geometrical property of the
1931 capacitor, called its capacitance.
1937 %q{The symbol for a capacitor.}
1944 %q{Some capacitors.}
1948 Based on this definition, the units of capacitance must be
1949 coulombs squared per joule, and this combination is more
1950 conveniently abbreviated as the farad\index{farad!defined},
1951 $1\ \zu{F}=1\ \zu{C}^2/\zu{J}$.
1952 ``Condenser'' is a less formal term
1953 for a capacitor. Note that the labels printed on capacitors
1954 often use MF to mean $\mu\zu{F}$, even though MF should really
1955 be the symbol for megafarads, not microfarads. Confusion
1956 doesn't result from this nonstandard notation, since
1957 picofarad and microfarad values are the most common, and it
1958 wasn't until the 1990's that even millifarad and farad
1959 values became available in practical physical sizes. Figure
1960 \figref{capsymbol} shows the symbol used in schematics
1961 to represent a capacitor.
1963 \begin{eg}{A parallel-plate capacitor}\label{eg:capparplate}
1965 Suppose a capacitor consists of two parallel metal plates with
1966 area $A$, and the gap between them is $h$. The gap is small
1967 compared to the dimensions of the plates. What is the
1971 Since the plates are metal, the charges on each plate are free to
1972 move, and will tend to cluster themselves more densely near the edges due to the
1973 mutual repulsion of the other charges in the same plate. However, it turns out
1974 that if the gap is small, this is a small effect, so we can get away with
1975 assuming uniform charge density on each plate.
1976 The result of example \ref{eg:numparplate} then applies, and
1977 for the region between the plates, we have $E=4\pi k\sigma=4\pi kq/ A$ and
1978 $U_{e} = (1/8\pi k) E^2 Ah$. Substituting the
1979 first expression into the second, we find $U_{e}=2\pi kq^2 h/ A$.
1980 Comparing this to the definition of capacitance, we end up with
1990 Two common geometries for inductors. The cylindrical
1991 shape on the left is called a solenoid.\index{solenoid}
1999 %q{The symbol for an inductor.}
2011 <% begin_sec("Inductors") %>
2013 create a magnetic field, so in fact every current-carrying
2014 wire in a circuit acts as an inductor!
2015 However, this type of ``stray'' inductance is typically
2016 negligible, just as we can usually ignore the stray resistance of
2017 our wires and only take into account the actual resistors.
2018 To store any appreciable amount of magnetic energy, one
2019 usually uses a coil of wire designed specifically to be an inductor.
2020 All the loops' contribution to the magnetic field add together
2021 to make a stronger field. Unlike
2022 capacitors and resistors, practical inductors are easy to make by
2023 hand. One can for instance spool some wire around a short wooden dowel.
2024 An inductor like this, in the form cylindrical coil of wire, is called
2025 a solenoid, \figref{coilshapes}, and a stylized solenoid, \figref{indsymbol}, is the
2026 symbol used to represent an inductor in a circuit regardless
2027 of its actual geometry.
2029 How much energy does an inductor store?
2030 The energy density is proportional
2031 to the square of the magnetic field strength, which is in
2032 turn proportional to the current flowing through the
2033 coiled wire, so the energy stored in the inductor must be
2034 proportional to $I^2$. We write $L/2$ for the constant of
2035 proportionality, giving
2037 U_L = \frac{L}{2}I^2 \qquad .
2040 As in the definition of capacitance, we have a
2041 factor of 1/2, which is purely a matter of definition. The
2042 quantity $L$ is called the \emph{inductance}\index{inductance!defined} of the
2043 inductor, and we see that its units must be joules per
2044 ampere squared. This clumsy combination of units is more
2045 commonly abbreviated as the henry, 1 henry = 1 $\zu{J}/\zu{A}^2$.
2046 Rather than memorizing this definition, it makes more sense
2047 to derive it when needed from the definition of inductance.
2048 Many people know inductors simply as ``coils,'' or
2049 ``chokes,'' and will not understand you if you refer to an
2050 ``inductor,'' but they will still refer to $L$ as the
2051 ``inductance,'' not the ``coilance'' or ``chokeance!''
2053 There is a lumped circuit approximation for inductors, just like the one
2054 for capacitors (p. \pageref{lumped-circuit-approx}).
2055 For a capacitor, this means assuming that
2056 the electric fields are completely internal, so
2057 that components only interact via currents that flow through wires,
2058 not due to the physical overlapping of their fields in space. Similarly
2059 for an inductor, the lumped circuit approximation is the assumption
2060 that the magnetic fields are completely internal.
2066 %q{Inductances in series add.}
2070 \begin{eg}{Identical inductances in series}
2071 If two inductors are placed in series, any current that
2072 passes through the combined double inductor must pass
2073 through both its parts.
2074 If we assume the lumped circuit approximation, the two inductors' fields
2075 don't interfere with each other, so the energy is doubled for a given current.
2077 definition of inductance, the inductance is doubled as well.
2078 In general, inductances in series add, just like resistances.
2079 The same kind of reasoning also shows that the inductance of
2080 a solenoid is approximately proportional to its length,
2081 assuming the number of turns per unit length is kept
2082 constant. (This is only approximately true, because putting
2083 two solenoids end-to-end causes the fields just outside
2084 their mouths to overlap and add together in a complicated manner.
2085 In other words, the lumped-circuit approximation may not be very
2093 %q{Capacitances in parallel add.}
2097 \begin{eg}{Identical capacitances in parallel}
2098 When two identical capacitances are placed in parallel, any
2099 charge deposited at the terminals of the combined double
2100 capacitor will divide itself evenly between the two parts.
2101 The electric fields surrounding each capacitor will be half
2102 the intensity, and therefore store one quarter the energy.
2103 Two capacitors, each storing one quarter the energy, give
2104 half the total energy storage. Since capacitance is
2105 inversely related to energy storage, this implies that
2106 identical capacitances in parallel give double the
2107 capacitance. In general, capacitances in parallel add. This
2108 is unlike the behavior of inductors and resistors, for which
2109 series configurations give addition.
2111 This is consistent with the result of example \ref{eg:capparplate},
2112 which had the capacitance of a single parallel-plate capacitor
2113 proportional to the area of the plates.
2115 parallel-plate capacitors, and we combine them in parallel
2116 and bring them very close together side by side, we have
2117 produced a single capacitor with plates of double the area, and
2118 it has approximately double the capacitance, subject to any
2119 violation of the lumped-circuit approximation due to the interaction
2120 of the fields where the edges of the capacitors are joined together.
2123 Inductances in parallel and capacitances in series are explored
2124 in homework problems \ref{hw:parallelinductors} and
2125 \ref{hw:seriescapacitors}.
2131 %q{A variable capacitor.}
2135 \begin{eg}{A variable capacitor}
2136 Figure \figref{variablecap}/1 shows the construction of a
2137 variable capacitor out of two parallel semicircles of metal.
2138 One plate is fixed, while the other can be rotated about
2139 their common axis with a knob. The opposite charges on the
2140 two plates are attracted to one another, and therefore tend
2141 to gather in the overlapping area. This overlapping area,
2142 then, is the only area that effectively contributes to the
2143 capacitance, and turning the knob changes the capacitance.
2144 The simple design can only provide very small capacitance
2145 values, so in practice one usually uses a bank of capacitors,
2146 wired in parallel, with all the moving parts on the same shaft.
2153 Suppose that two parallel-plate capacitors are wired in parallel, and
2154 are placed very close together, side by side, so that the lumped circuit
2155 approximation is not very accurate. Will the resulting capacitance be
2156 too small, or too big? Could you twist the circuit into a different shape
2157 and make the effect be the other way around, or make the effect vanish?
2158 How about the case of two inductors in series?
2165 %q{Discussion question \ref{dq:dielectric}.}
2169 \begin{dq}\label{dq:dielectric}
2170 Most practical capacitors do not have an air gap or vacuum gap between the plates; instead,
2171 they have an insulating substance called a dielectric. We can think of the molecules in this substance
2172 as dipoles that are free to rotate (at least a little), but that are not free to move
2173 around, since it is a solid. The figure shows a highly stylized and unrealistic way of
2174 visualizing this. We imagine that all the dipoles are intially turned sideways, (1),
2175 and that as the capacitor is charged, they all respond by turning through a certain angle, (2).
2176 (In reality, the scene might be much more random, and the alignment effect much weaker.)
2178 For simplicity, imagine inserting just one electric dipole
2179 into the vacuum gap. For a given amount of charge on
2180 the plates, how does this affect the amount of
2181 energy stored in the electric field? How does this affect the capacitance?
2183 Now redo the analysis in terms of the mechanical work needed in order to
2184 charge up the plates.
2187 % ------------------------------------------
2190 <% begin_sec("Oscillations") %>
2192 Figure \figref{lrc} shows the simplest possible oscillating circuit.
2193 For any useful application it would actually need to include
2194 more components. For example, if it was a radio tuner, it
2195 would need to be connected to an antenna and an amplifier.
2196 Nevertheless, all the essential physics is there.
2202 %q{A series LRC circuit.}
2209 %q{A mechanical analogy for the LRC circuit.}
2213 We can analyze it without any sweat or tears whatsoever,
2214 simply by constructing an analogy with a mechanical system.
2215 In a mechanical oscillator, \figref{lrcanalogy}, we have two forms of stored energy,
2217 U_{spring} &= \frac{1}{2}kx^2 &(1) \\
2218 K &= \frac{1}{2}mv^2 \qquad . \qquad &(2)
2221 In the case of a mechanical oscillator,
2222 we have usually assumed a friction force of the form that
2223 turns out to give the nicest mathematical results,
2224 $F=-bv$. In the circuit, the dissipation of energy
2225 into heat occurs via the resistor, with no mechanical force
2226 involved, so in order to make the analogy, we need to restate
2227 the role of the friction force in terms of energy. The power
2228 dissipated by friction equals the mechanical work it does in
2229 a time interval $\der t$, divided by $\der t$,
2230 $P=W/\der t=F\der x/\der t=Fv=-bv^2$, so
2232 \text{rate of heat dissipation} = -bv^2 \qquad . \qquad (3)
2235 <% self_check('lrcsigns',<<-'SELF_CHECK'
2236 Equation (1) has $x$ squared, and equations (2) and (3) have
2237 $v$ squared. Because they're squared, the results don't
2238 depend on whether these variables are positive or negative.
2239 Does this make physical sense?
2243 In the circuit, the stored forms of energy are
2245 U_C &= \frac{1}{2C}q^2 &(1') \\
2246 U_L &= \frac{1}{2}LI^2 \qquad , \qquad &(2')
2248 and the rate of heat dissipation in the resistor is
2250 \text{rate of heat dissipation} = -RI^2 \qquad . \qquad (3')
2252 Comparing the two sets of equations, we first form analogies
2253 between quantities that represent the state of the system at
2254 some moment in time:
2256 x &\leftrightarrow q\\
2257 v &\leftrightarrow I\\
2259 <% self_check('xvqi',<<-'SELF_CHECK'
2260 How is $v$ related mathematically to $x$? How
2261 is $I$ connected to $q$? Are the two relationships analogous?
2265 Next we relate the ones that describe the system's
2266 permanent characteristics:
2268 k &\leftrightarrow 1/C\\
2269 m &\leftrightarrow L\\
2270 b &\leftrightarrow R\\
2273 Since the mechanical
2274 system naturally oscillates with a frequency\footnote{As in chapter \ref{ch:2},
2275 we use the word ``frequency'' to mean either $f$ or $\omega=2\pi f$
2276 when the context makes it clear which is being referred to.}
2277 $\omega\approx\sqrt{k/m}$ , we can immediately solve the
2278 electrical version by analogy, giving
2280 \omega \approx \frac{1}{\sqrt{LC}} \qquad .
2283 Since the resistance $R$ is analogous to $b$ in the mechanical case, we
2284 find that the $Q$ (quality factor, not charge) of the
2285 resonance is inversely proportional to $R$, and the width of
2286 the resonance is directly proportional to $R$.
2288 \begin{eg}{Tuning a radio receiver}
2289 A radio receiver uses this kind of circuit to pick out the
2290 desired station. Since the receiver resonates at a
2291 particular frequency, stations whose frequencies are far off
2292 will not excite any response in the circuit. The value of
2293 $R$ has to be small enough so that only one station at a
2294 time is picked up, but big enough so that the tuner isn't
2295 too touchy. The resonant frequency can be tuned by adjusting
2296 either $L$ or $C$, but variable capacitors are easier to
2297 build than variable inductors.
2299 \begin{eg}{A numerical calculation}
2300 The phone company sends more than one conversation at a
2301 time over the same wire, which is accomplished by shifting
2302 each voice signal into different range of frequencies during
2303 transmission. The number of signals per wire can be maximized by making
2304 each range of frequencies (known as a bandwidth) as small as
2305 possible. It turns out that only a relatively narrow range
2306 of frequencies is necessary in order to make a human voice
2307 intelligible, so the phone company filters out all the
2308 extreme highs and lows. (This is why your phone voice sounds
2309 different from your normal voice.)
2312 If the filter consists of an LRC circuit with a
2313 broad resonance centered around 1.0 kHz, and the capacitor
2314 is 1 $\mu\zu{F}$ (microfarad), what inductance value must be used?
2317 Solving for $L$, we have
2319 L &= \frac{1}{ C\omega^2} \\
2320 &= \frac{1}{(10^{-6}\ \zu{F})(2\pi\times10^3\ \zu{s}^{-1})^2} \\
2321 &= 2.5\times10^{-3}\ \zu{F}^{-1}\zu{s}^2
2323 Checking that these really are the same units as henries is
2324 a little tedious, but it builds character:
2326 \zu{F}^{-1}\zu{s}^2 &= (\zu{C}^2/\zu{J})^{-1}\zu{s}^2 \\
2327 &= \zu{J}\cdot\zu{C}^{-2}\zu{s}^2 \\
2328 &= \zu{J}/\zu{A}^2 \\
2331 The result is 25 mH (millihenries).
2333 This is actually quite a large inductance value, and would
2334 require a big, heavy, expensive coil. In fact, there is a
2335 trick for making this kind of circuit small and cheap. There
2336 is a kind of silicon chip called an op-amp, which, among
2337 other things, can be used to simulate the behavior of an
2338 inductor. The main limitation of the op-amp is that it is
2339 restricted to low-power applications.\index{op-amp}\index{operational amplifier (op-amp)}
2343 % ------------------------------------------
2345 <% begin_sec("Voltage and current") %>
2347 What is physically happening in one of these oscillating
2348 circuits? Let's first look at the mechanical case, and then
2349 draw the analogy to the circuit. For simplicity, let's
2350 ignore the existence of damping, so there is no friction in
2351 the mechanical oscillator, and no resistance in the electrical one.
2353 Suppose we take the mechanical oscillator and pull the mass
2354 away from equilibrium, then release it. Since friction tends
2355 to resist the spring's force, we might naively expect that
2356 having zero friction would allow the mass to leap instantaneously
2357 to the equilibrium position. This can't happen, however,
2358 because the mass would have to have infinite velocity in
2359 order to make such an instantaneous leap. Infinite velocity
2360 would require infinite kinetic energy, but the only kind of
2361 energy that is available for conversion to kinetic is the
2362 energy stored in the spring, and that is finite,
2363 not infinite. At each step on its way back to equilibrium,
2364 the mass's velocity is controlled exactly by the amount of
2365 the spring's energy that has so far been converted into kinetic energy. After the mass
2366 reaches equilibrium, it overshoots due to its own momentum.
2367 It performs identical oscillations on both sides of
2368 equilibrium, and it never loses amplitude because friction
2369 is not available to convert mechanical energy into heat.
