2 require "../eruby_util.rb"
15 <% begin_sec("The magnetic field",nil,'b-field') %>\index{magnetic field}\index{field!magnetic}
17 <% begin_sec("No magnetic monopoles") %>\index{monopoles!magnetic}
24 Breaking a bar magnet in half doesn't create two monopoles, it creates two
32 'break-bar-magnet-atoms',
33 %q{An explanation at the atomic level.}
37 If you could play with a handful of electric dipoles and a
38 handful of bar magnets, they would appear very similar. For
39 instance, a pair of bar magnets wants to align themselves
40 head-to-tail, and a pair of electric dipoles does the same
41 thing. (It is unfortunately not that easy to make a
42 permanent electric dipole that can be handled like this,
43 since the charge tends to leak.)
45 You would eventually notice an important difference between
46 the two types of objects, however. The electric dipoles can
47 be broken apart to form isolated positive charges and
48 negative charges. The two-ended device can be broken into
49 parts that are not two-ended. But if you break a bar magnet
50 in half, \figref{break-bar-magnet}, you will find that you have simply made two
51 smaller two-ended objects.
53 The reason for this behavior is not hard to divine from our
54 microscopic picture of permanent iron magnets. An electric
55 dipole has extra positive ``stuff'' concentrated in one end
56 and extra negative in the other. The bar magnet, on the
57 other hand, gets its magnetic properties not from an
58 imbalance of magnetic ``stuff'' at the two ends but from the
59 orientation of the rotation of its electrons. One end is the
60 one from which we could look down the axis and see the
61 electrons rotating clockwise, and the other is the one from
62 which they would appear to go counterclockwise. There is no
63 difference between the ``stuff'' in one end of the
64 magnet and the other, \figref{break-bar-magnet-atoms}.
66 Nobody has ever succeeded in isolating a single magnetic
67 pole. In technical language, we say that magnetic \index{monopoles!magnetic}monopoles
68 do not seem to exist. Electric monopoles \emph{do} exist
69 --- that's what charges are.
71 Electric and magnetic forces seem similar in many ways. Both
72 act at a distance, both can be either attractive or
73 repulsive, and both are intimately related to the property
74 of matter called charge. (Recall that magnetism is an
75 interaction between moving charges.) Physicists's aesthetic
76 senses have been offended for a long time because this
77 seeming symmetry is broken by the existence of electric
78 monopoles and the absence of magnetic ones. Perhaps some
79 exotic form of matter exists, composed of particles that are
80 magnetic monopoles. If such particles could be found in
81 cosmic rays or moon rocks, it would be evidence that the
82 apparent asymmetry was only an asymmetry in the composition
83 of the universe, not in the laws of physics. For these
84 admittedly subjective reasons, there have been several
85 searches for magnetic monopoles. Experiments have been
86 performed, with negative results, to look for magnetic
87 monopoles embedded in ordinary matter. Soviet physicists in
88 the 1960's made exciting claims that they had created and
89 detected magnetic monopoles in particle accelerators, but
90 there was no success in attempts to reproduce the results
91 there or at other accelerators. The most recent search for
92 magnetic monopoles, done by reanalyzing data from the search
93 for the top quark at Fermilab, turned up no candidates,
94 which shows that either monopoles don't exist in nature or
95 they are extremely massive and thus hard to create in accelerators.
98 <% begin_sec("Definition of the magnetic field") %>
104 %q{The unit of magnetic field, the tesla, is named after Serbian-American inventor Nikola Tesla.},
111 'current-loop-dipole',
113 A standard dipole made from a square loop of wire shorting
114 across a battery. It acts very much like a bar magnet, but its strength is more
123 'current-loop-aligns',
124 %q{A dipole tends to align itself to the surrounding magnetic field.},
130 Since magnetic monopoles don't seem to exist, it would not
131 make much sense to define a magnetic field in terms of the
132 force on a test monopole. Instead, we follow the philosophy
133 of the alternative definition of the electric field, and
134 define the field in terms of the torque on a magnetic test
135 dipole. This is exactly what a magnetic compass does: the
136 needle is a little iron magnet which acts like a magnetic
137 dipole and shows us the direction of the earth's magnetic field.
139 To define the strength of a magnetic field, however, we need
140 some way of defining the strength of a test dipole, i.e., we
141 need a definition of the magnetic dipole moment. We could
142 use an iron permanent magnet constructed according to
143 certain specifications, but such an object is really an
144 extremely complex system consisting of many iron atoms, only
145 some of which are aligned. A more fundamental standard
146 dipole is a square current loop. This could be little
147 resistive circuit consisting of a square of wire shorting across a battery.
149 We will find that such a loop, when placed in a magnetic
150 field, experiences a torque that tends to align plane so
151 that its face points in a certain direction. (Since the loop
152 is symmetric, it doesn't care if we rotate it like a wheel
153 without changing the plane in which it lies.) It is this
154 preferred facing direction that we will end up defining as
155 the direction of the magnetic field.
157 Experiments show if the loop is out of alignment with the
158 field, the torque on it is proportional to the amount of
159 current, and also to the interior area of the loop. The
160 proportionality to current makes sense, since magnetic
161 forces are interactions between moving charges, and current
162 is a measure of the motion of charge. The proportionality to
163 the loop's area is also not hard to understand, because
164 increasing the length of the sides of the square increases
165 both the amount of charge contained in this circular
166 ``river'' and the amount of leverage supplied for making
167 torque. Two separate physical reasons for a proportionality
168 to length result in an overall proportionality to length
169 squared, which is the same as the area of the loop. For
170 these reasons, we define the magnetic dipole moment of a
171 square current loop as
173 D_m = IA \qquad , \hfill\shoveright{\text{[definition of the magnetic}}\\
174 \text{ dipole moment of a square current loop]}
176 We now define the \index{magnetic field!defined}magnetic
177 field in a manner entirely analogous to the second
178 definition of the electric field:
180 \begin{lessimportant}[definition of the magnetic field]
181 The magnetic field vector, $\vc{B}$, at any location in space is
182 defined by observing the torque exerted on a magnetic test
183 dipole $D_{mt}$ consisting of a square current loop. The
184 field's magnitude is $|\vc{B}| =\tau/D_{mt} \sin \theta $, where
185 $\theta $ is the angle by which the loop is misaligned. The
186 direction of the field is perpendicular to the loop; of the
187 two perpendiculars, we choose the one such that if we look
188 along it, the loop's current is counterclockwise.
191 We find from this definition that the magnetic field has
192 units of $\zu{N}\unitdot\munit/\zu{A}\unitdot\munit^2=\zu{N}/\zu{A}\unitdot\munit$. This unwieldy combination of
193 units is abbreviated as the \index{tesla (unit)}tesla, 1
194 $\zu{T}=1\ \zu{N}/\zu{A}\unitdot\munit$. Refrain from memorizing the part about the
195 counterclockwise direction at the end; in section \ref{sec:em-waves} we'll
196 see how to understand this in terms of more basic principles.
201 'iron-filings-around-magnet',
202 %q{The magnetic field pattern of a bar magnet. This picture was made by putting iron filings on
203 a piece of paper, and bringing a bar magnet up underneath it. Note how the field pattern
204 passes across the body of the magnet, forming closed loops, as in figure \figref{electric-versus-magnetic-dipole}/2.
205 There are no sources or sinks.}
211 'electric-versus-magnetic-dipole',
212 %q{Electric fields, 1, have sources and sinks, but magnetic fields, 2, don't.}
219 %q{Transformation of the fields.}
224 The nonexistence of magnetic monopoles means that unlike an
225 electric field, \figref{electric-versus-magnetic-dipole}/1, a magnetic one, \figref{electric-versus-magnetic-dipole}/2, can never have
226 sources or sinks. The magnetic field vectors lead in paths
227 that loop back on themselves, without ever converging or
228 diverging at a point.
230 <% begin_sec("Relativity") %>
231 The definition of the tesla as $1\ \zu{N}/\zu{A}\unitdot\munit$ looks messy, but relativity suggests
232 a simple explanation. We saw in __subsection_or_section(magnetic-interactions)
233 that a particular mixture of electric and magnetic fields appears to be a different mixture in a
234 different frame of reference. In a system of units designed for relativity, $\vc{E}$ and $\vc{B}$
235 have the same units. The SI, which predates relativity, wasn't designed this way, which is also why
236 $c$ is some number with units of $\munit/\sunit$ rather than simply equaling 1. The SI units of the
237 $\vc{E}$ and $\vc{B}$ fields are \emph{almost} the same: they differ only by a factor of $\munit/\sunit$.