2371 Now with the electrical oscillator, the analog of position
2372 is charge. Pulling the mass away from equilibrium is like
2373 depositing charges $+q$ and $-q$ on the plates of the
2374 capacitor. Since resistance tends to resist the flow of
2375 charge, we might imagine that with no friction present, the
2376 charge would instantly flow through the inductor (which is,
2377 after all, just a piece of wire), and the capacitor would
2378 discharge instantly. However, such an instant discharge is
2379 impossible, because it would require infinite current for
2380 one instant. Infinite current would create infinite magnetic
2381 fields surrounding the inductor, and these fields would have
2382 infinite energy. Instead, the rate of flow of current is
2383 controlled at each instant by the relationship between the
2384 amount of energy stored in the magnetic field and the amount
2385 of current that must exist in order to have that strong a
2386 field. After the capacitor reaches $q=0$, it overshoots. The
2387 circuit has its own kind of electrical ``inertia,'' because
2388 if charge was to stop flowing, there would have to be zero
2389 current through the inductor. But the current in the
2390 inductor must be related to the amount of energy stored in
2391 its magnetic fields. When the capacitor is at $q=0$, all the
2392 circuit's energy is in the inductor, so it must therefore
2393 have strong magnetic fields surrounding it and quite a bit
2394 of current going through it.
2396 The only thing that might seem spooky here is that we used
2397 to speak as if the current in the inductor caused the
2398 magnetic field, but now it sounds as if the field causes the
2399 current. Actually this is symptomatic of the elusive nature
2400 of cause and effect in physics. It's equally valid to think
2401 of the cause and effect relationship in either way. This may
2402 seem unsatisfying, however, and for example does not really
2403 get at the question of what brings about a voltage
2404 difference across the resistor (in the case where the
2405 resistance is finite); there must be such a voltage
2406 difference, because without one, Ohm's law would predict
2407 zero current through the resistor.
2409 Voltage, then, is what is really missing from our story so far.
2411 Let's start by studying the voltage across a capacitor.
2412 Voltage is electrical potential energy per unit charge, so
2413 the voltage difference between the two plates of the
2414 capacitor is related to the amount by which its energy would
2415 increase if we increased the absolute values of the charges on the plates
2416 from $q$ to $q+\der q$:
2418 V_C &= (U_{q+\der q}-U_q)/\der q \\
2419 &= \frac{\der U_C}{\der q} \\
2420 &= \frac{\der}{\der q}\left(\frac{1}{2C}q^2\right) \\
2424 use this as the definition of capacitance. This equation, by
2425 the way, probably explains the historical reason why $C$ was
2426 defined so that the energy was \emph{inversely} proportional
2427 to $C$ for a given value of $C$: the people who invented the
2428 definition were thinking of a capacitor as a device for
2429 storing charge rather than energy, and the amount of charge
2430 stored for a fixed voltage (the charge ``capacity'') is
2431 proportional to $C$.
2437 %q{The inductor releases energy and gives it to the black box.}
2441 In the case of an inductor, we know that if there is a
2442 steady, constant current flowing through it, then the
2443 magnetic field is constant, and so is the amount of energy
2444 stored; no energy is being exchanged between the inductor and
2445 any other circuit element. But what if the current is changing? The magnetic
2446 field is proportional to the current, so a change in one
2447 implies a change in the other. For concreteness, let's
2448 imagine that the magnetic field and the current are both
2449 decreasing. The energy stored in the magnetic field is
2450 therefore decreasing, and by conservation of energy, this
2451 energy can't just go away --- some other circuit element must
2452 be taking energy from the inductor. The simplest example,
2453 shown in figure \figref{indv}, is a series circuit consisting
2454 of the inductor plus one other circuit element. It doesn't matter
2455 what this other circuit element is, so we just call it a black box,
2456 but if you like, we can think of it as a resistor, in which case
2457 the energy lost by the inductor is being turned into heat by
2459 The junction rule tells us that both circuit
2460 elements have the same current through them, so $I$ could refer
2461 to either one, and likewise the loop rule tells us
2462 $V_{inductor}+V_{black\ box}=0$, so the two voltage drops
2463 have the same absolute value, which we can refer to as $V$.
2464 Whatever the black box is, the rate at which it
2465 is taking energy from the inductor is given by $|P|=|IV|$, so
2467 |IV| &= \left|\frac{\der U_L}{\der t}\right| \\
2468 &= \left|\frac{\der}{\der t}\left( \frac{1}{2}LI^2\right) \right| \\
2469 &= \left|LI\frac{\der I}{\der t}\right| \qquad ,\\
2471 |V| &= \left|L\frac{\der I}{\der t}\right| \qquad , \\
2473 which in many books is taken to be the definition of inductance. The
2474 direction of the voltage drop (plus or minus sign) is such
2475 that the inductor resists the change in current.
2477 There's one very intriguing thing about this result. Suppose, for
2478 concreteness, that the black box in figure \figref{indv} is a resistor,
2479 and that the inductor's energy is decreasing, and being converted into heat
2480 in the resistor. The voltage drop across the resistor indicates that
2481 it has an electric field across it, which is driving the current.
2482 But where is this electric field coming from? There are no charges anywhere
2483 that could be creating it! What we've discovered is one special case of
2484 a more general principle, the principle of induction: a changing magnetic
2485 field creates an electric field, which is in addition to any electric
2486 field created by charges.\index{induction} (The reverse is also true:
2487 any electric field that changes over time creates a magnetic field.)
2488 Induction forms the basis for such technologies as the generator\index{generator}
2489 and the transformer,\index{transformer} and ultimately it leads to the
2490 existence of light, which is a wave pattern in the electric and magnetic fields.
2491 These are all topics for chapter \ref{ch:em}, but it's truly remarkable
2492 that we could come to this conclusion without yet having learned any details about
2493 magnetism.\label{inductorinduction}
2499 Electric fields made by charges, 1, and by
2500 changing magnetic fields, 2 and 3.
2508 The cartoons in figure \figref{inductor-voltage} compares electric fields made by charges, 1,
2509 to electric fields made by changing magnetic fields, 2-3. In \figref{inductor-voltage}/1, two
2510 physicists are in a room whose ceiling is positively charged and whose floor is negatively
2511 charged. The physicist on the bottom throws a positively charged bowling ball into the curved
2512 pipe. The physicist at the top uses a radar gun to measure the speed of the ball as it comes
2513 out of the pipe. They find that the ball has slowed down by the time it gets to the top. By measuring the change in the
2514 ball's kinetic energy, the two physicists are acting just like a voltmeter. They conclude that the
2515 top of the tube is at a higher voltage than the bottom of the pipe. A difference in voltage
2516 indicates an electric field, and this field is clearly being caused by the charges in the
2519 In \figref{inductor-voltage}/2, there are no charges anywhere in the room except for the
2520 charged bowling ball. Moving charges make magnetic fields, so there is a magnetic field
2521 surrounding the helical pipe while the ball is moving through it. A magnetic field has been
2522 created where there was none before, and that field has energy. Where could the energy have
2523 come from? It can only have come from the ball itself, so the ball must be losing kinetic energy.
2524 The two physicists working together are again acting as a voltmeter, and again they conclude
2525 that there is a voltage difference between the top and bottom of the pipe. This indicates
2526 an electric field, but this electric field can't have been created by any charges, because
2527 there aren't any in the room. This electric field was created by the change in the magnetic field.
2529 The bottom physicist keeps on throwing balls into the pipe, until the pipe is full of balls,
2530 \figref{inductor-voltage}/3, and finally a steady current is established.
2531 While the pipe was filling up with balls, the energy in the
2532 magnetic field was steadily increasing, and that energy was being stolen from the
2533 balls' kinetic energy. But once a steady current is established, the energy in the magnetic
2534 field is no longer changing. The balls no longer have to give up energy in order to
2535 build up the field, and the physicist at the top finds that the balls are exiting the
2536 pipe at full speed again. There is no voltage difference any more. Although there is
2537 a current, $\der I/\der t$ is zero.
2543 'Ballasts for fluorescent lights. Top: a big, heavy inductor used as a ballast in an
2544 old-fashioned fluorescent bulb. Bottom: a small solid-state ballast, built into the base
2545 of a modern compact fluorescent bulb.'
2549 \begin{eg}{Ballasts}\label{eg:ballast}\index{ballast}\index{gas discharge tube}\index{fluorescent light}
2550 In a gas discharge tube, such as
2551 a neon sign, enough voltage is applied to a tube full of gas to ionize some of the atoms
2552 in the gas. Once ions have been created, the voltage accelerates them, and they
2553 strike other atoms, ionizing them as well and resulting in a chain reaction. This is a spark,
2554 like a bolt of lightning. But once the spark starts up,
2555 the device begins to act as though it has no resistance: more and more current flows, without
2556 the need to apply any more voltage. The power, $P=IV$, would grow without limit, and the tube
2557 would burn itself out.
2559 The simplest solution is to connect an inductor, known as the ``ballast,'' in series with the
2560 tube, and run the whole thing on an AC voltage. During each cycle, as the voltage reaches the
2561 point where the chain reaction begins, there is a surge of current, but the inductor resists
2562 such a sudden change of current, and the energy that would otherwise have burned out the bulb
2563 is instead channeled into building a magnetic field.
2565 A common household fluorescent lightbulb consists of a gas discharge tube in which the
2566 glass is coated with a fluorescent material. The gas in the tube emits ultraviolet light,
2567 which is absorbed by the coating, and the coating then glows in the visible spectrum.
2569 Until recently, it was common for a fluroescent light's ballast to be a simple inductor,
2570 and for the whole device to be operated at the 60 Hz frequency of the electrical power lines. This caused
2571 the lights to flicker annoyingly at 120 Hz, and could also cause an audible hum, since
2572 the magnetic field surrounding the inductor could exert mechanical forces on things.
2573 These days, the trend is toward using a solid-state circuit that mimics the behavior
2574 of an inductor, but at a frequency in the kilohertz range, eliminating the flicker and
2575 hum. Modern compact fluorescent bulbs electronic have ballasts built into their bases,
2576 so they can be used as plug-in replacements for incandescent bulbs. A compact fluorescent bulb
2577 uses about 1/4 the electricity of an incandescent bulb, lasts ten times longer, and
2578 saves \$30 worth of electricity over its lifetime.
2584 What happens when the physicist at the bottom in figure \figref{inductor-voltage}/3 starts
2585 getting tired, and decreases the current?
2592 % ------------------------------------------
2594 <% begin_sec("Decay") %>
2596 Up until now I've soft-pedaled the fact that by changing the
2597 characteristics of an oscillator, it is possible to produce
2598 non-oscillatory behavior. For example, imagine taking the
2599 mass-on-a-spring system and making the spring weaker and
2600 weaker. In the limit of small $k$, it's as though there was
2601 no spring whatsoever, and the behavior of the system is that
2602 if you kick the mass, it simply starts slowing down. For
2603 friction proportional to $v$, as we've been assuming, the
2604 result is that the velocity approaches zero, but never
2605 actually reaches zero. This is unrealistic for the
2606 mechanical oscillator, which will not have vanishing
2607 friction at low velocities, but it is quite realistic in the
2608 case of an electrical circuit, for which the voltage drop
2609 across the resistor really does approach zero as the
2610 current approaches zero.
2612 We do not even have to reduce $k$ to exactly zero in order
2613 to get non-oscillatory behavior. There is actually a finite,
2614 critical value below which the behavior changes, so that the
2615 mass never even makes it through one cycle. This is the case
2616 of overdamping, discussed on page \pageref{overdamped}.\index{oscillations!overdamped!electrical}%
2617 \index{overdamped oscillations!electrical}\index{damped oscillations!overdamped!electrical}
2619 Electrical circuits can exhibit all the same behavior. For
2620 simplicity we will analyze only the cases of LRC circuits
2621 with $L=0$ or $C=0$.
2623 <% begin_sec("The RC circuit") %>\index{RC circuit}
2624 We first analyze the RC circuit, \figref{rc}. In reality one would
2625 have to ``kick'' the circuit, for example by briefly
2626 inserting a battery, in order to get any interesting
2627 behavior. We start with Ohm's law and the equation for the
2628 voltage across a capacitor:
2642 The loop rule tells us
2644 V_R + V_C = 0 \qquad ,
2646 and combining the three equations results in a relationship between $q$ and $I$:
2650 The negative sign tells us that the current tends to reduce
2651 the charge on the capacitor, i.e., to discharge it. It makes
2652 sense that the current is proportional to $q$\/: if $q$ is
2653 large, then the attractive forces between the $+q$ and $-q$
2654 charges on the plates of the capacitor are large, and
2655 charges will flow more quickly through the resistor in order
2656 to reunite. If there was zero charge on the capacitor
2657 plates, there would be no reason for current to flow. Since
2658 amperes, the unit of current, are the same as coulombs per
2659 second, it appears that the quantity $RC$ must have
2660 units of seconds, and you can check for yourself that this
2661 is correct. $RC$ is therefore referred to as the time
2662 constant of the circuit.\index{RC time constant}\index{time constant!RC}
2664 How exactly do $I$ and $q$ vary with time? Rewriting $I$ as
2665 $\der q/\der t$, we have
2667 \frac{\der q}{\der t} = -\frac{1}{RC}q \qquad .
2669 We need a function $q(t)$ whose derivative
2670 equals itself, but multiplied by a negative constant.
2671 A function of the form $ae^t$, where $e=2.718...$ is the base of natural logarithms,
2672 is the only one that has its derivative equal to itself, and
2673 $ae^{bt}$ has its derivative equal to itself multiplied by $b$. Thus
2676 q = q_\zu{o}\exp\left(-\frac{t}{RC}\right) \qquad .
2684 Over a time interval $RC$, the charge on the capacitor
2685 is reduced by a factor of $e$.
2699 'spark-gap-transmitter',
2700 'Example \ref{eg:spark-gap-transmitter}.'
2705 <% begin_sec("The RL circuit") %>\index{RL circuit}
2706 The RL circuit, \figref{rl}, can be attacked by similar methods, and
2707 it can easily be shown that it gives
2709 I = I_\zu{o}\exp\left(-\frac{R}{L}t\right) \qquad .
2711 The RL time constant equals $L/R$.
2713 \begin{eg}{Death by solenoid; spark plugs}\index{spark plug}
2714 When we suddenly break an RL circuit, what will happen?
2715 It might seem that we're faced with a paradox, since we only
2716 have two forms of energy, magnetic energy and heat, and if the
2717 current stops suddenly, the magnetic field must collapse suddenly.
2718 But where does the lost magnetic energy go? It can't go into
2719 resistive heating of the resistor, because the circuit has now
2720 been broken, and current can't flow!
2722 The way out of this conundrum is to recognize that the open gap
2723 in the circuit has a resistance which is large, but not infinite.
2724 This large resistance causes the RL time constant $L/ R$ to be very
2725 small. The current thus continues to flow for a very brief time, and
2726 flows straight across the air gap where the circuit has been opened.
2727 In other words, there is a spark!
2729 We can determine based on several different lines of
2730 reasoning that the voltage drop from one end of the spark to the other must be
2731 very large. First, the air's resistance is large, so $V= IR$ requires a large
2732 voltage. We can also reason that all the energy in the magnetic field is
2733 being dissipated in a short time, so the power dissipated in the spark, $P= IV$,
2734 is large, and this requires a large value of $V$. ($I$ isn't large --- it is
2735 decreasing from its initial value.) Yet a third way to reach the same result
2736 is to consider the equation $V_{L}=\der I/\der t$\/: since the time constant
2737 is short, the time derivative $\der I/\der t$ is large.
2739 This is exactly how a car's spark plugs work. Another application is to
2740 electrical safety: it can be dangerous to break an inductive circuit
2741 suddenly, because so much energy is released in a short time. There is
2742 also no guarantee that the spark will discharge across the air gap; it
2743 might go through your body instead, since your body might have a lower
2747 \begin{eg}{A spark-gap radio transmitter}\label{eg:spark-gap-transmitter}
2748 Figure \figref{spark-gap-transmitter} shows a primitive type of radio transmitter, called
2749 a spark gap transmitter, used to send Morse code around the turn of the twentieth century. The high voltage
2750 source, V, is typically about 10,000 volts. When the telegraph
2751 switch, S, is closed, the RC
2752 circuit on the left starts charging up. An increasing voltage difference develops between the electrodes of the spark gap, G.
2753 When this voltage difference gets large enough, the electric field in the air between the electrodes
2754 causes a spark, partially discharging the RC circuit, but charging the LC circuit on the right.
2755 The LC circuit then oscillates at its resonant frequency (typically about 1 MHz), but the energy of these oscillations is
2756 rapidly radiated away by the antenna, A, which sends out radio waves (chapter \ref{ch:em}).
2763 A gopher gnaws through one of the wires in the DC lighting system in
2764 your front yard, and the lights turn off. At the instant when the circuit becomes open, we can
2765 consider the bare ends of the wire to be like the plates of a capacitor,
2766 with an air gap (or gopher gap) between them. What kind of capacitance
2767 value are we talking about here? What would this tell you about the
2772 % ------------------------------------------
2775 <% begin_sec("Review of Complex Numbers",nil,'complex-numbers') %>\index{complex numbers}
2776 For a more detailed treatment of complex numbers, see ch. 3 of
2777 James Nearing's free book at \\
2778 http://www.physics.miami.edu/nearing/mathmethods/.
2784 %q{Visualizing complex numbers as points in a plane.}
2792 Addition of complex numbers is just like addition of vectors,
2793 although the real and imaginary axes don't actually represent directions in space.
2799 'complex-conjugate',
2800 %q{A complex number and its conjugate.}
2805 We assume there is a number, $i$, such that $i^2=-1$.
2806 The square roots of $-1$ are then $i$ and $-i$. (In electrical engineering work,
2807 where $i$ stands for current, $j$ is sometimes used instead.) This gives rise
2808 to a number system, called the complex numbers, containing the real numbers as a subset.
2809 Any complex number $z$ can be written in the form $z=a+bi$, where $a$ and $b$ are
2810 real, and $a$ and $b$ are then referred to as the real and imaginary parts of $z$.
2811 A number with a zero real part is called an imaginary number.
2812 The complex numbers can be visualized as a plane, with the real number line placed
2813 horizontally like the $x$ axis of the familiar $x-y$ plane, and the imaginary numbers running
2814 along the $y$ axis. The complex numbers are complete in a way that the real numbers
2815 aren't: every nonzero complex number has two square roots. For example, 1 is
2816 a real number, so it is also a member of the complex numbers, and its square roots
2817 are $-1$ and 1. Likewise, $-1$ has square roots $i$ and $-i$, and the number $i$
2818 has square roots $1/\sqrt{2}+i/\sqrt{2}$ and $-1/\sqrt{2}-i/\sqrt{2}$.
2820 Complex numbers can be added and subtracted by adding or subtracting their real
2821 and imaginary parts. Geometrically, this is the same as vector addition.
2823 The complex numbers $a+bi$ and $a-bi$, lying at equal distances above and below the
2824 real axis, are called complex conjugates. The results of the quadratic formula
2825 are either both real, or complex conjugates of each other.
2826 The complex conjugate of a number $z$ is notated as $\bar{z}$ or
2829 The complex numbers obey all the same rules of arithmetic as the reals, except that
2830 they can't be ordered along a single line. That is, it's not possible to say whether
2831 one complex number is greater than another. We can compare them in terms of their
2832 magnitudes (their distances from the origin), but two distinct complex numbers may
2833 have the same magnitude, so, for example, we can't say whether $1$ is greater than
2834 $i$ or $i$ is greater than $1$.
2836 \begin{eg}{A square root of $i$}\label{eg:sqrt-i}
2837 \egquestion Prove that $1/\sqrt{2}+i/\sqrt{2}$ is a square root of $i$.
2839 \eganswer Our proof can use any ordinary rules of arithmetic, except for
2842 (\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})^2
2843 & = \frac{1}{\sqrt{2}}\cdot\frac{1}{\sqrt{2}}
2844 +\frac{1}{\sqrt{2}}\cdot\frac{i}{\sqrt{2}}
2845 +\frac{i}{\sqrt{2}}\cdot\frac{1}{\sqrt{2}}
2846 +\frac{i}{\sqrt{2}}\cdot\frac{i}{\sqrt{2}} \\
2847 &= \frac{1}{2}(1+i+i-1) \\
2856 A complex number can be described in terms of its magnitude and
2864 'complex-multiplication',
2865 %q{The argument of $uv$ is the sum of the arguments of $u$ and $v$.}
2870 Example \ref{eg:sqrt-i} showed one method of multiplying complex numbers.
2871 However, there is another nice interpretation of complex multiplication.
2872 We define the argument of a complex number as its angle in the complex plane, measured
2873 counterclockwise from the positive real axis.
2874 Multiplying two complex numbers then corresponds to multiplying their magnitudes,
2875 and adding their arguments.
2877 <% self_check('complex-square-root',<<-'SELF_CHECK'
2878 Using this interpretation of multiplication, how could you find the
2879 square roots of a complex number?
2884 \begin{eg}{An identity}
2885 The magnitude $|z|$ of a complex number $z$ obeys
2886 the identity $|z|^2=z\bar{z}$. To prove this, we first note that $\bar{z}$
2887 has the same magnitude as $z$, since flipping it to the other side of the
2888 real axis doesn't change its distance from the origin. Multiplying $z$ by
2889 $\bar{z}$ gives a result whose magnitude is found by multiplying their
2890 magnitudes, so the magnitude of
2891 $z\bar{z}$ must therefore equal $|z|^2$. Now we just have to prove that
2892 $z\bar{z}$ is a positive real number. But if, for example, $z$ lies counterclockwise
2893 from the real axis, then $\bar{z}$ lies clockwise from it. If $z$ has a positive
2894 argument, then $\bar{z}$ has a negative one, or vice-versa. The sum of their arguments is therefore
2895 zero, so the result has an argument of zero, and is on the positive real axis.
2896 \footnote{I cheated a little. If $z$'s argument
2897 is 30 degrees, then we could say $\bar{z}$'s was -30, but we could also call it
2898 330. That's OK, because 330+30 gives 360, and an argument of
2899 360 is the same as an argument of zero.}
2902 This whole system was built up in order to make every number have square roots.
2903 What about cube roots, fourth roots, and so on? Does it get even more weird when
2904 you want to do those as well? No. The complex number system we've already discussed
2905 is sufficient to handle all of them. The nicest way of thinking about it is in terms
2906 of roots of polynomials. In the real number system, the polynomial $x^2-1$ has
2907 two roots, i.e., two values of $x$ (plus and minus one) that we can plug in to the
2908 polynomial and get zero. Because it has these two real roots, we can rewrite the
2909 polynomial as $(x-1)(x+1)$. However, the polynomial $x^2+1$ has no real roots. It's
2910 ugly that in the real number system, some second-order polynomials have two
2911 roots, and can be factored, while others can't. In the complex number system,
2912 they all can. For instance, $x^2+1$ has roots $i$ and $-i$, and can be factored
2913 as $(x-i)(x+i)$. In general, the fundamental theorem of algebra\index{fundamental theorem of algebra}
2914 states that in the complex number system,
2915 any nth-order polynomial can be factored completely
2916 into $n$ linear factors, and we can also say that it has $n$ complex roots,
2917 with the understanding that some of the roots may be the same. For instance,
2918 the fourth-order polynomial $x^4+x^2$ can be factored as $(x-i)(x+i)(x-0)(x-0)$,
2919 and we say that it has four roots, $i$, $-i$, 0, and 0, two of which happen to be
2920 the same. This is a sensible way to think about it, because in real life, numbers are
2921 always approximations anyway, and if we make tiny, random changes
2922 to the coefficients of this polynomial,
2923 it will have four distinct roots, of which two just happen to be very close
2929 Find $\arg i$, $\arg(-i)$, and $\arg 37$, where $\arg z$ denotes the argument of the complex number $z$.
2933 Visualize the following multiplications in the complex plane using the interpretation of multiplication
2934 in terms of multiplying magnitudes and adding arguments: $(i)(i)=-1$, $(i)(-i)=1$, $(-i)(-i)=-1$.
2938 If we visualize $z$ as a point in the complex plane, how should we visualize $-z$? What does this mean
2939 in terms of arguments? Give similar interpretations for $z^2$ and $\sqrt{z}$.
2943 Find four different complex numbers $z$ such that $z^4=1$.
2947 Compute the following. For the final two, use the magnitude and argument, not the real and imaginary parts.
2950 \arg(1+i) \quad , \quad
2951 \left|\frac{1}{1+i}\right| \quad , \quad
2952 \arg\left(\frac{1}{1+i}\right) \quad , \quad
2954 From these, find the real and imaginary parts of $1/(1+i)$.
2958 <% begin_sec("Euler's formula") %>\index{Euler's formula}\index{Euler, Leonhard}
2959 Having expanded our horizons to include the complex numbers, it's natural to want to extend
2960 functions we knew and loved from the world of real numbers so that they can also operate on
2961 complex numbers. The only really natural way to do this in general is to use Taylor series.
2962 A particularly beautiful thing happens with the functions $e^x$, $\sin x$, and $\cos x$:
2964 e^x &= 1 + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \ldots \\
2965 \cos x &= 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4 - \ldots \\
2966 \sin x &= x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \ldots
2968 If $x=i\phi$ is an imaginary number, we have
2970 e^{i\phi} = \cos \phi + i \sin \phi \qquad ,
2972 a result known as Euler's formula.\index{Euler's formula}
2973 The geometrical interpretation in the complex
2974 plane is shown in figure \figref{euler}.
2977 Although the result may seem like something out of a freak show at first,
2978 applying the definition of the exponential function
2979 makes it clear how natural it is:
2981 e^x = \lim_{n\rightarrow \infty} \left(1+\frac{x}{n}\right)^n \qquad .
2983 When $x=i\phi$ is imaginary, the quantity $(1+i\phi/n)$ represents a number
2984 lying just above 1 in the complex plane. For large $n$, $(1+i\phi/n)$
2985 becomes very close to the unit circle, and its argument is the small
2986 angle $\phi/n$. Raising this number to the nth power multiplies its
2987 argument by $n$, giving a number with an argument of $\phi$.
2992 'The complex number $e^{i\phi}$ lies on the unit circle.'
2999 'Leonhard Euler (1707-1783)'
3004 Euler's formula is used frequently in physics and engineering.
3006 \begin{eg}{Trig functions in terms of complex exponentials}
3007 \egquestion Write the sine and cosine functions in terms of exponentials.
3009 \eganswer Euler's formula for $x=-i\phi$ gives $\cos \phi - i \sin \phi$,
3010 since $\cos(-\theta)=\cos\theta$, and $\sin(-\theta)=-\sin\theta$.
3012 \cos x &= \frac{e^{ix}+e^{-ix}}{2} \\
3013 \sin x &= \frac{e^{ix}-e^{-ix}}{2i}
3017 \begin{eg}{A hard integral made easy}
3018 \egquestion Evaluate
3020 \int e^x \cos x \der x
3023 \eganswer This seemingly impossible integral becomes easy if we rewrite
3024 the cosine in terms of exponentials:
3026 \int e^x & \cos x \der x \\
3027 &= \int e^x \left(\frac{e^{ix}+e^{-ix}}{2}\right) \der x \\
3028 &= \frac{1}{2} \int (e^{(1+i)x}+e^{(1-i)x})\der x \\
3029 &= \frac{1}{2} \left( \frac{e^{(1+i)x}}{1+i}+\frac{e^{(1-i)x}}{1-i} \right)+ c
3032 Since this result is the integral of a real-valued function, we'd like it to be
3033 real, and in fact it is, since the first and second terms are complex conjugates of
3034 one another. If we wanted to, we could use Euler's theorem to convert it back to
3035 a manifestly real result.\footnote{In general, the use of complex number techniques to
3036 do an integral could result in a complex number, but that complex number would
3037 be a constant, which could be subsumed within the usual constant of integration.}
3042 <% begin_sec("Impedance",nil,'impedance') %>\index{impedance}
3044 So far we have been thinking in terms of the free oscillations
3045 of a circuit. This is like a mechanical oscillator that has
3046 been kicked but then left to oscillate on its own without
3047 any external force to keep the vibrations from dying out.
3048 Suppose an LRC circuit is driven with a sinusoidally varying
3049 voltage, such as will occur when a radio tuner is hooked up
3050 to a receiving antenna. We know that a current will flow in
3051 the circuit, and we know that there will be resonant
3052 behavior, but it is not necessarily simple to relate current
3053 to voltage in the most general case. Let's start instead
3054 with the special cases of LRC circuits consisting of only a
3055 resistance, only a capacitance, or only an inductance. We
3056 are interested only in the steady-state response.
3058 The purely resistive case is easy. Ohm's law gives
3060 I = \frac{V}{R} \qquad .
3063 In the purely capacitive case, the relation $V=q/C$ lets us calculate
3065 I &= \frac{\der q}{\der t} \\
3066 &= C \frac{\der V}{\der t} \qquad .
3069 This is partly analogous to Ohm's law. For example, if we double the amplitude of a sinusoidally
3070 varying AC voltage, the derivative $\der V/\der t$ will also double, and the amplitude of the
3071 sinusoidally varying current will also double. However, it is not true that $I=V/R$, because
3072 taking the derivative of a sinusoidal function shifts its phase by 90 degrees.
3073 If the voltage varies as, for example, $V(t)=V_\zu{o}\sin (\omega t)$,
3074 then the current will be $I(t)=\omega C V_\zu{o}\cos (\omega t)$.
3075 The amplitude of the current is $\omega C V_\zu{o}$, which is proportional to $V_\zu{o}$, but
3076 it's not true that $I(t)=V(t)/R$ for some constant $R$.
3078 A second problem that crops up is that our entire analysis of DC resistive circuits was built
3079 on the foundation of the loop rule and the junction rule, both of which are statements about
3080 sums. To apply the junction rule to an AC circuit, for exampe, we would say that the sum of the
3081 sine waves describing the currents coming into the junction is equal (at every moment in time)
3082 to the sum of the sine waves going out. Now sinusoidal functions have a remarkable property, which is that if you
3083 add two different sinusoidal functions having the same frequency, the result
3084 is also a sinusoid with that frequency.
3085 For example, $\cos\omega t+\sin\omega t=\sqrt{2}\sin(\omega t+\pi/4)$, which can be proved
3086 using trig identities. The trig identities can get very cumbersome, however, and there is
3087 a much easier technique involving complex numbers.
3089 Figure \figref{polar} shows a useful way to visualize what's going on.
3090 When a circuit is oscillating at a frequency $\omega$, we use points in
3091 the plane to represent sinusoidal functions with various phases and
3094 <% self_check('represent-as-complex',<<-'SELF_CHECK'
3095 Which of the following functions can be represented in this way? $\cos(6t-4)$, $\cos^2t$, $\tan t$
3103 %q{In a capacitor, the current is $90\degunit$ ahead of the voltage in phase.}
3110 %q{Representing functions with points in polar coordinates.}
3118 %q{Adding two sinusoidal functions.}
3123 The simplest examples of how to
3124 visualize this in polar coordinates are ones like $\cos \omega t+\cos \omega t=2\cos \omega t$,
3125 where everything has the same phase, so all the points
3126 lie along a single line in the polar plot, and addition is just like adding numbers
3128 The less trivial example $\cos\omega t+\sin\omega t=\sqrt{2}\sin(\omega t+\pi/4)$,
3129 can be visualized as in figure \figref{sinpluscos}.
3131 Figure \figref{sinpluscos} suggests that
3132 all of this can be tied together nicely if we identify our plane with the plane
3133 of complex numbers. For example, the complex numbers 1 and $i$ represent the
3134 functions $\sin\omega t$ and $\cos\omega t$.