239 Figure \figref{e-b-lorentz} shows that something similar to the parallelogram diagrams developed in
240 ch.~\ref{ch:relativity} also works as a way of representing the transformation of the
241 $\vc{E}$ and $\vc{B}$ fields from one frame of reference to another. One frame is moving relative
242 to the other in the $x$ direction, and this mixes certain components of the fields. A dot on the
243 graph represents a particular set of fields, which are seen by each observer according to her
245 <% end_sec() %> % Relativity
248 <% begin_sec("Calculating magnetic fields and forces",4,'calculating-magnetism') %>
250 <% begin_sec("Magnetostatics") %>\index{magnetostatics}
252 Our study of the electric field built on our previous
253 understanding of electric forces, which was ultimately based
254 on Coulomb's law for the electric force between two point
255 charges. Since magnetism is ultimately an interaction
256 between currents, i.e., between moving charges, it is
257 reasonable to wish for a magnetic analog of Coulomb's law,
258 an equation that would tell us the magnetic force between
259 any two moving point charges.
261 Such a law, unfortunately, does not exist. Coulomb's law
262 describes the special case of electrostatics: if a set of
263 charges is sitting around and not moving, it tells us the
264 interactions among them. Coulomb's law fails if the charges
265 are in motion, since it does not incorporate any allowance
266 for the time delay in the outward propagation of a change in
267 the locations of the charges.
269 A pair of moving point charges will certainly exert magnetic
270 forces on one another, but their magnetic fields are like
271 the v-shaped bow waves left by boats. Each point charge
272 experiences a magnetic field that originated from the other
273 charge when it was at some previous position. There is no
274 way to construct a force law that tells us the force between
275 them based only on their current positions in space.
277 There is, however, a science of magnetostatics that covers a
278 great many important cases. Magnetostatics describes
279 magnetic forces among currents in the special case where the
280 currents are steady and continuous, leading to magnetic
281 fields throughout space that do not change over time.
283 The magnetic field of a long, straight wire is one example that we can
284 say something about without resorting to fancy mathematics. We saw in examples \ref{eg:line-charge-radial-dependence}
285 on p.~\pageref{eg:line-charge-radial-dependence} and \ref{eg:line-charge-gauss} on p.~\pageref{eg:line-charge-gauss} that the \emph{electric} field of a
286 uniform line of charge is $E=2kq/Lr$, where $r$ is the distance from the line and $q/L$ is the charge per unit length.
287 In a frame of reference moving at velocity $v$ parallel to the line, this electric field will be observed
288 as a combination of electric and magnetic fields. It therefore follows that the magnetic
289 field of a long, straight, current-carrying wire must be proportional to $1/r$. We also expect that it will
290 be proportional to the Coulomb constant, which sets the strength of electric and magnetic
291 interactions, and to the current $I$ in the wire. The complete expression turns out to be
292 $B=(k/c^2)(2I/r)$. This is identical to the expression for $E$ except for replacement of
293 $q/L$ with $I$ and an additional factor of $1/c^2$.
294 The latter occurs because magnetism is a purely relativistic effect, and the relativistic
295 length contraction depends on $v^2/c^2$.
299 Figure \figref{magnetic-field-equations} shows the equations for some of the more commonly
300 encountered configurations, with illustrations of their
301 field patterns. They all have a factor of $k/c^2$ in front, which shows that magnetism
302 is just electricity ($k$) seen through the lens of relativity ($1/c^2$). A convenient
303 feature of SI units is that $k/c^2$ has a numerical value of exactly $10^{-7}$, with units of $\nunit/\zu{A}^2$.
307 'magnetic-field-equations',
308 %q{Some magnetic fields.}
314 \noindent\textit{Field created by a long, straight wire carrying current $I$:}
316 B = \frac{k}{c^2}\cdot\frac{2I}{r}
318 Here $r$ is the distance from the center of the wire. The field vectors trace
319 circles in planes perpendicular to the wire, going clockwise when viewed from along
320 the direction of the current.
322 \noindent\textit{Field created by a single circular loop of current:}\\
323 The field vectors form a dipole-like pattern, coming through the loop and back
324 around on the outside. Each oval path traced out by the field vectors appears clockwise
325 if viewed from along the direction the current is going when it punches through it.
326 There is no simple equation for a field at an arbitrary point in space, but
327 for a point lying \emph{along the central axis} perpendicular to the loop,
330 B = \frac{k}{c^2}\cdot 2\pi Ib^2\left(b^2+z^2\right)^{-3/2} \qquad ,
332 where $b$ is the radius of the loop and $z$ is the distance of the point
333 from the plane of the loop.
335 \noindent\textit{Field created by a solenoid (cylindrical coil):}\\
336 The field pattern is similar to that of a single loop, but for a long solenoid
337 the paths of the field vectors become very straight on the inside of the coil
338 and on the outside immediately next to the coil. For a sufficiently long solenoid,
339 the interior field also becomes very nearly uniform, with a magnitude of
341 B = \frac{k}{c^2}\cdot 4\pi I N/\ell \qquad ,
343 where $N$ is the number of turns of wire and $\ell$ is the length of the
344 solenoid. The field near the mouths or outside the coil is not constant, and
345 is more difficult to calculate. For a long solenoid, the exterior field is
346 much smaller than the interior field.
349 Don't memorize the equations!
352 <% begin_sec("Force on a charge moving through a magnetic field",4) %>
354 We now know how to calculate magnetic fields in some typical
355 situations, but one might also like to be able to calculate
356 magnetic forces, such as the force of a solenoid on a moving
357 charged particle, or the force between two parallel
358 current-carrying wires.
360 We will restrict ourselves to the case of the force on a
361 charged particle moving through a magnetic field, which
362 allows us to calculate the force between two objects when
363 one is a moving charged particle and the other is one whose
364 magnetic field we know how to find. An example is the use of
365 solenoids inside a TV tube to guide the electron beam as
368 Experiments show that the magnetic force on a moving charged
369 particle has a magnitude given by
371 |\vc{F}| = q|\vc{v}||\vc{B}|\sin\theta \qquad ,
373 where $\vc{v}$ is the velocity vector of the particle, and
374 $\theta $ is the angle between the $\vc{v}$ and $\vc{B}$ vectors.
375 Unlike electric and gravitational forces, magnetic forces do
376 not lie along the same line as the field vector. The force
377 is always \emph{perpendicular} to both $\vc{v}$ and $\vc{B}$. Given
378 two vectors, there is only one line perpendicular to both of
379 them, so the force vector points in one of the two possible
380 directions along this line. For a positively charged
381 particle, the direction of the force vector can be found as follows.
382 First, position the $\vc{v}$ and $\vc{B}$ vectors with their tails
383 together. The direction of $\vc{F}$ is such
384 that if you sight along it, the $\vc{B}$ vector is clockwise from
385 the $\vc{v}$ vector; for a negatively charged particle the
386 direction of the force is reversed. Note that since the
387 force is perpendicular to the particle's motion, the
388 magnetic field never does work on it.\label{geom-mag-force-on-charge}
392 'battery-wire-magnet',
394 Example \ref{eg:battery-wire-magnet}.
401 \enlargethispage{-2\baselineskip}
403 \begin{eg}{Magnetic levitation}\label{eg:battery-wire-magnet}
404 In figure \figref{battery-wire-magnet}, a small, disk-shaped permanent magnet is stuck on the side of a
405 battery, and a wire is clasped loosely around the battery, shorting it.
406 A large current flows through the wire. The electrons moving through the
407 wire feel a force from the magnetic field made by the permanent magnet,
408 and this force levitates the wire.
410 From the photo, it's possible to find the direction of the magnetic field made by the permanent magnet.
411 The electrons in the copper wire are negatively charged, so they flow from the negative (flat) terminal
412 of the battery to the positive terminal (the one with the bump, in front). As the electrons pass by the
413 permanent magnet, we can imagine that they would experience a field either toward the magnet, or away from
414 it, depending on which way the magnet was flipped when it was stuck onto the battery.
415 Imagine sighting along the upward force vector, which you could do if
416 you were a tiny bug lying on your back underneath the wire. Since the electrons are negatively charged,
417 the $\vc{B}$ vector must be counterclockwise from the $\vc{v}$ vector, which means toward the magnet.
422 \begin{eg}{A circular orbit}\label{eg:circular-orbit}
423 Magnetic forces cause a beam of electrons to move in a circle.
424 The beam is created in a vacuum tube, in which a small amount of
425 hydrogen gas has been left. A few of the electrons strike hydrogen
426 molecules, creating light and letting us see the beam. A magnetic
427 field is produced by passing a current (meter) through the circular
428 coils of wire in front of and behind the tube. In the bottom figure,
429 with the magnetic field turned on, the force perpendicular to the
430 electrons' direction of motion causes them to move in a circle.