3135 In figure \figref{capvi}, for example, the voltage across the capacitor
3136 is a sine wave multiplied by a number that gives its amplitude, so we associate that
3137 function with a number $\tilde{V}$ lying on the real axis. Its magnitude, $|\tilde{V}|$,
3138 gives the amplitude in units of volts, while its argument $\arg \tilde{V}$, gives its
3139 phase angle, which is zero. The current is a multiple of a sine wave, so we identify
3140 it with a number $\tilde{I}$ lying on the imaginary axis.
3141 We have $\arg\tilde{I}=90\degunit$, and $|\tilde{I}|$ is the amplitude of the current,
3142 in units of amperes. But comparing with our result above, we have $|\tilde{I}|=\omega C|\tilde{V}|$.
3143 Bringing together the phase and magnitude information, we have $\tilde{I}=i\omega C\tilde{V}$.
3144 This looks very much like Ohm's law, so we write
3146 \tilde{I} = \frac{\tilde{V}}{Z_C} \qquad ,
3150 Z_C = -\frac{i}{\omega C} \qquad , \qquad \text{[impedance of a capacitor]}
3152 having units of ohms, is called the \emph{impedance} of
3153 the capacitor at this frequency.
3156 the impedance becomes infinite at zero frequency. Zero
3157 frequency means that it would take an infinite time before
3158 the voltage would change by any amount. In other words, this
3159 is like a situation where the capacitor has been connected
3160 across the terminals of a battery and been allowed to settle
3161 down to a state where there is constant charge on both
3162 terminals. Since the electric fields between the plates are
3163 constant, there is no energy being added to or taken out of
3164 the field. A capacitor that can't exchange energy with any
3165 other circuit component is nothing more than a broken (open) circuit.
3167 Note that we have two types of complex numbers: those that represent sinusoidal functions
3168 of time, and those that represent impedances.
3169 The ones that represent sinusoidal functions have tildes on top, which look like little sine
3172 <% self_check('caplabel',<<-'SELF_CHECK'
3173 Why can't a capacitor have its impedance printed on it along
3174 with its capacitance?
3183 The current through an inductor lags behind the voltage by
3184 a phase angle of $90\degunit$.
3189 Similar math (but this time with an integral instead
3190 of a derivative) gives
3192 Z_L = i\omega L \qquad \text{[impedance of an inductor]}
3194 for an inductor.\index{impedance!of an inductor} It makes sense that the inductor has lower
3195 impedance at lower frequencies, since at zero frequency
3196 there is no change in the magnetic field over time. No
3197 energy is added to or released from the magnetic field, so
3198 there are no induction effects, and the inductor acts just
3199 like a piece of wire with negligible resistance. The term
3200 ``choke'' for an inductor refers to its ability to ``choke
3201 out'' high frequencies.
3203 The phase relationships shown in figures \figref{capvi} and \figref{indvi}
3204 can be remembered using my own mnemonic, ``eVIL,'' which shows that the voltage
3205 (V) leads the current (I) in an inductive circuit, while the opposite is
3206 true in a capacitive one. A more traditional mnemonic is ``ELI the ICE man,''
3207 which uses the notation E for emf, a concept closely related to voltage
3208 (see p. \pageref{emf-term-introduced}).
3210 Summarizing, the impedances of resistors, capacitors, and inductors are
3213 Z_C &= -\frac{i}{\omega C}\\
3214 Z_L &= i\omega L \qquad .
3217 \begin{eg}{Low-pass and high-pass filters}
3218 An LRC circuit only responds to a certain range (band) of
3219 frequencies centered around its resonant frequency. As a
3220 filter, this is known as a bandpass filter. If you turn down
3221 both the bass and the treble on your stereo, you have
3222 created a bandpass filter.
3224 To create a high-pass or low-pass filter, we only need to
3225 insert a capacitor or inductor, respectively, in series. For
3226 instance, a very basic surge protector for a computer could
3227 be constructed by inserting an inductor in series with the
3228 computer. The desired 60 Hz power from the wall is
3229 relatively low in frequency, while the surges that can
3230 damage your computer show much more rapid time variation.
3231 Even if the surges are not sinusoidal signals, we can think
3232 of a rapid ``spike'' qualitatively as if it was very high in
3233 frequency --- like a high-frequency sine wave, it changes very rapidly.
3235 Inductors tend to be big, heavy, expensive circuit
3236 elements, so a simple surge protector would be more likely
3237 to consist of a capacitor in \emph{parallel} with the
3238 computer. (In fact one would normally just connect one side
3239 of the power circuit to ground via a capacitor.) The
3240 capacitor has a very high impedance at the low frequency of
3241 the desired 60 Hz signal, so it siphons off very little of
3242 the current. But for a high-frequency signal, the capacitor's
3243 impedance is very small, and it acts like a zero-impedance,
3244 easy path into which the current is diverted.
3247 The main things to be careful about with impedance are that
3248 (1) the concept only applies to a circuit that is being
3249 driven sinusoidally, (2) the impedance of an inductor or
3250 capacitor is frequency-dependent.
3255 Figure \figref{capvi} on page \pageref{fig:capvi}
3256 shows the voltage and current for a capacitor.
3257 Sketch the $q$-$t$ graph, and use it to give a physical
3258 explanation of the phase relationship between the voltage and current. For example,
3259 why is the current zero when the voltage is at a maximum or minimum?
3263 Figure \figref{indvi} on page \pageref{fig:indvi}
3264 shows the voltage and current for an inductor. The power is
3265 considered to be positive when energy is being put into the inductor's magnetic field.
3266 Sketch the graph of the power, and then the graph of $U$, the energy stored in the magnetic field, and use it to give a physical
3267 explanation of the $P$-$t$ graph. In particular, discuss why the frequency is doubled on
3272 Relate the features of the graph in figure \figref{indvi} on page \pageref{fig:indvi}
3273 to the story told in cartoons in figure \figref{inductor-voltage}/2-3 on page \pageref{fig:inductor-voltage}.
3277 <% begin_sec("Power") %>
3278 How much power is delivered when an oscillating voltage
3279 is applied to an impedance? The equation $P=IV$ is generally true,
3280 since voltage is defined as energy per unit charge, and current is
3281 defined as charge per unit time: multiplying them gives energy
3282 per unit time. In a DC circuit, all three quantities were constant,
3283 but in an oscillating (AC) circuit, all three display time variation.
3284 <% begin_sec("A resistor") %>
3285 First let's examine the case of
3286 a resistor. For instance, you're probably reading this book from
3287 a piece of paper illuminated by a glowing lightbulb, which is
3288 driven by an oscillating voltage with amplitude $V_\zu{o}$.
3289 In the special case of a resistor, we know that $I$ and $V$ are in phase.
3290 For example, if $V$ varies as $V_\zu{o}\cos \omega t$, then $I$ will be a cosine as well,
3291 $I_\zu{o}\cos \omega t$. The power is then $I_\zu{o}V_\zu{o}\cos^2\omega t$, which
3292 is always positive,\footnote{A resistor always turns electrical energy into heat. It never
3293 turns heat into electrical energy!} and varies between 0 and $I_\zu{o}V_\zu{o}$.
3294 Even if the time variation was $\cos\omega t$ or $\sin(\omega t+\pi/4)$,
3295 we would still have a maximum power of $I_\zu{o}V_\zu{o}$, because both the voltage and the
3296 current would reach their maxima at the same time. In a lightbulb, the moment of maximum
3297 power is when the circuit is most rapidly heating the filament. At the
3298 instant when $P=0$, a quarter of a cycle later, no current is flowing, and no electrical
3299 energy is being turned into heat. Throughout the whole cycle, the filament is getting rid
3300 of energy by radiating light.\footnote{To many people, the word ``radiation'' implies nuclear
3301 contamination. Actually, the word simply means something that ``radiates'' outward.
3302 Natural sunlight is ``radiation.'' So is the light from a lightbulb, or the infrared light
3303 being emitted by your skin right now.}
3304 Since the circuit oscillates at a frequency\footnote{Note that this time ``frequency''
3305 means $f$, not $\omega$! Physicists and engineers generally use $\omega$ because it
3306 simplifies the equations, but electricians and technicians always use $f$. The 60 Hz
3307 frequency is for the U.S.} of $60\ \zu{Hz}$, the temperature doesn't really have time to
3308 cycle up or down very much over the 1/60 s period of the oscillation, and we don't notice
3309 any significant variation in the brightness of the light, even with a short-exposure
3317 Power in a resistor: the rate at which electrical energy is being
3318 converted into heat.
3323 Thus, what we really want to know is the average power, ``average'' meaning
3324 the average over one full cycle. Since we're covering a whole cycle with our average, it
3325 doesn't matter what phase we assume. Let's use a cosine. The total amount of energy
3326 transferred over one cycle is
3329 &= \int_0^T \frac{\der E}{\der t} \der t \qquad , \\
3330 \intertext{where $T=2\pi/\omega$ is the period.}
3331 E &= \int_0^T P \der t \\
3332 &= \int_0^T P \der t \\
3333 &= \int_0^T I_\zu{o}V_\zu{o} \cos^2\omega t \der t \\
3334 &= I_\zu{o}V_\zu{o} \int_0^T \cos^2\omega t \der t \\
3335 &= I_\zu{o}V_\zu{o} \int_0^T \frac{1}{2} \left(1+\cos 2\omega t\right) \der t \\
3336 \intertext{The reason for using the trig identity $\cos^2 x=(1+\cos 2 x)/2$ in the last step is that
3337 it lets us get the answer without doing a hard integral. Over the course of one full
3338 cycle, the quantity $\cos 2\omega t$ goes positive, negative, positive, and negative again,
3339 so the integral of it is zero. We then have}
3340 E &= I_\zu{o}V_\zu{o} \int_0^T \frac{1}{2} \der t \\
3341 &= \frac{I_\zu{o}V_\zu{o}T}{2}
3343 The average power is
3345 P_{av} &= \frac{\text{energy transferred in one full cycle}}{\text{time for one full cycle}} \\
3346 &= \frac{I_\zu{o}V_\zu{o}T/2}{T} \\
3347 &= \frac{I_\zu{o}V_\zu{o}}{2} \qquad ,\\
3349 i.e., the average is half the maximum. The power
3350 varies from $0$ to $I_\zu{o}V_\zu{o}$, and it spends equal amounts of time above and below the
3351 maximum, so it isn't surprising that the average power is half-way in between zero and
3352 the maximum. Summarizing, we have
3354 P_{av} &= \frac{I_\zu{o}V_\zu{o}}{2} \qquad \text{[average power in a resistor]}\\
3359 <% begin_sec("RMS quantities") %>
3360 Suppose one day the electric company decided to start supplying your electricity as
3361 DC rather than AC. How would the DC voltage have to be related to the amplitude
3362 $V_\zu{o}$ of the AC voltage previously used if they wanted your lightbulbs to
3363 have the same brightness as before? The resistance of the bulb, $R$, is a fixed
3364 value, so we need to relate the power to the voltage and the resistance, eliminating
3365 the current. In the DC case, this gives $P=IV=(V/R)V=V^2/R$. (For DC, $P$ and $P_{av}$
3366 are the same.) In the AC case, $P_{av} = I_\zu{o}V_\zu{o}/2=V_\zu{o}^2/2R$.
3367 Since there is no factor of 1/2 in the DC case, the same power could be provided
3368 with a DC voltage that was smaller by a factor of $1/\sqrt{2}$.
3369 Although you will hear people say that household voltage in the U.S. is 110 V,
3370 its amplitude is actually $(110\ \zu{V})\times\sqrt{2}\approx160\ \zu{V}$. The reason
3371 for referring to $V_\zu{o}/\sqrt{2}$ as ``the'' voltage is that people who are naive
3372 about AC circuits can plug $V_\zu{o}/\sqrt{2}$ into a familiar DC equation like
3373 $P=V^2/R$ and get the right \emph{average} answer. The quantity $V_\zu{o}/\sqrt{2}$
3374 is called the ``RMS'' voltage\index{root mean square}\index{RMS (root mean square)},
3375 which stands for ``root mean square.'' The idea is that if you square the function
3376 $V(t)$, take its average (mean) over one cycle, and then take the square root of that
3377 average, you get $V_\zu{o}/\sqrt{2}$. Many digital meters provide RMS readouts for
3378 measuring AC voltages and currents.
3380 <% begin_sec("A capacitor") %>
3381 For a capacitor, the calculation starts out the same, but ends up with a
3382 twist. If the voltage varies as a cosine, $V_\zu{o}\cos \omega t$, then
3383 the relation $I=C\der V/\der t$ tells us that the current will be some
3384 constant multiplied by minus the sine, $-V_\zu{o}\sin \omega t$.
3385 The integral we did in the case of a resistor now becomes
3387 E = \int_0^T -I_\zu{o}V_\zu{o} \sin \omega t \cos \omega t \der t \qquad ,\\
3389 and based on figure \figref{capvip}, you can easily
3390 convince yourself that over the course of one full cycle, the power spends two quarter-cycles
3391 being negative and two being positive. In other words, the average power is zero!
3398 Power in a capacitor: the rate at which energy is being
3399 stored in (+) or removed from (-) the electric field.
3404 Why is this? It makes sense if you think in terms of energy. A resistor converts
3405 electrical energy to heat, never the other way around. A capacitor, however, merely
3406 stores electrical energy in an electric field and then gives it back.
3409 P_{av} &= 0 \qquad \text{[average power in a capacitor]}\\
3411 Notice that although the average power is zero, the power at any given instant is
3412 \emph{not} typically zero, as shown in figure \figref{capvip}.
3413 The capacitor \emph{does}
3414 transfer energy: it's just that after borrowing some energy, it always pays it
3415 back in the next quarter-cycle.
3417 <% begin_sec("An inductor") %>
3418 The analysis for an inductor is similar to that for a capacitor: the power averaged
3419 over one cycle is zero. Again, we're merely storing energy temporarily in a field
3420 (this time a magnetic field) and getting it back later.
3427 <% begin_sec("Impedance Matching") %>\index{impedance matching}
3433 We wish to maximize the power delivered to
3434 the load, $Z_\zu{o}$, by adjusting its impedance.
3439 Figure \figref{zmatch} shows a commonly encountered situation: we wish
3440 to maximize the average
3441 power, $P_{av}$, delivered to the load for a fixed value of $V_\zu{o}$,
3442 the amplitude of the oscillating driving voltage. We assume that the
3443 impedance of the transmission line, $Z_T$ is a fixed value, over
3444 which we have no control, but we are able to design the load, $Z_\zu{o}$, with any impedance
3445 we like. For now, we'll also assume that both impedances are resistive. For example,
3446 $Z_T$ could be the resistance of a long extension cord, and
3447 $Z_\zu{o}$ could be a lamp at the end of it. The result generalizes immediately,
3448 however, to any kind of impedance.
3449 For example, the load could be a stereo
3450 speaker's magnet coil, which is displays both inductance and resistance. (For a purely inductive
3451 or capacitive load, $P_{av}$ equals zero, so the problem isn't very interesting!)
3453 Since we're assuming both the load and the transmission line are resistive, their
3454 impedances add in series, and the amplitude of the current is given by
3456 I_\zu{o} &= \frac{V_\zu{o}}{Z_\zu{o}+Z_T} \qquad ,\\
3458 P_{av} &= I_\zu{o}V_\zu{o}/2 \\
3459 &= I_\zu{o}^2Z_\zu{o}/2 \\
3460 &= \frac{V_\zu{o}^2Z_\zu{o}}{\left(Z_\zu{o}+Z_T\right)^2}/2 \qquad .