437 Example \ref{eg:circular-orbit}.
444 \enlargethispage{-2\baselineskip}
448 \begin{eg}{Nervous-system effects during an MRI scan}\index{MRI scan}
449 During an MRI scan of the head, the patient's nervous system
450 is exposed to intense magnetic fields, and there are ions moving
451 around in the nerves. The resulting forces on the ions can cause
452 symptoms such as vertigo.
457 <% end_sec() %> % Force on a charge moving through a magnetic field
458 <% begin_sec("Energy in the magnetic field") %>\label{sec:b-energy}\index{energy!stored in magnetic field}
459 On p.~\pageref{sec:energy-in-fields} I gave equations for the energy stored in the gravitational and electric fields.
460 Since a magnetic field is essentially an electric field seen in a different frame of reference, we expect the magnetic-field
461 equation to be closely analogous to the electric version, and it is:
463 (\text{energy stored in the gravitational field per $\munit^3$}) &= -\frac{1}{8\pi G}|\vc{g}|^2\\
464 (\text{energy stored in the electric field per $\munit^3$}) &= \frac{1}{8\pi k}|\vc{E}|^2\\
465 (\text{energy stored in the magnetic field per $\munit^3$}) &= \frac{c^2}{8\pi k}|\vc{B}|^2
467 The idea here is that $k/c^2$ is the magnetic version of the electric quantity $k$, the $1/c^2$ representing the fact
468 that magnetism is a relativistic effect.
472 \begin{eg}{Getting killed by a solenoid}
473 Solenoids are very common electrical devices, but they can
474 be a hazard to someone who is working on them. Imagine a
475 solenoid that initially has a DC current passing through it.
476 The current creates a magnetic field inside and around it,
477 which contains energy. Now suppose that we break the
478 circuit. Since there is no longer a complete circuit,
479 current will quickly stop flowing, and the magnetic field
480 will collapse very quickly. The field had energy stored in
481 it, and even a small amount of energy can create a dangerous
482 power surge if released over a short enough time interval.
483 It is prudent not to fiddle with a solenoid that has current
484 flowing through it, since breaking the circuit could be
485 hazardous to your health.
487 As a typical numerical estimate, let's assume a 40 cm $\times$
488 40 cm $\times$ 40 cm solenoid with an interior magnetic field of
489 1.0 T (quite a strong field). For the sake of this rough
490 estimate, we ignore the exterior field, which is weak, and
491 assume that the solenoid is cubical in shape. The energy
492 stored in the field is
495 (\text{energy per unit volume})(\text{volume})
496 &= \frac{c^2}{8\pi k}|\vc{B}|^2V\\
497 &= 3\times10 ^4\ \junit
499 That's a lot of energy!
502 <% end_sec() %> % Energy in the magnetic field
504 <% begin_sec("The universal speed $c$") %>\label{sec:universal-speed-c}
505 Let's think a little more about the role of the 45-degree diagonal in the Lorentz transformation.
506 Slopes on these graphs are interpreted as velocities.
507 This line has a slope of 1 in relativistic units, but that slope corresponds to $c$ in ordinary metric units.
508 We already know that the relativistic distance unit must
509 be extremely large compared to the relativistic time unit, so $c$ must be extremely large.
510 Now note what happens when we perform a Lorentz transformation: this particular line gets stretched, but the new version
511 of the line lies right on top of the old one, and its slope stays the
512 same. In other words, if one observer says that something has a velocity equal to $c$, every other observer will agree
513 on that velocity as well. (The same thing happens with $-c$.)
519 %q{If you've flown in a jet plane, you can thank relativity for helping you to avoid
520 crashing into a mountain or an ocean. The figure shows a standard
521 piece of navigational equipment called a ring laser gyroscope. A beam of light is
522 split into two parts, sent around the perimeter of the device, and reunited. Since
523 light travels at the universal speed $c$, which is constant, we expect the two parts to come back together at the
524 same time. If they don't, it's evidence that the device has been rotating. The plane's
525 computer senses this and notes how much rotation has accumulated.}
530 <% begin_sec("Velocities don't simply add and subtract.") %>\label{relativistic-combination-of-vel}\index{velocity!addition of!relativistic}
531 This is counterintuitive, since we expect velocities to
532 add and subtract in relative motion. If a dog is running away from me at
533 5 m/s relative to the sidewalk, and I run after it at 3 m/s,
534 the dog's velocity in my frame of reference is 2 m/s.
535 According to everything we have learned about motion (p.~\pageref{vel-addition-newtonian}), the
536 dog must have different speeds in the two frames: 5 m/s in
537 the sidewalk's frame and 2 m/s in mine. But velocities are measured by dividing a distance by a time, and
538 both distance and time are distorted by relativistic effects, so we actually shouldn't expect the ordinary
539 arithmetic addition of velocities to hold in relativity; it's an approximation that's valid at velocities
540 that are small compared to $c$.
543 <% begin_sec("A universal speed limit") %>\label{c-as-speed-limit}
544 For example, suppose Janet takes a trip in a
545 spaceship, and accelerates until she is moving at $0.6c$ relative to the
546 earth. She then launches a space probe in the forward
547 direction at a speed relative to her ship of $0.6c$. We might think that the
548 probe was then moving at a velocity of $1.2c$, but in fact the answer is still less
549 than $c$ (problem \ref{hw:six-tenths-c-twice}, page \pageref{hw:six-tenths-c-twice}).
550 This is an example of a more general fact about relativity, which is that $c$ represents
551 a universal speed limit. This is required by causality, as shown in figure \figref{speed-limit}.
557 %q{A proof that causality imposes a universal speed limit. In the original frame of reference, represented by the square, event A happens a little before event B.
558 In the new frame, shown by the parallelogram, A happens after $t=0$, but B happens before $t=0$; that is,
559 B happens before A. The time ordering of the two events has been reversed. This can only happen because
560 events A and B are very close together in time and fairly far apart in space. The line segment
561 connecting A and B has a slope greater than 1, meaning that if we wanted to be present at both events, we would
562 have to travel at a speed greater than $c$ (which equals 1 in the units used on this graph). You will find that if
563 you pick any two points for which the slope of the line segment connecting them is less than 1, you can never get them
564 to straddle the new $t=0$ line in this funny, time-reversed way. Since different observers disagree on the time order
565 of events like A and B, causality requires that information never travel from A to B or from B to A; if it did, then
566 we would have time-travel paradoxes. The conclusion is that $c$ is the maximum speed of cause and effect in relativity.
572 <% begin_sec("Light travels at $c$.") %>\label{sec:light-travels-at-c}
573 Now consider a beam of light. We're used to talking casually about the ``speed of light,'' but what does that really
574 mean? Motion is relative, so normally if we want to talk about a velocity, we have to specify what it's measured
575 relative to. A sound wave has a certain speed relative to the air, and a water wave has its own speed relative to the
576 water. If we want to measure the speed of an ocean wave, for example, we should make sure to measure it in a frame
577 of reference at rest relative to the water. But light isn't a vibration of a physical medium; it can propagate through the near-perfect vacuum of outer space,
578 as when rays of sunlight travel to earth. This seems like a paradox: light is supposed to have a specific speed,
579 but there is no way to decide what frame of reference to measure it in. The way out of the paradox is that light
580 must travel at a velocity equal to $c$. Since all observers agree on a velocity of $c$, regardless of their frame
581 of reference, everything is consistent.
584 <% begin_sec("The Michelson-Morley experiment") %>
585 The constancy of the speed of light had in fact already been
586 observed when Einstein was an 8-year-old boy, but because nobody could
587 figure out how to interpret it, the result was largely ignored.
588 In 1887 Michelson and Morley set up a clever apparatus to
589 measure any difference in the speed of light beams traveling
590 east-west and north-south. The motion of the earth around
591 the sun at 110,000 km/hour (about 0.01\% of the speed of
592 light) is to our west during the day. Michelson and Morley
593 believed that light was a vibration of a mysterious medium called the ether, so they expected that the
594 speed of light would be a fixed value relative to the ether.
595 As the earth moved through the ether, they thought they
596 would observe an effect on the velocity of light along an
597 east-west line. For instance, if they released a beam of
598 light in a westward direction during the day, they expected
599 that it would move away from them at less than the normal
600 speed because the earth was chasing it through the ether.
601 They were surprised when they found that the expected 0.01\%
602 change in the speed of light did not occur.\index{Michelson-Morley experiment}\index{ether}
606 %q{The Michelson-Morley experiment, shown in photographs, and drawings from the original 1887 paper.
607 1. A simplified drawing of the apparatus. A beam of light from the source, s, is partially reflected and partially transmitted by the half-silvered
608 mirror $\zu{h}_1$. The two half-intensity parts of the beam are reflected by the mirrors at a and b, reunited,
609 and observed in the telescope, t. If the earth's surface was supposed to be moving through the ether,
610 then the times taken by the two light waves to pass through the moving ether would be unequal, and
611 the resulting time lag would be detectable by observing the interference between the waves when they were reunited.
612 2. In the real apparatus, the light beams were reflected multiple times. The effective length of each arm was
613 increased to 11 meters, which greatly improved its sensitivity to the small expected difference in the speed of light.
614 3. In an earlier version of the experiment, they had run into problems with its ``extreme sensitiveness to vibration,''
615 which was ``so great that it was impossible to see the interference fringes except at brief intervals \ldots even at
616 two o'clock in the morning.'' They therefore mounted the whole thing on a massive stone floating in a pool of mercury,
617 which also made it possible to rotate it easily. 4. A photo of the apparatus.
621 'sidecaption'=>false,
628 <% end_sec %> % The Michelson-Morley experiment
630 <% end_sec %> % Universality of $c$
639 The figure shows a famous thought experiment devised by Einstein. A train is moving at
640 constant velocity to the right when bolts of lightning strike the ground near its front and
641 back. Alice, standing on the dirt at the midpoint of the flashes, observes that the light
642 from the two flashes arrives simultaneously, so she says the two strikes must have occurred
643 simultaneously. Bob, meanwhile, is sitting aboard the train, at its middle.
644 He passes by Alice at the moment when Alice later figures out that the flashes happened.
645 Later, he receives flash 2, and then flash 1. He infers that since both flashes traveled
646 half the length of the train, flash 2 must have occurred first.
647 How can this be reconciled with Alice's belief that the flashes were simultaneous?
648 Explain using a graph.
666 Use a graph to resolve the following relativity paradox.
667 Relativity says that in one frame of reference, event A could happen before event B, but in someone else's
668 frame B would come before A. How can this be? Obviously the two people could meet up at A and talk
669 as they cruised past each other. Wouldn't they have to
670 agree on whether B had already happened?
675 \begin{dq}\label{dq:machine-gun-ftl}
676 The machine-gunner in the figure sends out a spray of bullets. Suppose that the bullets are being shot into
677 outer space, and that the distances traveled are trillions of miles (so that the human figure
678 in the diagram is not to scale). After a long time, the bullets reach the points shown with dots
679 which are all equally far from the gun. Their arrivals at those points are events A through E,
680 which happen at different times. The chain of impacts extends across space at a speed greater
681 than $c$. Does this violate special relativity?
687 %q{Discussion question \ref{dq:machine-gun-ftl}.
698 <% begin_sec("Induction",4,'induction') %>\index{induction}
700 <% begin_sec("The principle of induction") %>
701 Physicists of Michelson and Morley's generation thought that light was
702 a mechanical vibration of the ether, but we now know that it is a ripple
703 in the electric and magnetic fields. With hindsight, relativity essentially
706 \item Relativity requires that changes in any field propagate as waves at a finite speed (p.~\pageref{subsec:time-delays}).
707 \item Relativity says that if a wave has a fixed speed but is not a mechanical disturbance in a physical medium, then it
708 must travel at the universal velocity $c$ (p.~\pageref{sec:light-travels-at-c}).
710 What is less obvious is that there are not two separate kinds of waves, electric and magnetic.
711 In fact an electric wave can't exist without a magnetic one, or a magnetic one without an electric one.
712 This new fact follows from the principle of induction, which was discovered experimentally
713 by Faraday\index{Faraday, Michael} in 1831, seventy-five years before Einstein. Let's state Faraday's idea first, and
714 then see how something like it must follow
715 inevitably from relativity:
719 'induced-field-geometry',
721 The geometry of induced fields. The induced field tends to
722 form a whirlpool pattern around the change in the vector producing it.
723 Note how they circulate in opposite directions.
729 \begin{lessimportant}[the principle of induction]
730 Any electric field that changes over time will produce a
731 magnetic field in the space around it.
733 \noindent Any magnetic field that changes over time will produce an
734 electric field in the space around it.
737 The induced field tends to have a whirlpool pattern, as
738 shown in figure \figref{induced-field-geometry}, but the whirlpool image is not to be
739 taken too literally; the principle of induction really just
740 requires a field pattern such that, if one inserted a
741 paddlewheel in it, the paddlewheel would spin. All of the
742 field patterns shown in figure \figref{curl-concept} are ones that
743 could be created by induction; all have a counterclockwise ``curl'' to them.
748 %q{Three fields with counterclockwise ``curls.''},
760 Observer 1 is at rest with respect to the bar magnet, and
761 observes magnetic fields that have different strengths at different
762 distances from the magnet. Observer 2, hanging out in the region to the left of the
763 magnet, sees the magnet moving toward her, and detects that the
764 magnetic field in that region is getting stronger as time passes.
769 'narrowfigwidecaption'=>true,
777 Figure \figref{relativity} shows an example of the fundamental
778 reason why a changing $\vc{B}$ field must create an $\vc{E}$ field.
779 In section \ref{sec:magnetic-interactions} we established that according to
780 relativity, what one observer describes as a purely magnetic field,
781 an observer in a different state of motion describes as a mixture of magnetic and electric fields.
782 This is why there must be both an $\vc{E}$ and a $\vc{B}$ in observer 2's frame.
783 Observer 2 cannot explain the electric field as coming from any
784 charges. In frame 2, the $\vc{E}$ can only be explained as an effect
785 caused by the changing $\vc{B}$.
787 Observer 1 says, ``2 feels a changing $\vc{B}$ field because he's moving through
788 a static field.'' Observer 2 says, ``I feel a changing $\vc{B}$ because the
789 magnet is getting closer.''
791 Although this argument doesn't prove the ``whirlpool'' geometry, we can
792 verify that the fields I've drawn in figure \figref{relativity} are consistent
794 The $\Delta\vc{B}$ vector is upward,
795 and the electric field has a curliness to it: a paddlewheel inserted
796 in the electric field would spin clockwise as seen from above, since
797 the clockwise torque made by the strong electric field on the right is
798 greater than the counterclockwise torque made by the weaker electric
801 \begin{eg}{The generator}\index{generator}
802 A generator, \figref{generator}, consists of a permanent magnet that
803 rotates within a coil of wire. The magnet is turned by a
804 motor or crank, (not shown). As it spins, the nearby
805 magnetic field changes. According to the principle of
806 induction, this changing magnetic field results in an
807 electric field, which has a whirlpool pattern. This electric
808 field pattern creates a current that whips around the coils
809 of wire, and we can tap this current to light the lightbulb.
827 <% self_check('alternator',<<-'SELF_CHECK'
828 When you're driving a car, the engine recharges the
829 battery continuously using a device called an alternator,
830 which is really just a generator like the one shown on the
831 previous page, except that the coil rotates while the
832 permanent magnet is fixed in place. Why can't you use the
833 alternator to start the engine if your car's battery is dead?
837 \begin{eg}{The transformer}
838 In example \ref{hvtransmission} on p.~\pageref{hvtransmission} we discussed the advantages of transmitting
839 power over electrical lines using high voltages and low
840 currents. However, we don't want our wall sockets to operate
841 at 10000 volts! For this reason, the electric company uses a
842 device called a \index{transformer}transformer, \figref{transformer}, to
843 convert to lower voltages and higher currents inside your
844 house. The coil on the input side creates a magnetic field.
845 Transformers work with alternating current, so the magnetic
846 field surrounding the input coil is always changing. This
847 induces an electric field, which drives a current around the output coil.
849 If both coils were the same, the arrangement would be
850 symmetric, and the output would be the same as the input,
851 but an output coil with a smaller number of coils gives the
852 electric forces a smaller distance through which to push the
853 electrons. Less mechanical work per unit charge means a
854 lower voltage. Conservation of energy, however, guarantees
855 that the amount of power on the output side must equal the
856 amount put in originally, $I_{in}V_{in} = I_{out}V_{out}$,
857 so this reduced voltage must be accompanied by an increased current.
864 <% begin_sec("Electromagnetic waves",0,'em-waves') %>\index{electromagnetic waves}\index{waves!electromagnetic}
866 The most important consequence of induction is the existence
867 of electromagnetic waves. Whereas a gravitational wave would
868 consist of nothing more than a rippling of gravitational
869 fields, the principle of induction tells us that there can
870 be no purely electrical or purely magnetic waves. Instead,
871 we have waves in which there are both electric and magnetic
872 fields, such as the sinusoidal one shown in the figure.
873 Maxwell proved that such waves were a direct consequence of
874 his equations, and derived their properties mathematically.
875 The derivation would be beyond the mathematical level of
876 this book, so we will just state the results.