3461 \intertext{The maximum of this expression occurs where the derivative is zero,}
3462 0 &= \frac{1}{2}\frac{\der}{\der Z_\zu{o}}\left[\frac{V_\zu{o}^2Z_\zu{o}}{\left(Z_\zu{o}+Z_T\right)^2}\right] \\
3463 0 &= \frac{1}{2}\frac{\der}{\der Z_\zu{o}}\left[\frac{Z_\zu{o}}{\left(Z_\zu{o}+Z_T\right)^2}\right] \\
3464 0 &= \left(Z_\zu{o}+Z_T\right)^{-2}-2Z_\zu{o}\left(Z_\zu{o}+Z_T\right)^{-3} \\
3465 0 &= \left(Z_\zu{o}+Z_T\right)-2Z_\zu{o} \\
3468 In other words, to maximize the power delivered to the load, we should make the load's
3469 impedance the same as the transmission line's. This result may seem surprising at first,
3470 but it makes sense if you think about it. If the load's impedance is too high, it's like
3471 opening a switch and breaking the circuit; no power is delivered. On the other hand, it doesn't
3472 pay to make the load's impedance too small. Making it smaller does give more current, but
3473 no matter how small we make it, the current will still be limited by the transmission line's
3474 impedance. As the load's impedance approaches zero, the current approaches this fixed value,
3475 and the the power delivered, $I_\zu{o}^2Z_\zu{o}$, decreases in proportion to $Z_\zu{o}$.
3477 Maximizing the power transmission by matching $Z_T$ to $Z_\zu{o}$ is called \emph{impedance
3478 matching}. For example, an 8-ohm home stereo speaker will be correctly matched to
3479 a home stereo amplifier with an internal impedance of 8 ohms, and 4-ohm car speakers
3480 will be correctly matched to a car stereo with a 4-ohm internal impedance. You might think
3481 impedance matching would be unimportant because even if, for example, we used a car stereo
3482 to drive 8-ohm speakers, we could compensate for the mismatch simply by turning the volume
3483 knob higher. This is indeed one way to compensate for any impedance mismatch, but there
3484 is always a price to pay. When the impedances are matched, half the power is dissipated
3485 in the transmission line and half in the load. By connecting a 4-ohm amplifier to an
3486 8-ohm speaker, however, you would be setting up a situation in two watts were being dissipated
3487 as heat inside the amp for every amp being delivered to the speaker. In other words, you
3488 would be wasting energy, and perhaps burning out your amp when you turned up the volume to
3489 compensate for the mismatch.
3492 <% begin_sec("Impedances in series and parallel") %>
3493 How do impedances combine in series and parallel? The beauty of treating them
3494 as complex numbers is that they simply combine according to the same rules you've
3495 already learned as resistances.
3497 \begin{eg}{Series impedance}\label{eg:series-impedance}
3499 A capacitor and an inductor in series with each other are driven by a sinusoidally
3500 oscillating voltage. At what frequency is the current maximized?
3503 Impedances in series, like resistances in series, add. The capacitor and inductor
3504 act as if they were a single circuit element with an impedance
3507 &= i\omega L-\frac{ i}{\omega C} \qquad .\\
3508 \intertext{The current is then}
3509 \tilde{ I} = \frac{\tilde{ V}}{ i\omega L- i/\omega C} \qquad .
3511 We don't care about the phase of the current, only its amplitude, which is
3512 represented by the absolute value of the complex number $\tilde{ I}$, and
3513 this can be maximized by making
3514 $| i\omega L- i/\omega C|$ as small as possible.
3515 But there is some frequency at which this quantity is \emph{zero}\/ ---
3517 0 = i\omega L-\frac{ i}{\omega C}\\
3518 \frac{1}{\omega C} = \omega L\\
3519 \omega = \frac{1}{\sqrt{ LC}}
3521 At this frequency, the current is infinite! What is going on physically?
3522 This is an LRC circuit with $R=0$. It has a resonance at this frequency,
3523 and because there is no damping, the response at resonance is infinite.
3524 Of course, any real LRC circuit will have some damping, however small (cf. figure \figref{resonance} on page \pageref{fig:resonance}).
3526 \begin{eg}{Resonance with damping}\index{LRC circuit}
3528 What is the amplitude of the current in a series LRC circuit?
3531 Generalizing from example \ref{eg:series-impedance}, we add a third, real impedance:
3533 |\tilde{ I}| &= \frac{|\tilde{ V}|}{| Z|} \\
3534 &= \frac{|\tilde{ V}|}{| R+ i\omega L- i/\omega C|} \\
3535 &= \frac{|\tilde{ V}|}{\sqrt{ R^2+(\omega L-1/\omega C)^2}}
3537 This result would have taken pages of algebra without the complex number technique!
3540 \begin{eg}{A second-order stereo crossover filter}\label{eg:crossover}
3541 A stereo crossover filter ensures that the high frequencies
3542 go to the tweeter and the lows to the woofer. This can be
3543 accomplished simply by putting a single capacitor in series
3544 with the tweeter and a single inductor in series with the
3545 woofer. However, such a filter does not cut off very
3546 sharply. Suppose we model the speakers as resistors. (They
3547 really have inductance as well, since they have coils in
3548 them that serve as electromagnets to move the diaphragm that
3549 makes the sound.) Then the power they draw is $I^2 R$.
3550 Putting an inductor in series with the woofer, \figref{crossover}/1, gives a
3551 total impedance that at high frequencies is dominated by the
3552 inductor's, so the current is proportional to $\omega^{-1}$, and the
3553 power drawn by the woofer is proportional to $\omega^{-2}$.
3558 %q{Example \ref{eg:crossover}.}
3563 A second-order filter, like \figref{crossover}/2, is one that cuts off more
3564 sharply: at high frequencies, the power goes like $\omega^{-4}$. To
3565 analyze this circuit, we first calculate the total impedance:
3567 Z = Z_{L}+( Z_{C}^{-1}+ Z_R^{-1})^{-1}
3569 All the current passes through the inductor, so if the driving voltage being supplied
3570 on the left is $\tilde{ V}_d$, we have
3572 \tilde{ V}_d = \tilde{ I}_{L} Z \qquad ,
3576 \tilde{ V}_{L} = \tilde{ I}_{L} Z_L \qquad .
3578 The loop rule, applied to the outer perimeter of the circuit, gives
3580 \tilde{ V}_{d} = \tilde{ V}_{L}+\tilde{ V}_R \qquad .
3582 Straightforward algebra now results in
3584 \tilde{ V}_{R} = \frac{\tilde{ V}_{d}}%
3585 {1+ Z_L/ Z_{C}+ Z_{L}/ Z_R} \qquad .
3587 At high frequencies, the $Z_{L}/ Z_C$ term, which varies as
3588 $\omega^2$, dominates, so $\tilde{ V}_R$ and $\tilde{ I}_R$
3589 are proportional to $\omega^{-2}$,
3590 and the power is proportional to $\omega^{-4}$.
3599 <% begin_sec("Fields by Gauss' Law",4,'gauss') %>
3600 <% begin_sec("Gauss' law") %>
3601 The flea of subsection \ref{subsec:surfacefield} had a long and illustrious
3602 scientific career, and we're now going to pick up her story where we left off.
3603 This flea, whose name is Gauss\footnote{no relation to the human
3604 mathematician of the same name}, has derived the equation $E_\perp=2\pi k\sigma$ for the
3605 electric field very close to a charged surface with charge density $\sigma$.
3606 Next we will describe two improvements she is going to make to that equation.
3609 that the equation is not as useful as it could be, because it only gives the part of
3610 the field \emph{due to the surface}. If other charges are nearby, then their fields
3611 will add to this field as vectors, and the equation will not be true unless we carefully
3612 subtract out the field from the other charges. This is especially problematic for her
3613 because the planet on which she lives, known for obscure reasons as planet Flatcat, is
3614 itself electrically charged, and so are all the fleas
3615 --- the only thing that keeps them from floating off into
3616 outer space is that they are negatively charged, while Flatcat carries a positive charge, so they
3617 are electrically attracted to it. When Gauss found the original version of her equation,
3618 she wanted to demonstrate it to her skeptical colleagues in the laboratory, using electric
3619 field meters and charged pieces of metal foil. Even if she set up the measurements by
3620 remote control, so that her the charge on her own body would be too far away to have any
3621 effect, they would be disrupted by the ambient field of planet Flatcat. Finally, however,
3622 she realized that she could improve her equation by rewriting it as follows:
3624 E_{outward,\ on\ side\ 1}+E_{outward,\ on\ side\ 2} = 4\pi k\sigma \qquad .
3626 The tricky thing here is that ``outward'' means a different thing, depending on which
3627 side of the foil we're on. On the left side, ``outward'' means to the left, while on
3628 the right side, ``outward'' is right. A positively charged piece of metal foil has
3629 a field that points leftward on the left side, and rightward on its right side, so
3630 the two contributions of $2\pi k\sigma$ are both positive, and we get $4\pi k\sigma$.
3631 On the other hand, suppose there is a field created by other charges, not by the charged foil,
3632 that happens to point to the right. On the right side, this externally created field is in the same
3633 direction as the foil's field, but on the left side, the it \emph{reduces} the strength
3634 of the leftward field created by the foil. The increase in one term of the equation
3635 balances the decrease in the other term. This new version of the equation is thus
3636 exactly correct regardless of what externally generated fields are present!
3638 Her next innovation starts by multiplying the equation on both sides by the area,
3639 $A$, of one side of the foil:
3641 \left(E_{outward,\ on\ side\ 1}+E_{outward,\ on\ side\ 2}\right)A &= 4\pi k\sigma A \\
3643 E_{outward,\ on\ side\ 1}A+E_{outward,\ on\ side\ 2}A &= 4\pi kq \qquad , \\
3645 where $q$ is the charge of the foil. The reason for this modification is that
3646 she can now make the whole thing more attractive by
3647 defining a new vector, the area vector \vc{A}. As shown in figure
3648 \figref{avectorflat}, she defines an area vector
3649 for side 1 which has magnitude $A$ and points outward from side 1, and
3650 an area vector for side 2 which has the same magnitude and points outward from
3651 that side, which is in the opposite direction. The dot product of two vectors,
3652 $\vc{u}\cdot\vc{v}$, can be interpreted as $u_{parallel\ to\ v}|\vc{v}|$, and she
3653 can therefore rewrite her equation as
3655 \vc{E}_1\cdot\vc{A}_1+\vc{E}_2\cdot\vc{A}_2 = 4\pi k q \qquad .
3657 The quantity on the left side of this equation is called the \emph{flux} through
3658 the surface, written $\Phi$.\index{flux!defined}
3665 The area vector is defined to be perpendicular to the surface,
3666 in the outward direction. Its magnitude tells how much the area is.
3674 %q{Gauss contemplates a map of the known world.}
3678 Gauss now writes a grant proposal to her favorite funding agency, the BSGS
3679 (Blood-Suckers' Geological Survey), and it is quickly approved. Her audacious
3680 plan is to send out exploring teams to chart the electric fields of the whole
3681 planet of Flatcat, and thereby determine the total electric charge of the planet.
3682 The fleas' world is commonly assumed to be a flat disk, and its size is known
3683 to be finite, since the sun passes behind it at sunset and comes back around on the other
3684 side at dawn. The most daring part of the plan is that it requires surveying not just
3685 the known side of the planet but the uncharted Far Side as well. No flea has ever
3686 actually gone around the edge and returned to tell the tale, but Gauss assures them
3687 that they won't fall off --- their negatively charged bodies will be attracted
3688 to the disk no matter which side they are on.
3690 Of course it is possible that the electric charge of planet Flatcat is not perfectly
3691 uniform, but that isn't a problem. As discussed in subsection \ref{subsec:surfacefield},
3692 as long as one is very close to the surface, the field only depends on the \emph{local}
3693 charge density. In fact, a side-benefit of Gauss's program of exploration is that any
3694 such local irregularities will be mapped out. But what the newspapers find exciting is
3695 the idea that once all the teams get back from their voyages and tabulate their data,
3696 the \emph{total} charge of the planet will have been determined for the first time.
3697 Each surveying team is assigned to visit a certain list of republics, duchies, city-states, and
3698 so on. They are to record each territory's electric field vector, as well as its area.
3699 Because the electric field may be nonuniform, the final equation for determining the
3700 planet's electric charge will have many terms, not just one for each side of the planet:
3702 \Phi = \sum \vc{E}_j\cdot\vc{A}_j = 4\pi k q_{total}
3705 Gauss herself leads one of the expeditions, which heads due east, toward the distant
3706 Tail Kingdom, known only from fables and the occasional account from a
3707 caravan of traders. A strange thing happens, however. Gauss embarks from her
3708 college town in the wetlands of the Tongue Republic, travels straight east,
3709 passes right through the Tail Kingdom, and one day finds herself right back at
3710 home, all without ever seeing the edge of the world! What can have happened?
3711 All at once she realizes that the world isn't flat.
3718 Each part of the surface has its own area vector. Note the
3719 differences in lengths of the vectors, corresponding to the unequal areas.
3724 Now what? The surveying teams all return, the data are tabulated, and
3725 the result for the total charge of Flatcat is $(1/4\pi k)\sum \vc{E}_j\cdot\vc{A}_j=37\ \zu{nC}$
3726 (units of nanocoulombs). But the equation was derived under the assumption that
3727 Flatcat was a disk. If Flatcat is really round, then the result may be completely
3728 wrong. Gauss and two of her grad students go to their favorite bar, and decide to keep
3729 on ordering Bloody Marys until they either solve their problems or forget them. One
3730 student suggests that perhaps Flatcat really is a disk, but the edges are rounded. Maybe
3731 the surveying teams really did flip over the edge at some point, but just didn't realize
3732 it. Under this assumption, the original equation will be approximately valid, and
3733 37 nC really is the total charge of Flatcat.
3740 An area vector can be defined for a sufficiently small part of
3746 A second student, named Newton, suggests that they take seriously the possibility that
3747 Flatcat is a sphere. In this scenario, their planet's surface is really curved, but the surveying
3748 teams just didn't notice the curvature, since they were close to the surface, and the
3749 surface was so big compared to them. They divided up the surface into a patchwork, and
3750 each patch was fairly small compared to the whole planet, so each patch was nearly flat.
3751 Since the patch is nearly flat, it makes sense to define an area vector that is perpendicular
3752 to it. In general, this is how we define the direction of an area vector, as shown
3753 in figure \figref{avector}. This only works if the areas are small. For instance, there
3754 would be no way to define an area vector for an entire sphere, since ``outward'' is
3755 in more than one direction.
3757 If Flatcat is a sphere, then
3758 the inside of the sphere must be vast, and there is no way of knowing
3759 exactly how the charge is arranged below the surface. However, the survey teams all found
3760 that the electric field was approximately perpendicular to the surface everywhere, and
3761 that its strength didn't change very much from one location to another. The simplest
3762 explanation is that the charge is all concentrated in one small lump at the center of the sphere.
3763 They have no way of knowing if this is really the case, but it's a hypothesis that allows them
3764 to see how much their 37 nC result would change if they assumed a different geometry.
3765 Making this assumption, Newton performs the following simple computation on a napkin.
3766 The field at the surface is related to the charge at the center\label{napkin}
3771 |\vc{E}| &= \frac{kq_{total}}{r^2} \qquad ,
3772 \intertext{where $r$ is the radius of Flatcat. The flux is then}
3773 \Phi &= \sum \vc{E}_j\cdot\vc{A}_j \qquad , \\\\
3774 \intertext{and since the $\vc{E}_j$ and $\vc{A}_j$ vectors are parallel, the dot
3775 product equals $|\vc{E}_j||\vc{A}_j|$, so}
3776 \Phi &= \sum \frac{kq_{total}}{r^2}|\vc{A}_j| \qquad .\\\\
3777 \intertext{But the field strength is always the same, so we can take it outside the sum,
3779 \Phi &= \frac{kq_{total}}{r^2} \sum |\vc{A}_j| \\\\
3780 &= \frac{kq_{total}}{r^2} A_{total} \\\\
3781 &= \frac{kq_{total}}{r^2} 4\pi r^2 \\\\
3782 &= 4\pi kq_{total} \qquad .