878 \enlargethispage{-\baselineskip}
883 %q{An electromagnetic wave.},
892 A sinusoidal electromagnetic wave has the geometry shown above.
893 The $\vc{E}$ and $\vc{B}$ fields are perpendicular
894 to the direction of motion, and are also perpendicular to
895 each other. If you look along the direction of motion of the
896 wave, the $\vc{B}$ vector is always 90 degrees clockwise from the
897 $\vc{E}$ vector. In a plane wave, the magnitudes of the two fields are related by
898 $|\vc{E}|=c|\vc{B}|$.
900 How is an electromagnetic wave created? It could be emitted,
901 for example, by an electron orbiting an atom or currents
902 going back and forth in a transmitting antenna. In general
903 any accelerating charge will create an electromagnetic wave,
904 although only a current that varies sinusoidally with time
905 will create a sinusoidal wave. Once created, the wave
906 spreads out through space without any need for charges or
907 currents along the way to keep it going. As the electric
908 field oscillates back and forth, it induces the magnetic
909 field, and the oscillating magnetic field in turn creates
910 the electric field. The whole wave pattern propagates
911 through empty space at the velocity $c$.
913 \begin{eg}{Einstein's motorcycle}\label{eg:einstein-motorcycle}
914 As a teenage physics student, Einstein imagined the following paradox.
915 (See p.~\pageref{qualitative-doppler-light}.) What if
916 he could get on a motorcycle and ride at speed $c$, alongside a beam of light?
917 In his frame of reference, he observes constant electric and magnetic fields.
918 But only a \emph{changing} electric field can induce a magnetic field, and
919 only a \emph{changing} magnetic field can induce an electric field. The laws
920 of physics are violated in his frame, and this seems
921 to violate the principle that all frames of reference are equally valid.
923 The resolution of the paradox is that $c$ is a universal speed limit, so the
924 motorcycle can't be accelerated to $c$. Observers can never be at rest relative
925 to a light wave, so no observer can have a frame of reference in which a light
926 wave is observed to be at rest.
929 \begin{eg}{Reflection}\label{eg:em-wave-refl-conductor}
930 The wave in figure \figref{em-wave-refl-conductor} hits a silvered mirror.
931 The metal is a good conductor, so it has constant voltage throughout, and
932 the electric field equals zero inside it: the wave
933 doesn't penetrate and is 100\%
934 reflected. If the electric field is to be zero at the surface as well, the reflected wave must have
935 its electric field inverted (p.~\pageref{wave-inversion}), so that the incident and reflected
938 But the magnetic field of the reflected wave is \emph{not} inverted. This is because the reflected
939 wave, when viewed along its leftward direction of propagation, needs to have its $\vc{B}$ vector
940 90 degrees clockwise from its $\vc{E}$ vector.
943 <% begin_sec("Polarization") %>
946 Two electromagnetic waves traveling in the same direction
947 through space can differ by having their electric and
948 magnetic fields in different directions, a property of the
949 wave called its polarization.
953 \enlargethispage{-\baselineskip}
955 <% begin_sec("Light is an electromagnetic wave") %>
957 Once Maxwell had derived the existence of electromagnetic
958 waves, he became certain that they were the same phenomenon
959 as light. Both are transverse waves (i.e., the vibration is
960 perpendicular to the direction the wave is moving), and the
961 velocity is the same.
965 'em-wave-refl-conductor',
966 %q{Example \ref{eg:em-wave-refl-conductor}. The incident and reflected waves are drawn offset from
967 each other for clarity, but are actually on top of each other so that their fields superpose.}
974 %q{Heinrich Hertz (1857-1894).},
980 Heinrich \index{Hertz, Heinrich}Hertz (for whom the unit of
981 frequency is named) verified Maxwell's ideas experimentally.
982 Hertz was the first to succeed in producing, detecting, and
983 studying electromagnetic waves in detail using antennas and
984 electric circuits. To produce the waves, he had to make
985 electric currents oscillate very rapidly in a circuit. In
986 fact, there was really no hope of making the current reverse
987 directions at the frequencies of $10^{15}$ Hz possessed by
988 visible light. The fastest electrical oscillations he could
989 produce were $10^9$ Hz, which would give a wavelength of
990 about 30 cm. He succeeded in showing that, just like light,
991 the waves he produced were polarizable, and could be
992 reflected and refracted (i.e., bent, as by a lens), and he
993 built devices such as parabolic mirrors that worked
994 according to the same optical principles as those employing
995 light. Hertz's results were convincing evidence that light
996 and electromagnetic waves were one and the same.
999 <% begin_sec("The electromagnetic spectrum") %>\index{spectrum!electromagnetic}\index{electromagnetic spectrum}
1001 Today, electromagnetic waves with frequencies in the range
1002 employed by Hertz are known as radio waves. Any remaining
1003 doubts that the ``Hertzian waves,'' as they were then
1004 called, were the same type of wave as light waves were soon
1005 dispelled by experiments in the whole range of frequencies
1006 in between, as well as the frequencies outside that range.
1007 In analogy to the spectrum of visible light, we speak of the
1008 entire electromagnetic spectrum, of which the visible
1009 spectrum is one segment.
1015 {'width'=>'fullpage'}
1019 The terminology for the various parts of the spectrum is
1020 worth memorizing, and is most easily learned by recognizing
1021 the logical relationships between the wavelengths and the
1022 properties of the waves with which you are already familiar.
1023 Radio waves have wavelengths that are comparable to the
1024 size of a radio antenna, i.e., meters to tens of meters.
1025 Microwaves were named that because they have much shorter
1026 wavelengths than radio waves; when food heats unevenly in a
1027 microwave oven, the small distances between neighboring hot
1028 and cold spots is half of one wavelength of the standing
1029 wave the oven creates. The infrared, visible, and
1030 ultraviolet obviously have much shorter wavelengths, because
1031 otherwise the wave nature of light would have been as
1032 obvious to humans as the wave nature of ocean waves. To
1033 remember that ultraviolet, x-rays, and gamma rays all lie on
1034 the short-wavelength side of visible, recall that all three
1035 of these can cause cancer. (As we'll discuss later in the
1036 course, there is a basic physical reason why the cancer-causing
1037 disruption of DNA can only be caused by very short-wavelength
1038 electromagnetic waves. Contrary to popular belief,
1039 microwaves cannot cause cancer, which is why we have
1040 microwave ovens and not x-ray ovens!)
1044 \begin{eg}{Switching frames of reference}\label{eg:null-field}
1045 If we switch to a different frame of
1046 reference, a legal light wave should still be legal. Consider the requirement $\vc{E}=c\vc{B}$,
1047 in the case where observer 1 says observer 2 is trying to run away from the wave.
1048 In figure \figref{e-b-lorentz} on p.~\pageref{fig:e-b-lorentz}, we saw that the familiar parallelogram
1049 graphs described the transformation of electric and magnetic fields from one frame of
1050 reference to another. These pictures are intended to be used in units where $c=1$, so
1051 the requirement for the fields becomes $\vc{E}=\vc{B}$, and such a combination of fields is represented
1052 by a dot on the diagonal, which is the same line in both frames.
1063 Example \ref{eg:null-field}: an electromagnetic wave that is legal in one frame of reference is legal in another.
1064 As in figure \figref{e-b-lorentz} on p.~\pageref{fig:e-b-lorentz}, the each frame of reference is
1065 in motion relative to the other along the $x$ axis.
1066 If the wave's electric field
1067 is aligned with the $y$ axis, and its magnetic field with $z$, then $x$ is also the direction in which the
1068 wave is moving, as required for our example.
1076 \begin{eg}{Why the sky is blue}
1077 When sunlight enters the upper atmosphere, a particular air mole\-cule
1078 finds itself being washed over by an electromagnetic wave of frequency $f$.
1079 The molecule's charged
1080 particles (nuclei and electrons) act like oscillators being driven by
1081 an oscillating force, and respond by vibrating at the same frequency $f$.
1082 Energy is sucked out of the incoming beam of sunlight and converted into the
1083 kinetic energy of the oscillating particles. However, these particles are
1084 accelerating, so they act like little radio antennas that put the energy back
1085 out as spherical waves of light that spread out in all directions.
1086 An object oscillating at a frequency $f$ has an acceleration proportional to
1087 $f^2$, and an accelerating charged particle creates an electromagnetic wave
1088 whose fields are proportional to its acceleration, so the field of the
1089 reradiated spherical wave is proportional to $f^2$. The energy of a field
1090 is proportional to the square of the field, so the energy of the reradiated wave
1091 is proportional to $f^4$. Since blue light has about twice the frequency of
1092 red light, this process is about $2^4=16$ times as strong for blue as for
1093 red, and that's why the sky is blue.
1098 <% end_sec() %> % The electromagnetic spectrum
1103 'maxwellian-momentum-of-light',
1104 %q{An electromagnetic wave strikes an ohmic surface. The wave's electric field causes currents to flow up and down.
1105 The wave's magnetic field then acts on these currents, producing a force in the direction of the wave's propagation.
1106 This is a pre-relativistic argument that light must possess inertia.},
1109 'narrowfigwidecaption'=>true,
1119 'nichols-radiometer',
1120 %q{A simplified drawing of the 1903 experiment by Nichols and Hull that verified the predicted momentum
1121 of light waves. Two circular mirrors were hung from a fine quartz fiber, inside an evacuated
1122 bell jar. A 150 mW beam of light was shone on one of the mirrors for 6 s, producing a tiny rotation,
1123 which was measurable by an optical lever (not shown). The force was within 0.6\%
1124 of the theoretically predicted value
1125 (problem \ref{hw:ultrarelativistic} on p.~\pageref{hw:ultrarelativistic})
1126 of $0.001\ \mu\nunit$.
1127 For comparison, a short clipping of a single human hair weighs $\sim 1\ \mu\nunit$.},
1130 'narrowfigwidecaption'=>true,
1136 <% begin_sec("Momentum of light") %>\index{light!momentum of}\index{electromagnetic wave!momentum of}\index{momentum!of light}
1138 Newton defined momentum as $mv$, which would imply that light, which has no mass,
1139 should have no momentum. But Newton's laws only work at speeds small compared to
1140 the speed of light, and light travels \emph{at} the speed of light.
1141 In fact, it's straightforward to show that electromagnetic waves have momentum. If a light wave strikes
1142 an ohmic surface, as in figure \figref{maxwellian-momentum-of-light}, the wave's electric field causes charges to vibrate back and forth
1143 in the surface. These currents then
1144 experience a magnetic force from the wave's magnetic field, and application of the geometrical rule on p.~\pageref{geom-mag-force-on-charge} shows that
1145 the resulting force is in the direction of propagation of the wave. A light wave
1146 has momentum and inertia. This is explored further in
1147 problem \ref{hw:ultrarelativistic} on p.~\pageref{hw:ultrarelativistic}. Figure
1148 \figref{nichols-radiometer} shows an experimental confirmation.
1150 <% end_sec() %> % Momentum of light
1151 <% end_sec() %> % Electromagnetic waves
1152 <% begin_sec("Symmetry and handedness",nil,'',{'optional'=>true}) %>\index{symmetry}\index{handedness}
1154 Imagine that you establish
1155 radio contact with an alien on another planet. Neither of
1156 you even knows where the other one's planet is, and you
1157 aren't able to establish any landmarks that you both
1158 recognize. You manage to learn quite a bit of each other's
1159 languages, but you're stumped when you try to establish the
1160 definitions of left and right (or, equivalently, clockwise
1161 and counterclockwise). Is there any way to do it?
1165 '../../../share/mechanics/figs/swan-lake-symmetry',
1166 %q{In this scene from Swan Lake, the choreography has a symmetry with respect to left and right.}
1169 \label{fig:swan-lake-symmetry}
1172 If there was any way to do it without reference to external
1173 landmarks, then it would imply that the laws of physics
1174 themselves were asymmetric, which would be strange. Why
1175 should they distinguish left from right? The gravitational
1176 field pattern surrounding a star or planet looks the same in
1177 a mirror, and the same goes for electric fields.
1178 The field patterns shown in section \ref{sec:calculating-magnetism} seem to violate this
1179 principle, but do they really? Could you use these patterns
1180 to explain left and right to the alien? In fact, the answer
1181 is no. If you look back at the definition of the magnetic
1182 field in section \ref{sec:b-field}, it also contains a reference to
1183 handedness: the counterclockwise direction of the loop's
1184 current as viewed along the magnetic field. The aliens might
1185 have reversed their definition of the magnetic field, in
1186 which case their drawings of field patterns would look like
1187 mirror images of ours.
1191 '../../../share/mechanics/figs/c-s-wu-with-beamline',
1196 \label{fig:c-s-wu-with-beamline}
1202 A graphical representation of the Lorentz transformation for a velocity of $(3/5)c$. The long diagonal is stretched by a factor of two, the
1203 short one is half its former length, and the area is the same as before.
1210 '../../../lm/vw/figs/doppler',
1212 The pattern of waves made by a point source moving to the right
1213 across the water. Note the shorter wavelength of the forward-emitted waves and the longer
1214 wavelength of the backward-going ones.
1216 {'suffix'=>'2'} # doesn't work, gets labeled by subsec number ... why?
1221 Until the middle of the twentieth century, physicists
1222 assumed that any reasonable set of physical laws would have
1223 to have this kind of symmetry between left and right. An
1224 asymmetry would be grotesque. Whatever their aesthetic
1225 feelings, they had to change their opinions about reality
1226 when experiments by C.S.~Wu et al.~showed that the \index{weak nuclear
1227 force}weak \index{nuclear forces}nuclear force (section \ref{sec:energy-in-fields})
1228 violates right-left symmetry! It is still a mystery why
1229 right-left symmetry is observed so scrupulously in general,
1230 but is violated by one particular type of physical process.
1233 <% begin_sec("Doppler shifts and clock time",nil,'doppler-and-clock',{'optional'=>true}) %>
1235 Figure \figref{six-tenths-c} shows our now-familiar method of visualizing a Lorentz transformation,
1236 in a case where the numbers come out to be particularly simple. This diagram has two geometrical features that we have referred to
1237 before without digging into their physical significance: the \emph{stretch factor} of the diagonals, and the \emph{area}. In this section
1238 we'll see that the former can be related to the Doppler effect, and the latter to clock time.
1241 <% begin_sec("Doppler shifts of light",nil,'doppler-light') %>\index{light!Doppler effect for}\index{Doppler effect!for light}
1242 When Doppler shifts happen to ripples on a pond or the sound waves from an airplane, they can depend on the relative motion of three
1243 different objects: the source, the receiver, and the medium. But light waves don't have a medium. Therefore Doppler shifts of
1244 light can only depend on the relative motion of the source and observer.
1246 One simple case is the one in which the relative motion of the source and the receiver is perpendicular to the line connecting them.
1247 That is, the motion is transverse.
1248 Nonrelativistic Doppler shifts happen because the distance between the source and receiver is changing, so in nonrelativistic
1249 physics we don't expect any Doppler shift at all when the motion is transverse, and this is what is in fact observed to high precision.
1250 For example, the photo
1251 % Can't get it to label correctly, so can't do figref.
1252 shows shortened and lengthened wavelengths to the right and left, along the source's line of motion,
1253 but an observer above or below the source measures just the normal, unshifted wavelength and frequency. But relativistically, we have
1254 a time dilation effect, so for light waves emitted transversely, there is a Doppler shift of $1/\mygamma$ in frequency (or $\mygamma$ in wavelength).
1256 The other simple case is the one in which the relative motion of the source and receiver is longitudinal, i.e., they are either approaching or
1257 receding from one another. For example, distant galaxies are receding from our galaxy due to the expansion of the universe, and this expansion was
1258 originally detected because Doppler shifts toward the red (low-frequency) end of the spectrum were observed.
1260 Nonrelativistically, we would expect
1261 the light from such a galaxy to be Doppler shifted down in frequency by some factor, which would depend on the relative velocities of three different
1262 objects: the source, the wave's medium, and the receiver. Relativistically, things get simpler, because light isn't a vibration of a physical medium,
1263 so the Doppler shift can only depend on a single velocity $v$, which is the rate at which the separation between the source and the receiver is
1266 The square in figure \figref{doppler-geometry} is the ``graph paper'' used by someone who considers the source to be at rest, while
1267 the parallelogram plays a similar role for the receiver. The figure is drawn for the case where $v=3/5$ (in units where $c=1$),
1268 and in this case the stretch factor of the long diagonal is 2. To keep the area the same, the short diagonal has to be squished to half its
1269 original size. But now it's a matter of simple geometry to show that OP equals half the width of the square, and this tells us that the Doppler
1270 shift is a factor of 1/2 in frequency. That is, the squish factor of the short diagonal is interpreted as the Doppler shift.
1271 To get this as a general equation for velocities other than 3/5, one can show by straightforward fiddling with the result of
1272 part c of problem \ref{hw:gamma-derivation} on p.~\pageref{hw:gamma-derivation} that
1273 the Doppler shift is
1275 D(v) = \sqrt{\frac{1-v}{1+v}} \qquad .
1277 Here $v>0$ is the case where the source and receiver are getting farther apart, $v<0$ the case where they are approaching.
1278 (This is the opposite of the sign convention used in section \ref{sec:doppler}. It is convenient to change conventions here so that we can
1279 use positive values of $v$ in the case of cosmological red-shifts, which are the most important application.)