3788 |\vc{E}| &= \frac{kq_{total}}{r^2} \qquad ,
3790 where $r$ is the radius of Flatcat. The flux is then
3792 \Phi &= \sum \vc{E}_j\cdot\vc{A}_j \qquad , \\
3794 and since the $\vc{E}_j$ and $\vc{A}_j$ vectors are parallel, the dot
3795 product equals $|\vc{E}_j||\vc{A}_j|$, so
3797 \Phi &= \sum \frac{kq_{total}}{r^2}|\vc{A}_j| \qquad .\\
3799 But the field strength is always the same, so we can take it outside the sum,
3802 \Phi &= \frac{kq_{total}}{r^2} \sum |\vc{A}_j| \\
3803 &= \frac{kq_{total}}{r^2} A_{total} \\
3804 &= \frac{kq_{total}}{r^2} 4\pi r^2 \\
3805 &= 4\pi kq_{total} \qquad .
3810 Not only have all the factors of $r$ canceled out, but the result is the same
3813 Everyone is pleasantly surprised by this apparent mathematical coincidence, but
3814 is it anything more than that? For instance, what if the charge wasn't concentrated
3815 at the center, but instead was evenly distributed throughout Flatcat's interior
3816 volume? Newton, however, is familiar with a result called the shell theorem
3817 (page \pageref{shelltheoremsubsection}), which states that the field of a uniformly
3818 charged sphere is the same as if all the charge had been concentrated at its
3819 center.\footnote{Newton's human namesake actually proved this for gravity, not
3820 electricity, but they're both $1/r^2$ forces, so the proof works equally well in
3821 both cases.} We now have three different assumptions about the shape of Flatcat and
3822 the arrangement of the charges inside it, and all three lead to exactly the
3823 \emph{same} mathematical result, $\Phi = 4\pi kq_{total}$. This is starting to
3824 look like more than a coincidence. In fact, there is a general mathematical theorem,
3825 called Gauss' theorem, which states the following:
3827 For any region of space, the flux through the surface equals $4\pi kq_{in}$, where
3828 $q_{in}$ is the total charge in that region.\index{flux!in Gauss' theorem}\index{Gauss' theorem}
3830 Don't memorize the factor of $4\pi$ in front --- you can rederive it any time you
3831 need to, by considering a spherical surface centered on a point charge.
3833 Note that although region and its surface had a definite physical existence in our
3834 story --- they are the planet Flatcat and the surface of planet Flatcat --- Gauss'
3835 law is true for any region and surface we choose, and in general, the Gaussian
3836 surface has no direct physical significance. It's simply a computational tool.
3838 Rather than proving Gauss' theorem and then presenting some examples and applications,
3839 it turns out to be easier to show some examples that demonstrate its salient properties.
3840 Having understood these properties, the proof becomes quite simple.
3842 <% self_check('influx',<<-'SELF_CHECK'
3843 Suppose we have a negative point charge, whose field points
3844 inward, and we pick a Gaussian surface which is a sphere centered on that charge.
3845 How does Gauss' theorem apply here?
3851 <% begin_sec("Additivity of flux") %>
3852 Figure \figref{addflux} shows two two different ways in which flux is additive. Figure
3853 \figref{addflux}/1, additivity by charge, shows that we can break down a charge distribution
3854 into two or more parts, and the flux equals the sum of the fluxes due to the individual
3855 charges. This follows directly from the fact that the flux is defined in terms of a dot
3856 product, $\vc{E}\cdot\vc{A}$, and the dot product has the additive property
3857 $(\vc{a}+\vc{b})\cdot\vc{c}=\vc{a}\cdot\vc{c}+\vc{b}\cdot\vc{c}$.\index{flux!additivity by charge}
3864 1. The flux due to two charges equals the sum of the fluxes from each one.
3865 2. When two regions are joined together, the flux through the new region equals the sum
3866 of the fluxes through the two parts.
3871 To understand additivity of flux by region, \figref{addflux}/2, we have to consider
3872 the parts of the two surfaces that were eliminated when they were joined together,
3873 like knocking out a wall to make two small apartments into one big one. Although the
3874 two regions shared this wall before it was removed, the area vectors were opposite:
3875 the direction that is outward from one region is inward with respect to the other.
3876 Thus if the field on the wall contributes positive flux to one region, it contributes
3877 an equal amount of negative flux to the other region, and we can therefore eliminate
3878 the wall to join the two regions, without changing the total flux.
3879 \index{flux!additivity by region}
3883 <% begin_sec("Zero flux from outside charges") %>\label{tinycubeproof}
3884 A third important property of Gauss' theorem is that it only refers to the charge
3885 \emph{inside} the region we choose to discuss. In other words, it asserts that any
3886 charge outside the region contributes zero to the flux. This makes at least some sense, because
3887 a charge outside the region will have field vectors pointing into the surface on one
3888 side, and out of the surface on the other. Certainly there
3889 should be at least partial cancellation between the
3890 negative (inward) flux on one side and the positive (outward) flux on the other. But why
3891 should this cancellation be exact?
3893 To see the reason for this perfect cancellation,
3894 we can imagine space as being built out of tiny cubes, and we can think of any charge
3895 distribution as being composed of point charges. The additivity-by-charge property tells us that
3896 any charge distribution can be handled by considering its point charges individually,
3897 and the additivity-by-region property tells us that if we have a single point charge
3898 outside a big region, we can break the region down into tiny cubes. If we can prove
3899 that the flux through such a tiny cube really does cancel exactly, then the same must
3900 be true for any region, which we could build out of such cubes, and any charge distribution,
3901 which we can build out of point charges.
3903 For simplicity, we will carry out this calculation only in the special case shown
3904 in figure \figref{pointflux}, where the charge lies along one axis of the cube.
3905 Let the sides of the cube have length
3906 $2b$, so that the area of each side is $(2b)^2=4b^2$. The cube extends a
3907 distance $b$ above, below, in front of, and behind
3908 the horizontal $x$ axis. There is a distance $d-b$ from the charge to the left
3909 side, and $d+b$ to the right side.
3915 %q{The flux through a tiny cube due to a point charge.}
3921 will be one negative flux, through the left side, and five
3922 positive ones. Of these positive ones, the one through the right side is very nearly
3923 the same in magnitude as the negative flux through the left side, but just a little less
3924 because the field is weaker on the right, due to the greater distance from the charge.
3925 The fluxes through the other four sides are very small, since the field is nearly
3926 perpendicular to their area vectors, and the dot product $\vc{E}_j\cdot\vc{A}_j$ is
3927 zero if the two vectors are perpendicular. In the limit where $b$ is very small, we can approximate
3928 the flux by evaluating the field at the center of each of the cube's six sides,
3929 giving\label{tinycube}
3931 \Phi &= \Phi_{left}+4\Phi_{side}+\Phi_{right} \\
3932 &= |\vc{E}_{left}||\vc{A}_{left}|\cos 180\degunit
3933 +4|\vc{E}_{side}||\vc{A}_{side}|\cos \theta_{side} \\
3934 & \quad +|\vc{E}_{right}||\vc{A}_{right}|\cos 0\degunit \qquad ,\\
3935 \intertext{and a little trig gives $\cos\theta_{side}\approx b/d$, so}
3936 \Phi &= -|\vc{E}_{left}||\vc{A}_{left}|
3937 +4|\vc{E}_{side}||\vc{A}_{side}|\frac{b}{d}
3938 +|\vc{E}_{right}||\vc{A}_{right}|\\
3939 &= \left(4b^2\right)\left(-|\vc{E}_{left}|
3940 +4|\vc{E}_{side}|\frac{b}{d}
3941 +|\vc{E}_{right}|\right)\\
3942 &= \left(4b^2\right)\left(-\frac{kq}{(d-b)^2}
3943 +4\frac{kq}{d^2}\frac{b}{d}
3944 +\frac{kq}{(d+b)^2}\right)\\
3945 &= \left(\frac{4kqb^2}{d^2}\right)\left(-\frac{1}{(1-b/d)^2}
3947 +\frac{1}{(1+b/d)^2}\right) \qquad .\\
3948 \intertext{Using the approximation $(1+\epsilon)^{-2}\approx 1-2\epsilon$ for small $\epsilon$, this
3950 \Phi &= \left(\frac{4kqb^2}{d^2}\right)\left(-1-\frac{2b}{d}
3952 +1-\frac{2b}{d}\right) \\
3955 Thus in the limit of a very small cube, $b\ll d$, we have proved that the flux due to
3956 this exterior charge is zero. The proof can be extended to the case where the
3957 charge is not along any axis of the
3958 cube,\footnote{The math gets messy for the off-axis case. This part of the proof can be
3959 completed more easily and transparently using the techniques of section
3960 \ref{sec:gaussdiff}, and that is exactly we'll do in example \ref{eg:divpointcharge}
3961 on page \pageref{eg:divpointcharge}.}
3962 and based on additivity we then have a proof that the flux due to an outside charge is
3965 \begin{eg}{No charge on the interior of a conductor}\label{eg:no-charge-on-interior}
3966 I asserted on p.~\pageref{assert-no-charge-on-interior} that for a perfect conductor in equilibrium,
3967 excess charge is found only at the surface, never in the interior. This can be proved using Gauss's theorem.
3968 Suppose that a charge $q$ existed at some point in the interior, and it was in stable equilibrium. For concreteness,
3969 let's say $q$ is positive. If its
3970 equilibrium is to be stable, then we need an electric field everywhere around it that points inward like
3971 a pincushion, so that if the charge were to be perturbed slightly, the field would bring it back to its
3972 equilibrium position. Since Newton's third law forbids objects from making forces on themselves, this field
3973 would have to be the field contributed by all the other charges, not by $q$ itself.
3974 But this is impossible, because this kind of inward-pointing pincushion pattern would have
3975 a nonzero (negative) flux through the pincushion, but Gauss's theorem says we can't have flux
3976 from outside charges.
3985 'dq-gauss-examples',
3986 %q{Discussion question \ref{dq:gauss-example1}-\ref{dq:gauss-example4}.},
3994 \begin{dq}\label{dq:gauss-example1}
3995 One question that might naturally occur to you about Gauss's law is what happens for charge that is exactly on the
3996 surface --- should it be counted toward the enclosed charge, or not? If charges can be perfect, infinitesimal points,
3997 then this could be a physically meaningful question. Suppose we approach this question by way of a limit: start with charge $q$ spread out over a sphere of finite size, and then make the size of the sphere approach zero.
3998 The figure shows a uniformly charged sphere that's exactly half-way in and half-way out of the cubical Gaussian surface.
3999 What is the flux through the cube, compared to what it would be if the charge was entirely enclosed? (There are at least
4000 three ways to find this flux: by direct integration, by Gauss's law, or by the additivity of flux by region.)
4004 The dipole is completely enclosed in the cube. What does Gauss's law say about the flux through the cube? If you imagine
4005 the dipole's field pattern, can you verify that this makes sense?
4008 \begin{dq}\label{dq:gauss-example3}
4009 The wire passes in through one side of the cube and out through the other. If the current through the wire is
4010 increasing, then the wire will act like an inductor, and there will be a voltage difference between its ends.
4011 (The inductance will be relatively small, since the wire isn't coiled up, and the $\Delta V$ will therefore
4012 also be fairly small, but still not zero.) The $\Delta V$ implies the
4013 existence of electric fields, and yet Gauss's law says the flux must be zero, since there is no charge inside
4014 the cube. Why isn't Gauss's law violated?
4017 \begin{dq}\label{dq:gauss-example4}
4018 The charge has been loitering near the edge of the cube, but is then suddenly hit with a mallet, causing it to
4019 fly off toward the left side of the cube. We haven't yet discussed in detail how disturbances in the electric and magnetic fields
4020 ripple outward through space, but it turns out that they do so at the speed of light. (In fact, that's what light
4021 is: ripples in the electric and magnetic fields.) Because the charge is closer to the left side of the cube,
4022 the change in the electric field occurs there before the information reaches the right side.
4023 This would seem certain to lead to a violation of Gauss's law. How can the ideas explored in
4024 discussion question \ref{dq:gauss-example3} show the resolution to this paradox?
4029 <% begin_sec("Proof of Gauss' theorem") %>
4030 With the computational machinery we've developed, it is now simple to prove Gauss'
4031 theorem. Based on additivity by charge, it suffices to prove the law for a point charge.
4032 We have already proved Gauss' law for a point charge in the case where the point
4033 charge is outside the region. If we can prove it for the inside case, then we're all done.
4039 %q{Completing the proof of Gauss' theorem.}
4043 If the charge is inside, we reason as follows. First, we forget about the actual Gaussian
4044 surface of interest, and instead construct a spherical one, centered on the charge.
4045 For the case of a sphere, we've already seen the proof written on a napkin by
4046 the flea named Newton (page \pageref{napkin}). Now wherever the actual surface sticks
4047 out beyond the sphere, we glue appropriately shaped pieces onto the sphere. In the
4048 example shown in figure \figref{gaussproof}, we have to add two Mickey Mouse ears.
4049 Since these added pieces do not contain the point charge, the flux through them is zero,
4050 and additivity of flux by region therefore tells us that the total flux is not changed
4051 when we make this alteration. Likewise, we need to chisel out any regions where the
4052 sphere sticks out beyond the actual surface. Again, there is no change in flux, since
4053 the region being altered doesn't contain the point charge. This proves that the flux
4054 through the Gaussian surface of interest is the same as the flux through the sphere,
4055 and since we've already proved that that flux equals $4\pi kq_{in}$, our proof of
4056 Gauss' theorem is complete.\index{Gauss' theorem!proof of}
4061 A critical part of the proof of Gauss' theorem was the proof that a tiny
4062 cube has zero flux through it due to an external charge. Discuss qualitatively
4063 why this proof would fail if Coulomb's law was a $1/r$ or $1/r^3$ law.
4068 <% begin_sec("Gauss' law as a fundamental law of physics") %>
4069 Note that the proof of Gauss' theorem depended on the computation on the napkin
4070 discussed on page \ref{napkin}. The crucial point in this computation was that
4071 the electric field of a point charge falls off like $1/r^2$, and since the
4072 area of a sphere is proportional to $r^2$, the result is independent of $r$.
4073 The $1/r^2$ variation of the field also came into play on page \pageref{tinycube}
4074 in the proof that the flux due to an outside charge is zero.
4075 In other words, if we discover some other force of nature which is proportional to
4076 $1/r^3$ or $r$, then Gauss' theorem will not apply to that force. Gauss' theorem
4077 is not true for nuclear forces, which fall off exponentially with distance. However, this
4078 is the \emph{only} assumption we had to make about the nature of the field. Since
4079 gravity, for instance, also has fields that fall off as $1/r^2$, Gauss' theorem
4080 is equally valid for gravity --- we just have to replace mass with
4081 charge, change the Coulomb constant $k$ to the gravitational constant
4082 $G$, and insert a minus sign because the gravitational fields around a (positive)
4083 mass point inward.\index{Gauss' theorem!for gravity}
4085 Gauss' theorem can only be proved if we assume a $1/r^2$ field, and the converse
4086 is also true: any field that satisfies Gauss' theorem must be a $1/r^2$ field.
4087 Thus although we previously thought of Coulomb's law as the fundamental law
4088 of nature describing electric forces, it is equally valid to think of Gauss'
4089 theorem as the basic law of nature for electricity. From this point of view,
4090 Gauss' theorem is not a mathematical fact but an experimentally testable statement
4091 about nature, so we'll refer to it as Gauss' \emph{law}, just as we speak of
4092 Coulomb's \emph{law} or Newton's \emph{law} of gravity.\index{Gauss' law}
4094 If Gauss' law is equivalent to Coulomb's law, why not just use Coulomb's law?
4095 First, there are some cases where calculating a field is easy with Gauss'
4096 law, and hard with Coulomb's law. More importantly, Gauss' law and Coulomb's
4097 law are only mathematically equivalent under the assumption that all our charges
4098 are standing still, and all our fields are constant over time, i.e., in
4099 the study of electrostatics, as opposed to electrodynamics. As we broaden our
4100 scope to study generators, inductors, transformers, and radio antennas, we will
4101 encounter cases where Gauss' law is valid, but Coulomb's law is not.