1285 At event O, the source and the receiver are on top of each other, so as the source emits a wave crest, it is received without any time delay.
1286 At P, the source emits another wave crest, and at Q the receiver receives it.
1292 Suppose that Alice stays at home on earth while her twin Betty takes off in her rocket ship at 3/5 of the speed of light.
1293 When I first learned relativity, the thing that caused me the most pain was understanding how each observer could say that the
1294 other was the one whose time was slow. It seemed to me that if I could take a pill that would speed up my mind and my body,
1295 then naturally I would see everybody \emph{else} as being \emph{slow}. Shouldn't the same apply to relativity? But suppose Alice and Betty
1296 get on the radio and try to settle who is the fast one and who is the slow one. Each twin's voice sounds slooooowed doooowwwwn
1297 to the other. If Alice claps her hands twice, at a time interval of one second by her clock, Betty hears the hand-claps coming
1298 over the radio two seconds apart, but the situation is exactly symmetric, and Alice hears the same thing if Betty claps.
1299 Each twin analyzes the situation using a diagram identical to \figref{doppler-geometry}, and attributes her sister's
1300 observations to a complicated combination of time distortion, the time taken by the radio signals to propagate, and
1301 the motion of her twin relative to her.
1303 <% self_check('doppler-approaching',<<-'SELF_CHECK'
1304 Turn your book upside-down and reinterpret figure \figref{doppler-geometry}.
1308 \begin{eg}{A symmetry property of the Doppler effect}\label{eg:doppler-abc}
1309 Suppose that A and B are at rest relative to one another, but C is moving along the line between A and B. A transmits a signal
1310 to C, who then retransmits it to B. The signal accumulates two Doppler shifts, and the result is their product $D(v)D(-v)$. But
1311 this product must equal 1, so we must have $D(-v)D(v)=1$, which can be verified directly from the equation.
1314 \begin{eg}{The Ives-Stilwell experiment}\index{Ives-Stilwell experiment}
1315 The result of example \ref{eg:doppler-abc} was the basis of one of the earliest laboratory tests of special relativity, by Ives and Stilwell in 1938.
1316 They observed the light emitted by excited by a beam of $\zu{H}_2^+$ and $\zu{H}_3^+$ ions with speeds of a few tenths of a percent of $c$.
1317 Measuring the light from both ahead of and behind the beams, they found that the product of the Doppler shifts $D(v)D(-v)$ was equal to 1,
1318 as predicted by relativity. If relativity had been false, then one would have expected the product to differ from 1 by an amount that would
1319 have been detectable in their experiment. In 2003, Saathoff et al.~carried out an extremely precise version of the Ives-Stilwell technique
1320 with $\zu{Li}^+$ ions moving at 6.4\% of $c$. The frequencies observed, in units of MHz, were:
1322 \noindent\begin{tabular}{rl}
1323 $f_\zu{o}$ &= $546466918.8 \pm 0.4$ \\
1324 & \hfill (unshifted frequency)\\
1325 $f_\zu{o}D(-v)$ & = $582490203.44 \pm .09$ \\
1326 & \hfill (shifted frequency, forward)\\
1327 $f_\zu{o} D(v)$ & = $512671442.9 \pm 0.5$ \\
1328 & \hfill (shifted frequency, backward)\\
1329 $\sqrt{f_\zu{o}D(-v)\cdot f_\zu{o} D(v)}$ &= $546466918.6 \pm 0.3$
1332 \noindent The results show incredibly precise agreement between $f_\zu{o}$ and $\sqrt{f_\zu{o}D(-v)\cdot f_\zu{o} D(v)}$, as expected
1333 relativistically because $D(v)D(-v)$ is supposed to equal 1. The agreement extends to 9 significant figures, whereas
1334 if relativity had been false there should have been a relative disagreement of about $v^2=.004$, i.e., a discrepancy in the third significant figure.
1335 The spectacular agreement with theory has made this experiment a lightning rod for
1336 anti-relativity kooks.
1339 We saw on p.~\pageref{relativistic-combination-of-vel} that relativistic velocities should not be expected to be exactly additive,
1340 and problem \ref{hw:six-tenths-c-twice} on p.~\pageref{hw:six-tenths-c-twice} verifies this in the special case where A moves relative to B
1341 at $0.6c$ and B relative to C at $0.6c$ --- the result \emph{not} being $1.2c$.\index{velocity!addition of!relativistic}
1342 The relativistic Doppler shift provides a simple way of deriving a general equation for the relativistic combination of velocities;
1343 problem \ref{hw:rel-vel-addition} on p.~\pageref{hw:rel-vel-addition} guides you through the steps of this derivation.
1346 <% begin_sec("Clock time",nil,'proper-time',{'optional'=>true}) %>
1347 On p.~\pageref{fig:area-proof} we proved that the Lorentz transformation doesn't change the area of a shape in the $x$-$t$ plane.
1348 We used this only as a stepping stone toward the Lorentz transformation, but it is natural to wonder whether
1349 this kind of area has any physical interest of its own.
1351 The equal-area result is not relativistic, since the proof never appeals to property 5 on page \pageref{spacetime-properties}. Cases
1352 I and II on page \pageref{fig:three-cases} also have the equal-area property. We can see this clearly in a Galilean transformation like
1353 figure \figref{galilean-boost} on p.~\pageref{fig:galilean-boost}, where the distortion of the rectangle could be accomplished by cutting it into
1354 vertical slices and then displacing the slices upward without changing their areas.
1356 But the area does have a nice interpretation in the relativistic
1357 case. Suppose that we have events A (Charles VII is restored to the throne) and B (Joan of Arc is executed). Now imagine
1358 that technologically advanced aliens want to be present at both A and B, but in the interim they wish to fly away in their spaceship, be present
1359 at some other event P (perhaps a news conference at which they give an update on the events taking place on earth), but get back in
1360 time for B. Since nothing can go faster than $c$ (which we take to equal 1 in appropriate units), P cannot be too far away. The set of all possible events
1361 P forms a rectangle, figure \subfigref{light-rectangles}{1}, in the $x-t$ plane that has A and B at opposite corners and whose edges have slopes equal to $\pm 1$.
1362 We call this type of rectangle a light-rectangle, because its sides could represent the motion of rays of light.
1365 fig('light-rectangles',
1366 %q{1.~The gray light-rectangle represents the set of all events such as P that could be visited after A and before B.\\\\
1367 2.~The rectangle becomes a square in the frame in which A and B occur at the same location in space.\\\\
1368 3.~The area of the dashed square is $\tau^2$, so the area of the gray square is $\tau^2/2$.},
1376 The area of this rectangle will be the same regardless of one's frame of reference. In particular, we could choose a
1377 special frame of reference, panel 2 of the figure, such that A and B occur
1378 in the same place. (They do not occur at the same place, for example, in the sun's frame,
1379 because the earth is spinning and going around the sun.) Since the speed $c$, which equals 1 in our units,
1380 is the same in all frames of reference, and the sides of
1381 the rectangle had slopes $\pm 1$ in frame 1, they must still have slopes $\pm 1$ in frame 2.
1382 The rectangle becomes a square with its diagonals parallel to the $x$ and $t$ axes, and the length of these diagonals equals the
1383 time $\tau$ elapsed on a clock that is at rest in frame 2, i.e., a clock that glides through space at constant velocity from A to B,
1384 meeting up with the planet earth at the appointed time.
1385 As shown in panel 3 of the figure, the area of the gray regions can be interpreted as half the square of this gliding-clock time.
1386 If events A and B are separated by a distance $x$ and a time $t$, then in general $t^2-x^2$ gives the square of the gliding-clock time.\footnote{Proof: Based
1387 on units, the expression must have the form $(\ldots)t^2+(\ldots)tx+(\ldots)x^2$, where each $(\ldots)$ represents a unitless constant.
1388 The $tx$ coefficient must be zero by property 2 on p.~\pageref{spacetime-properties}. For consistency with figure \subfigref{light-rectangles}{3},
1389 the $t^2$ coefficient must equal 1. Since the area vanishes for $x=t$, the $x^2$ coefficient must equal $-1$.}
1391 When $|x|$ is greater than $|t|$, events A and B are so far apart in space and so close together in time that
1392 it would be impossible to have a cause and effect relationship between them, since $c=1$ is the maximum speed of cause and effect.
1393 In this situation $t^2-x^2$ is negative and cannot be interpreted as a clock time, but it can be interpreted as minus the square of the
1394 distance between A and B as measured by rulers at rest in a frame in which A and B are simultaneous.\label{interval-using-ruler}
1396 No matter what, $t^2-x^2$ is the same as measured in all frames of reference. Geometrically, it plays the same role
1397 in the $x$-$t$ plane that ruler measurements play in the Euclidean plane. In Euclidean
1398 geometry, the ruler-distance between any two points stays the same regardless of rotation, i.e., regardless of
1399 the angle from which we view the scene; according to the Pythagorean theorem, the square of this distance is $x^2+y^2$.