4104 <% begin_sec("Applications") %>
4105 Often we encounter situations where we have a static charge distribution, and
4106 we wish to determine the field. Although superposition is a generic
4107 strategy for solving this type of problem, if the charge distribution is
4108 symmetric in some way, then Gauss' law is often a far easier way to carry
4109 out the computation.
4111 <% begin_sec("Field of a long line of charge") %>
4112 Consider the field of an infinitely long line of charge, holding a uniform
4113 charge per unit length $\lambda$. Computing this
4114 field by brute-force superposition was fairly laborious (examples
4115 \ref{eg:chargedrodside} on page \pageref{eg:chargedrodside}
4116 and \ref{eg:pointlinesurface} on page \pageref{eg:pointlinesurface}).
4117 With Gauss' law it becomes a very simple calculation.
4123 %q{Applying Gauss' law to an infinite line of charge.}
4127 The problem has two types of symmetry. The line of charge, and therefore the resulting
4128 field pattern, look the same if we rotate them about the line. The second symmetry
4129 occurs because the line is infinite: if we slide the line along its own length,
4130 nothing changes. This sliding symmetry, known as a translation symmetry,
4131 tells us that the field must point directly away from the line at any given point.
4133 Based on these symmetries, we choose the Gaussian surface shown in
4134 figure \figref{gaussline}. If we want to know the field at a distance $R$ from
4135 the line, then we choose this surface to have a radius $R$, as shown in the
4136 figure. The length, $L$, of the surface is irrelevant.
4138 The field is parallel to the surface on the end caps,
4139 and therefore perpendicular to the end caps' area vectors, so there is no
4140 contribution to the flux. On the long, thin strips that make up the rest of
4141 the surface, the field is perpendicular to the surface, and therefore parallel
4142 to the area vector of each strip, so that the dot product occurring in the
4143 definition of the flux is $\vc{E}_j\cdot\vc{A}_j=|\vc{E}_j||\vc{A}_j||\cos\ 0\degunit=|\vc{E}_j||\vc{A}_j|$.
4146 4\pi k q_{in} &= \sum \vc{E}_j\cdot\vc{A}_j \\
4147 4\pi k \lambda L &= \sum |\vc{E}_j||\vc{A}_j| \qquad .\\
4148 \intertext{The magnitude of the field is the same on every strip, so we can take it outside the sum.}
4149 4\pi k \lambda L &= |\vc{E}| \sum |\vc{A}_j| \\
4150 \intertext{In the limit where the strips are infinitely narrow, the surface becomes a
4151 cylinder, with (area)=(circumference)(length)=$2\pi RL$.}
4152 4\pi k \lambda L &= |\vc{E}| \times 2\pi RL \\
4153 |\vc{E}| &= \frac{2k\lambda}{R} \\
4157 <% begin_sec("Field near a surface charge") %>
4158 As claimed earlier, the result $E=2\pi k\sigma$ for the field near a charged
4159 surface is a special case of Gauss' law. We choose a Gaussian surface of the shape
4160 shown in figure \figref{pillbox}, known as a Gaussian pillbox.\index{Gaussian pillbox}\index{pillbox!Gaussian}
4161 The exact shape of the flat end caps is unimportant.
4167 %q{Applying Gauss' law to an infinite charged surface.}
4171 The symmetry of the charge distribution tells us that the field points directly away
4172 from the surface, and is equally strong on both sides of the surface.
4173 This means that the end caps contribute equally to the flux, and the curved sides have zero
4174 flux through them. If the area of each end cap is $A$, then
4176 4\pi k q_{in} &= \vc{E}_1\cdot\vc{A}_1+\vc{E}_2\cdot\vc{A}_2 \qquad , \\
4177 \intertext{where the subscripts 1 and 2 refer to the two end caps. We have $\vc{A}_2=-\vc{A}_1$, so}
4178 4\pi k q_{in} &= \vc{E}_1\cdot\vc{A}_1-\vc{E}_2\cdot\vc{A}_1 \\
4179 4\pi k q_{in} &= \left(\vc{E}_1-\vc{E}_2\right)\cdot\vc{A}_1 \qquad , \\
4180 \intertext{and by symmetry the magnitudes of the two fields are equal, so}
4181 2|\vc{E}|A &= 4 \pi k \sigma A\\
4182 |\vc{E}| &= 2\pi k\sigma
4185 The symmetry between the two sides could be broken by the existence of other charges
4186 nearby, whose fields would add onto the field of the surface itself.
4187 Even then, Gauss's law still guarantees
4189 4\pi k q_{in} &= \left(\vc{E}_1-\vc{E}_2\right)\cdot\vc{A}_1 \qquad ,
4191 |\vc{E}_{\perp,1}-\vc{E}_{\perp,2}| &= 4\pi k \sigma \qquad ,
4193 where the subscript $\perp$ indicates the component of the field parallel
4194 to the surface (i.e., parallel to the area vectors). In other words,
4195 the electric field changes discontinuously when we pass through a charged
4196 surface; the discontinuity occurs in the component of the field perpendicular
4197 to the surface, and the amount of discontinuous change is
4198 $4\pi k \sigma$. This is a completely general statement that is true near
4199 any charged surface, regardless of the existence of other charges nearby.\label{e-discontinuity}
4206 <% begin_sec("Gauss' Law in Differential Form",4,'gaussdiff') %>
4207 Gauss' law is a bit spooky. It relates the field on the Gaussian surface
4208 to the charges inside the surface. What if the charges have been moving
4209 around, and the field at the surface right now is the one that was created
4210 by the charges in their previous locations? Gauss' law --- unlike
4211 Coulomb's law --- still works in cases like these, but it's far from
4212 obvious how the flux and the charges can still stay in agreement if
4213 the charges have been moving around.
4215 For this reason, it would be more physically attractive to restate Gauss'
4216 law in a different form, so that it related the behavior of the field
4217 at one point to the charges that were actually present at that point.
4218 This is essentially what we were doing in the fable of the flea named
4219 Gauss: the fleas' plan for surveying their planet was essentially one of
4220 dividing up the surface of their planet (which they believed was flat)
4221 into a patchwork, and then constructing \emph{small} a Gaussian pillbox
4222 around each \emph{small} patch. The equation $E_{\perp}=2\pi k\sigma$
4223 then related a particular property of the \emph{local} electric field to
4224 the \emph{local} charge density.
4230 %q{A tiny cubical Gaussian surface.}
4234 In general, charge distributions need not be confined to a flat surface ---
4235 life is three-dimensional --- but the general approach of defining
4236 very small Gaussian surfaces is still a good one. Our strategy is
4237 to divide up space into tiny cubes, like the one on page
4238 \pageref{tinycubeproof}. Each such cube constitutes a Gaussian surface,
4239 which may contain some charge. Again we approximate the field using
4240 its six values at the center of each of the six sides. Let the
4241 cube extend from $x$ to $x+\der x$, from $y$ to $y+\der y$, and from $y$ to $y+\der y$.
4243 The sides at $x$ and $x+\der x$ have area vectors $-\der y\der z\hat{\vc{x}}$
4244 and $\der y\der z\hat{\vc{x}}$, respectively.
4245 The flux through the side at $x$ is $-E_x(x)\der y\der z$, and the flux
4246 through the opposite side, at $x+\der x$ is $E_x(x+\der x)\der y\der z$.
4247 The sum of these is $(E_x(x+\der x)-E_x(x))\der y\der z$, and if the field was
4248 uniform, the flux through these two opposite sides would be zero. It will only
4250 field's $x$ component changes as a function of $x$.
4251 The difference $E_x(x+\der x)-E_x(x)$ can be rewritten as
4252 $\der E_x=(\der E_x)/(\der x)\der x$, so the contribution to the flux
4253 from these two sides of the cube ends up being
4255 \frac{\der E_x}{\der x}\der x\der y\der z \qquad .
4257 Doing the same for the other sides, we end up with a total flux
4259 \der \Phi &= \left(\frac{\der E_x}{\der x}+\frac{\der E_y}{\der y}
4260 +\frac{\der E_z}{\der z}\right)\der x\der y\der z \\
4261 &= \left(\frac{\der E_x}{\der x}+\frac{\der E_y}{\der y}
4262 +\frac{\der E_z}{\der z}\right)\der v \qquad ,\\
4263 \intertext{where $\der v$ is the volume of the cube. In evaluating each of these
4264 three derivatives, we are going to treat the other two variables as constants,
4265 to emphasize this we use the partial derivative notation $\partial$ introduced
4266 in chapter \ref{ch:3},}
4267 \der \Phi &= \left(\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}
4268 +\frac{\partial E_z}{\partial z}\right)\der v \qquad .\\
4269 \intertext{Using Gauss' law,}
4270 4\pi k q_{in} &= \left(\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}
4271 +\frac{\partial E_z}{\partial z}\right)\der v \qquad ,\\
4272 \intertext{and we introduce the notation $\rho$ (Greek letter rho) for the charge
4273 per unit volume, giving}
4274 4\pi k \rho &= \frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}
4275 +\frac{\partial E_z}{\partial z} \qquad .\\
4276 \intertext{The quantity on the right is called the \emph{divergence} of the electric
4277 field, written $\divg \vc{E}$. Using this notation, we have}
4278 \divg \vc{E} = 4\pi k \rho \qquad .
4280 This equation has all the same physical implications as Gauss' law. After all, we
4281 proved Gauss' law by breaking down space into little cubes like this. We therefore
4282 refer to it as the differential form of Gauss' law, as opposed to
4283 $\Phi=4\pi kq_{in}$, which is called the integral form.\index{Gauss' law!differential form}
4284 \index{partial derivative}\index{divergence}
4290 %q{A meter for measuring $\divg\vc{E}$.}
4294 Figure \figref{divmeter} shows an intuitive way of visualizing the meaning
4295 of the divergence. The meter consists of some electrically charged balls
4296 connected by springs. If the divergence is positive, then the whole cluster will
4297 expand, and it will contract its volume if it is placed at a point where the
4298 field has $\divg\vc{E}<0$. What if the field is constant? We know based on the
4299 definition of the divergence that we should have $\divg\vc{E}=0$ in this case,
4300 and the meter does give the right result: all the balls
4301 will feel a force in the same direction, but they will neither expand nor contract.
4307 %q{Example \ref{eg:divsine}.}
4311 \begin{eg}{Divergence of a sine wave}\label{eg:divsine}
4313 Figure \figref{sinewavevector} shows an electric field that varies as a sine
4314 wave. This is in fact what you'd see in a light wave: light is a wave pattern
4315 made of electric and magnetic fields. (The magnetic field would look similar,
4316 but would be in a plane perpendicular to the page.) What is the divergence
4317 of such a field, and what is the physical significance of the result?
4320 Intuitively, we can see that no matter where we put the div-meter in this
4321 field, it will neither expand nor contract. For instance, if we put it at
4322 the center of the figure, it will start spinning, but that's it.
4324 Mathematically, let the $x$ axis be to the right and let $y$ be up.
4325 The field is of the form
4327 \vc{E} = (\zu{sin} Kx)\: \hat{\vc{y}} \qquad ,
4329 where the constant $K$ is not to be confused with Coulomb's constant.
4330 Since the field has only a $y$ component, the only term in the divergence
4331 we need to evaluate is
4333 \vc{E} = \frac{\partial E_{y}}{\partial y} \qquad ,
4335 but this vanishes, because $E_y$ depends only on $x$, not $y$\/:
4336 we treat $y$ as a constant when evaluating the partial derivative
4337 $\partial E_{y}/\partial y$, and the derivative of an expression
4338 containing only constants must be zero.
4340 Physically this is a very important result: it tells us that a light wave
4341 can exist without any charges along the way to ``keep it going.'' In other
4342 words, light can travel through a vacuum, a region with no particles in it.
4343 If this wasn't true, we'd be dead, because the sun's light wouldn't be able
4344 to get to us through millions of kilometers of empty space!
4347 \begin{eg}{Electric field of a point charge}\label{eg:divpointcharge}
4348 The case of a point charge is tricky, because the field
4349 behaves badly right on top of the charge, blowing up and
4350 becoming discontinuous. At this point, we cannot use the
4351 component form of the divergence, since none of the
4352 derivatives are well defined. However, a little visualization
4353 using the original definition of the divergence will quickly
4354 convince us that div $E$ is infinite here, and that makes
4355 sense, because the density of charge has to be infinite at a
4356 point where there is a zero-size point of charge (finite
4357 charge in zero volume).
4359 At all other points, we have
4361 \vc{E} = \frac{ kq}{ r^2}\hat{\vc{r}} \qquad ,
4363 where $\hat{\vc{r}}=\vc{r}/ r=( x\hat{\vc{x}}+ y\hat{\vc{y}}+ z\hat{\vc{z}})/ r$
4364 is the unit vector pointing radially away from
4365 the charge. The field can therefore be written as
4367 \vc{E} &= \frac{ kq}{ r^3}\hat{\vc{r}} \\
4368 &= \frac{ kq( x\hat{\vc{x}}+ y\hat{\vc{y}}+ z\hat{\vc{z}})}%
4369 {\left( x^2+ y^2+ z^2\right)^\zu{3/2}} \qquad . \\
4370 \intertext{The three terms in the divergence are all similar, e.g.,}
4371 \frac{\partial E_{x}}{\partial x}
4372 &= kq\frac{\partial}{\partial x}%
4373 \left[\frac{ x}{\left( x^2+ y^2+ z^2\right)^\zu{3/2}}\right] \\
4374 &= kq\left[\frac{1}{\left( x^2+ y^2+ z^2\right)^\zu{3/2}}%
4375 -\frac{3}{2}\:\frac{2 x^2}{\left( x^2+ y^2+ z^2\right)^\zu{5/2}}\right] \\
4376 &= kq\left( r^{-3}-3 x^2 r^{-5}\right) \qquad .
4378 Straightforward algebra shows that adding in the other two
4379 terms results in zero, which makes sense, because there is no
4380 charge except at the origin.
4383 Gauss' law in differential form lends itself most easily to
4384 finding the charge density when we are give the field. What if we
4385 want to find the field given the charge density? As demonstrated
4386 in the following example, one technique that
4387 often works is to guess the general form of the field
4388 based on experience or physical intuition, and then
4389 try to use Gauss' law to find what specific version
4390 of that general form will be a solution.
4392 \begin{eg}{The field inside a uniform sphere of charge}\label{eg:divsphere}
4394 Find the field inside a uniform sphere of charge
4395 whose charge density is $\rho$. (This is very much like
4396 finding the gravitational field at some depth below the
4397 surface of the earth.)
4400 By symmetry we know that the field must be purely
4401 radial (in and out). We guess that the solution might be of the form
4403 \vc{E} = br^ p\hat{\vc{r}} \qquad ,
4405 where $r$ is the distance from the center, and $b$ and $p$
4406 are constants. A negative value of $p$ would indicate a
4407 field that was strongest at the center, while a positive $p$
4408 would give zero field at the center and stronger fields farther out.
4409 Physically, we know by symmetry that the field is zero at the center,
4410 so we expect $p$ to be positive.
4412 As in the example \ref{eg:divpointcharge},
4413 we rewrite $\hat{\vc{r}}$ as $\vc{r}/ r$,
4414 and to simplify the writing we
4417 \vc{E} = br^ n\vc{r} \qquad .
4419 Gauss' law in differential form is
4421 \divg\vc{E} = 4\pi k\rho \qquad ,
4423 so we want a field whose divergence is constant. For a field
4424 of the form we guessed, the divergence has terms in it like
4426 \frac{\partial E_{x}}{\partial x}
4427 &= \frac{\partial}{\partial x}\left( br^{n} x\right) \\
4428 &= b\left( nr^{ n-1}\frac{\partial r}{\partial x} x+%
4431 The partial derivative $\partial r/\partial x$ is easily calculated to be
4434 \frac{\partial E_{x}}{\partial x}
4435 = b\left( nr^{ n-2} x^2+%
4438 Adding in similar expressions for the other two terms in the
4439 divergence, and making use of $x^2+ y^2+ z^2= r^2$,
4442 \divg\vc{E} = b( n+3) r^ n \qquad .
4444 This can indeed be constant, but only if $n$ is 0 or $-3$,
4445 i.e., $p$ is 1 or $-2$.
4446 The second solution gives a divergence which is constant and \emph{zero}\/:
4447 this is the solution for the \emph{outside} of the sphere! The first
4448 solution, which has the field directly proportional to $r$,
4449 must be the one that applies to the inside of the sphere,
4450 which is what we care about right now.
4452 coefficient in front to the one in Gauss' law, the field is
4454 \vc{E} = \frac{4\pi k\rho}{3} r\:\hat{\vc{r}}\qquad .
4456 The field is zero at the center, and gets stronger and stronger as we approach
4465 'divmeterinsinewave',
4466 %q{Discussion question \ref{eg:divmeterinsinewave}.}
4471 \begin{dq}\label{eg:divmeterinsinewave}
4472 As suggested by the figure, discuss the results you would get by inserting the div-meter
4473 at various locations in the sine-wave field.
4476 \backofchapterboilerplate{efield}
4482 <% begin_hw('sparkplug',0) %>__incl(hw/sparkplug)<% end_hw() %>
4484 <% begin_hw('distantrsquared') %>__incl(hw/distantrsquared)<% end_hw() %>
4486 <% begin_hw('galaxy-mass') %>__incl(hw/galaxy-mass)<% end_hw() %>
4488 <% begin_hw('chargemotioninfield',0) %>__incl(hw/chargemotioninfield)<% end_hw() %>
4490 <% begin_hw('altedefunits',0) %>__incl(hw/altedefunits)<% end_hw() %>
4492 <% begin_hw('dipole-change-origin') %>__incl(hw/dipole-change-origin)<% end_hw() %>
4494 <% begin_hw('dipolechoiceoforigin') %>__incl(hw/dipolechoiceoforigin)<% end_hw() %>
4496 <% begin_hw('lineandsquaredipoles') %>__incl(hw/lineandsquaredipoles)<% end_hw() %>
4498 <% begin_hw('quadrupole') %>__incl(hw/quadrupole)<% end_hw() %>
4500 <% begin_hw('ringve',0) %>__incl(hw/ringve)<% end_hw() %>
4502 <% begin_hw('esquare') %>__incl(hw/esquare)<% end_hw() %>
4506 'lineandsquaredipoles',
4507 %q{Problem \ref{hw:lineandsquaredipoles}.}
4514 %q{Problem \ref{hw:esquare}.}
4521 %q{Problem \ref{hw:neuronfield}.}
4526 <% begin_hw('dipolefarfield',0) %>__incl(hw/dipolefarfield)<% end_hw() %>
4528 <% begin_hw('neuronfield') %>__incl(hw/neuronfield)<% end_hw() %>
4531 <% begin_hw('proton-in-nonuniform-field') %>__incl(hw/proton-in-nonuniform-field)<% end_hw() %>
4533 <% begin_hw('dipolev') %>__incl(hw/dipolev)<% end_hw() %>
4535 <% begin_hw('screened') %>__incl(hw/screened)<% end_hw() %>
4537 <% begin_hw('carbondioxide',0) %>
4538 A carbon dioxide molecule is structured like O-C-O, with
4539 all three atoms along a line. The oxygen atoms grab a little
4540 bit of extra negative charge, leaving the carbon positive.
4541 The molecule's symmetry, however, means that it has no
4542 overall dipole moment, unlike a V-shaped water molecule, for
4543 instance. Whereas the voltage of a dipole of magnitude
4544 $D$ is proportional to $D/r^2$ (see problem \ref{hw:dipolev}), it
4545 turns out that the voltage of a carbon dioxide molecule
4546 at a distant point along the molecule's
4547 axis equals $b/r^3$, where $r$ is the distance
4548 from the molecule and $b$ is a constant (cf. problem \ref{hw:quadrupole}). What would be the
4549 electric field of a carbon dioxide molecule at a point
4550 on the molecule's axis, at a distance $r$ from the molecule?\answercheck
4553 <% begin_hw('electron-cloud') %>__incl(hw/electron-cloud)<% end_hw() %>
4558 'hw-dipolemidplane',
4559 %q{Problem \ref{hw:dipolemidplane}.}
4563 <% begin_hw('dipolemidplane',2) %>__incl(hw/dipolemidplane)<% end_hw() %>
4569 %q{Problem \ref{hw:hyperbolic}.}
4573 <% begin_hw('hyperbolic',0) %>__incl(hw/hyperbolic)<% end_hw() %>
4575 <% begin_hw('quadraticvoltage') %>__incl(hw/quadraticvoltage)<% end_hw() %>
4577 <% begin_hw('estrips') %>__incl(hw/estrips)<% end_hw() %>
4582 'halfinfinitecylinder',
4583 %q{Problem \ref{hw:halfinfinitecylinder}.}
4587 <% begin_hw('halfinfinitecylinder') %>__incl(hw/halfinfinitecylinder)<% end_hw() %>
4589 <% begin_hw('lightning') %>
4590 In an electrical storm, the cloud and the ground act like
4591 a parallel-plate capacitor, which typically charges up due
4592 to frictional electricity in collisions of ice particles in
4593 the cold upper atmosphere. Lightning occurs when the
4594 magnitude of the electric field builds up to a critical
4595 value, $E_c$, at which air is ionized.\hwendpart
4596 (a) Treat the cloud as a flat square with sides of length
4597 $L$. If it is at a height $h$ above the ground, find the
4598 amount of energy released in the lightning strike.\answercheck\hwendpart
4599 (b) Based on your answer from part a, which is more
4600 dangerous, a lightning strike from a high-altitude cloud
4601 or a low-altitude one?\hwendpart
4602 (c) Make an order-of-magnitude estimate of the energy
4603 released by a typical lightning bolt, assuming reasonable
4604 values for its size and altitude. $E_c$ is about $10^6$ V/m.
4607 <% begin_hw('epointinfty') %>__incl(hw/epointinfty)<% end_hw() %>
4613 %q{Problem \ref{hw:neuronenergy}.},
4621 %q{Problem \ref{hw:cubecaps}.}
4625 <% begin_hw('neuronenergy') %>
4626 The neuron in the figure has been drawn fairly short, but
4627 some neurons in your spinal cord have tails (axons) up to a
4628 meter long. The inner and outer surfaces of the membrane act
4629 as the ``plates'' of a capacitor. (The fact that it has been
4630 rolled up into a cylinder has very little effect.) In order
4631 to function, the neuron must create a voltage difference $V$
4632 between the inner and outer surfaces of the membrane. Let
4633 the membrane's thickness, radius, and length be $t$, $r$, and $L$.
4634 (a) Calculate the energy that must be stored in the electric
4635 field for the neuron to do its job. (In real life, the
4636 membrane is made out of a substance called a dielectric,
4637 whose electrical properties increase the amount of energy
4638 that must be stored. For the sake of this analysis, ignore
4640 \hwhint{hwhint:neuronenergy}\answercheck\hwendpart
4641 (b) An organism's evolutionary fitness should be better if
4642 it needs less energy to operate its nervous system. Based on
4643 your answer to part a, what would you expect evolution to
4644 do to the dimensions $t$ and $r?$ What other constraints
4645 would keep these evolutionary trends from going too far?
4649 <% begin_hw('cubecaps') %>__incl(hw/cubecaps)<% end_hw() %>
4651 <% begin_hw('earthcap',0) %>__incl(hw/earthcap)<% end_hw() %>
4656 'hw-infinite-strip',
4657 %q{Problem \ref{hw:infinite-strip}.}
4661 <% begin_hw('infinite-strip') %>__incl(hw/infinite-strip)<% end_hw() %>
4663 <% begin_hw('charged-solid-cyl',1) %>__incl(hw/charged-solid-cyl)<% end_hw() %>
4665 <% begin_hw('vedgedisk',2) %>__incl(hw/vedgedisk)<% end_hw() %>
4667 <% begin_hw('capenergy',0) %>__incl(hw/capenergy)<% end_hw() %>
4669 <% begin_hw('seriescapacitors') %>__incl(hw/seriescapacitors)<% end_hw() %>
4671 <% begin_hw('complex-trig') %>__incl(hw/complex-trig)<% end_hw() %>
4673 <% begin_hw('lrcunits',0) %>
4674 (a) Show that the equation $V_L=L\der I/\der t$ has the right units.\hwendpart
4675 (b) Verify that $RC$ has units of time.\hwendpart
4676 (c) Verify that $L/R$ has units of time.
4680 <% begin_hw('parallelinductors') %>__incl(hw/parallelinductors)<% end_hw() %>
4682 <% begin_hw('i-to-the-i') %>__incl(hw/i-to-the-i)<% end_hw() %>
4684 <% begin_hw('strayimpedance',0) %>__incl(hw/strayimpedance)<% end_hw() %>
4686 <% begin_hw('lomega',0) %>
4687 Starting from the relation $V=L\der I/\der t$ for the
4688 voltage difference across an inductor, show that an inductor
4689 has an impedance equal to $L\omega$.
4692 <% begin_hw('charged-long-box',1) %>__incl(hw/charged-long-box)<% end_hw() %>
4694 <% begin_hw('charge-gun',1) %>__incl(hw/charge-gun)<% end_hw() %>
4696 <% begin_hw('fmradiolrc',0) %>__incl(hw/fmradiolrc)<% end_hw() %>
4698 <% begin_hw('rc-par-impedance') %>__incl(hw/rc-par-impedance)<% end_hw() %>
4700 <% begin_hw('lrc-z-plot') %>__incl(hw/lrc-z-plot)<% end_hw() %>
4702 <% begin_hw('series-to-parallel') %>__incl(hw/series-to-parallel)<% end_hw() %>
4704 <% begin_hw('coaxcap') %>__incl(hw/coaxcap)<% end_hw() %>
4706 <% begin_hw('gauss-const-e',1) %>__incl(hw/gauss-const-e)<% end_hw() %>
4708 <% begin_hw('lc-e-sharing',1) %>__incl(hw/lc-e-sharing)<% end_hw() %>
4710 <% begin_hw('solidchargedcylinder') %>__incl(hw/solidchargedcylinder)<% end_hw() %>
4712 <% begin_hw('gauss-simple-e',1) %>__incl(hw/gauss-simple-e)<% end_hw() %>
4714 <% begin_hw('gauss-em-wave',1) %>__incl(hw/gauss-em-wave)<% end_hw() %>
4716 <% begin_hw('charged-cylinder-div') %>__incl(hw/charged-cylinder-div)<% end_hw() %>
4718 <% begin_hw('divtrans') %>__incl(hw/divtrans)<% end_hw() %>
4720 <% begin_hw('div-rot-invariant',2) %>__incl(hw/div-rot-invariant)<% end_hw() %>
4722 <% begin_hw('cylindrical-charge-given-field') %>__incl(hw/cylindrical-charge-given-field)<% end_hw() %>
4723 % answer: c/2ku (units check)
4725 <% begin_hw('addition-theorem-for-sine') %>
4726 Use Euler's theorem to derive the addition theorems that express
4727 $\sin(a+b)$ and $\cos(a+b)$ in terms of the sines and cosines of
4732 <% begin_hw('cube-roots-of-unity') %>
4733 Find every complex number $z$ such that $z^3=1$.
4737 <% begin_hw('factor-cubic') %>
4738 Factor the expression $x^3-y^3$ into factors of the lowest possible
4739 order, using complex coefficients. (Hint: use the result of problem \ref{hw:cube-roots-of-unity}.) Then do the same using real
4750 % ====================================================================
4751 % ====================================================================
4752 % ====================================================================
4754 \extitle{A}{Field Vectors}
4756 \noindent Apparatus:
4757 \begin{indentedblock}
4770 At this point you've studied the gravitational field, $\vc{g}$,
4771 and the electric field, $\vc{E}$, but not the magnetic field,
4772 $\vc{B}$. However, they all have some of the same mathematical
4773 behavior: they act like vectors. Furthermore, magnetic
4774 fields are the easiest to manipulate in the lab. Manipulating
4775 gravitational fields directly would require futuristic
4776 technology capable of moving planet-sized masses around!
4777 Playing with electric fields is not as ridiculously
4778 difficult, but static electric charges tend to leak off
4779 through your body to ground, and static electricity effects
4780 are hard to measure numerically. Magnetic fields, on the
4781 other hand, are easy to make and control. Any moving charge,
4782 i.e., any current, makes a magnetic field.
4784 A practical device for making a strong magnetic field is
4785 simply a coil of wire, formally known as a solenoid. The
4786 field pattern surrounding the solenoid gets stronger or
4787 weaker in proportion to the amount of current passing through the wire.
4789 1. With a single solenoid connected to the power supply and
4790 laid with its axis horizontal, use a magnetic compass to
4791 explore the field pattern inside and outside it. The compass
4792 shows you the field vector's direction, but not its
4793 magnitude, at any point you choose. Note that the field the
4794 compass experiences is a combination (vector sum) of the
4795 solenoid's field and the earth's field.
4797 2. What happens when you bring the compass extremely far
4798 away from the solenoid?
4802 What does this tell you about the way the solenoid's field
4803 varies with distance?
4807 Thus although the compass doesn't tell you the field
4808 vector's magnitude numerically, you can get at least some
4809 general feel for how it depends on distance.
4813 3. The figure below is a cross-section of the solenoid
4814 in the plane containing its axis. Make a sea-of-arrows sketch of the magnetic field
4815 in this plane. The length
4816 of each arrow should at least approximately reflect the
4817 strength of the magnetic field at that point.
4823 \hspace{50mm}\anonymousinlinefig{map-solenoid-field}
4827 Does the field seem to have sources or sinks?
4829 4. What do you think would happen to your sketch if
4830 you reversed the wires?
4839 5. Now hook up the two solenoids in parallel. You are going
4840 to measure what happens when their two fields combine
4841 at a certain point in space. As you've seen already, the solenoids'
4842 nearby fields are much stronger than the earth's field; so
4843 although we now theoretically have three fields involved
4844 (the earth's plus the two solenoids'), it will be safe to
4845 ignore the earth's field. The basic idea here is to place
4846 the solenoids with their axes at some angle to each other,
4847 and put the compass at the intersection of their axes, so
4848 that it is the same distance from each solenoid. Since the
4849 geometry doesn't favor either solenoid, the only factor that
4850 would make one solenoid influence the compass more than the
4851 other is current. You can use the cut-off plastic cup as a
4852 little platform to bring the compass up to the same level as
4853 the solenoids' axes.
4855 a)What do you think will happen with the solenoids' axes at
4856 90 degrees to each other, and equal currents? Try it. Now
4857 represent the vector addition of the two magnetic fields
4858 with a diagram. Check your diagram with your instructor to
4859 make sure you're on the right track.
4861 \hspace{30mm}\anonymousinlinefig{crossed-fields}
4864 b) Now try to make a similar diagram of what would happen if
4865 you switched the wires on one of the solenoids.
4869 After predicting what the compass will do, try it and
4870 see if you were right.
4872 c)Now suppose you were to go back to the arrangement you had
4873 in part a, but you changed one of the currents to half its
4874 former value. Make a vector addition diagram, and use trig
4875 to predict the angle.
4877 \anonymousinlinefig{crossed-fields-half}
4879 Try it. To cut the current to one of the solenoids in half,
4880 an easy and accurate method is simply to put the third solenoid in
4881 series with it, and put that third solenoid so far away that its
4882 magnetic field doesn't have any significant effect on the compass.