1400 In the $x$-$t$ plane, $t^2-x^2$ stays the same regardless of the
1403 To avoid overloading the reader with terms to memorize, some commonly used terminology is
1404 relegated to problem \ref{hw:geroch-interval} on p.~\pageref{hw:geroch-interval}.
1405 <% end_sec() %> % Clock time
1406 <% end_sec() %> % Doppler shifts and clock time
1411 \vocabitem{magnetic field}{a field of force, defined in terms of the
1412 torque exerted on a test dipole}
1414 \vocabitem{magnetic dipole}{an object, such as a current loop, an atom,
1415 or a bar magnet, that experiences torques due to magnetic
1416 forces; the strength of magnetic dipoles is measured by
1417 comparison with a standard dipole consisting of a square
1418 loop of wire of a given size and carrying a given amount of current}
1420 \vocabitem{induction}{the production of an electric field by a changing
1421 magnetic field, or vice-versa}
1426 \notationitem{$\vc{B}$}{the magnetic field}
1428 \notationitem{$D_m$}{magnetic dipole moment}
1434 The magnetic field is defined in terms of
1435 the torque on a magnetic test dipole. It has no sources or
1436 sinks; magnetic field patterns never converge on or
1437 diverge from a point.
1439 Relativity dictates a maximum speed limit $c$ for cause and effect. This
1440 speed is the same in all frames of reference.
1442 Relativity requires that the magnetic and electric fields be intimately related. The
1443 principle of induction states that any changing electric
1444 field produces a magnetic field in the surrounding space,
1445 and vice-versa. These induced fields tend to form whirlpool patterns.
1447 The most important consequence of the principle of induction
1448 is that there are no purely magnetic or purely electric
1449 waves. Electromagnetic disturbances
1450 propagate outward at $c$ as combined magnetic and electric waves,
1451 with a well-defined relationship between the magnitudes
1452 and directions of the electric and magnetic fields. These electromagnetic waves are what light
1453 is made of, but other forms of electromagnetic waves exist
1454 besides visible light, including radio waves, x-rays, and gamma rays.
1458 %%===============================================================================
1462 <% begin_hw('six-tenths-c-twice') %>
1463 The figure illustrates a Lorentz transformation using the conventions employed in section \ref{sec:x-t-distortion}.
1464 For simplicity, the transformation chosen is one that lengthens one diagonal by a factor of 2. Since Lorentz transformations
1465 preserve area, the other diagonal is shortened by a factor of 2. Let the original frame of reference, depicted with the square,
1466 be A, and the new one B. (a) By measuring with a ruler on the figure, show that the velocity of frame B relative to frame A is $0.6c$.
1467 (b) Print out a copy of the page. With a ruler, draw a third parallelogram that represents a second successive Lorentz transformation, one
1468 that lengthens the long diagonal by another factor of 2. Call this third frame C. Use measurements with a ruler to determine
1469 frame C's velocity relative to frame A. Does it equal double the velocity found in part a? Explain why it should be expected to turn
1470 out the way it does.\answercheck
1471 % v=.6 exactly; combined v = tanh(2 atanh(.6))=.8824; verified graphically using inkscape that it's .8824
1476 \enlargethispage{3\baselineskip}
1480 'hw-six-tenths-c-twice',
1484 #'sidecaption'=>true
1493 <% begin_hw('side-by-side-lasers') %>__incl(../../share/relativity/hw/side-by-side-lasers)<% end_hw() %>
1497 <% begin_hw('nestedsolenoids') %>
1498 Consider two solenoids, one of which is smaller so that
1499 it can be put inside the other. Assume they are long enough
1500 so that each one only contributes significantly to the field
1501 inside itself, and the interior fields are nearly uniform.
1502 Consider the configuration where the small one is inside the
1503 big one with their currents circulating in the same
1504 direction, and a second configuration in which the currents
1505 circulate in opposite directions. Compare the energies of
1506 these configurations with the energy when the solenoids are
1507 far apart. Based on this reasoning, which configuration is
1508 stable, and in which configuration will the little solenoid
1509 tend to get twisted around or spit out? [Hint: A stable
1510 system has low energy; energy would have to be added to
1511 change its configuration.]
1519 'hw-nested-wire-loops',
1520 %q{Problem \ref{hw:nested-wire-loops}.}
1524 <% begin_hw('nested-wire-loops') %>__incl(hw/nested-wire-loops)<% end_hw() %>
1528 <% begin_hw('atom') %>
1529 One model of the hydrogen atom has the electron circling
1530 around the proton at a speed of $2.2\times10^6$ m/s, in an
1531 orbit with a radius of 0.05 nm. (Although the electron and
1532 proton really orbit around their common center of mass, the
1533 center of mass is very close to the proton, since it is 2000
1534 times more massive. For this problem, assume the proton is
1535 stationary.) In homework problem \ref{hw:lpcurrent} on page \pageref{hw:lpcurrent}, you
1536 calculated the electric current created.\hwendpart
1537 (a) Now estimate the magnetic field created at the center
1538 of the atom by the electron. We are treating the circling
1539 electron as a current loop, even though it's only a single particle.\answercheck\hwendpart
1540 (b) Does the proton experience a nonzero force from the
1541 electron's magnetic field? Explain.\hwendpart
1542 (c) Does the electron experience a magnetic field from
1543 the proton? Explain.\hwendpart
1544 (d) Does the electron experience a magnetic field created by
1545 its own current? Explain.\hwendpart
1546 (e) Is there an electric force acting between the proton
1547 and electron? If so, calculate it.\answercheck\hwendpart
1548 (f) Is there a gravitational force acting between the proton
1549 and electron? If so, calculate it.\hwendpart
1550 (g) An inward force is required to keep the electron in its
1551 orbit -- otherwise it would obey Newton's first law and go
1552 straight, leaving the atom. Based on your answers to the
1553 previous parts, which force or forces (electric, magnetic
1554 and gravitational) contributes significantly to this inward force?\hwendpart
1555 [Based on a problem by Arnold Arons.]
1560 <% begin_hw('velocityfilter') %>__incl(hw/velocityfilter)<% end_hw() %>
1564 <% begin_hw('solenoid-u-quadruple-i') %>__incl(hw/solenoid-u-quadruple-i)<% end_hw() %>
1568 <% begin_hw('mystery-magnet',2) %>__incl(hw/mystery-magnet)<% end_hw() %>
1572 <% begin_hw('solenoid-sex') %>
1573 Consider two solenoids, one of which is smaller so that
1574 it can be put inside the other. Assume they are long enough
1575 to act like ideal solenoids, so that each one only
1576 contributes significantly to the field inside itself, and
1577 the interior fields are nearly uniform. Consider the
1578 configuration where the small one is partly inside and
1579 partly hanging out of the big one, with their currents
1580 circulating in the same direction. Their axes are constrained to coincide.
1582 (a) Find the magnetic potential energy as a function of the
1583 length $x$ of the part of the small solenoid that is inside
1584 the big one. (Your equation will include other relevant
1585 variables describing the two solenoids.)
1587 (b) Based on your answer to part (a), find the force acting
1588 between the solenoids.
1594 %q{Problem \ref{hw:wire-box}.},
1603 <% begin_hw('wire-box') %>__incl(hw/wire-box)<% end_hw() %>
1607 <% begin_hw('force-on-wire-in-b') %>__incl(hw/force-on-wire-in-b)<% end_hw() %>
1611 <% begin_hw('force-between-wires') %>__incl(hw/force-between-wires)<% end_hw() %>
1615 <% begin_hw('cyclic-vbf') %>__incl(hw/cyclic-vbf)<% end_hw() %>
1619 <% begin_hw('dipoleshape',2) %>__incl(hw/dipoleshape)<% end_hw() %>
1626 'hw-helmholtz-coil',
1627 %q{Problem \ref{hw:helmholtzcoil}.}
1631 <% begin_hw('helmholtzcoil') %>
1632 A Helmholtz coil is defined as a pair of identical
1633 circular coils separated by a distance, $h$, equal to their
1634 radius, $b$. (Each coil may have more than one turn of
1635 wire.) Current circulates in the same direction in each
1636 coil, so the fields tend to reinforce each other in the
1637 interior region. This configuration has the advantage of
1638 being fairly open, so that other apparatus can be easily
1639 placed inside and subjected to the field while remaining
1640 visible from the outside. The choice of $h=b$ results in the
1641 most uniform possible field near the center. (a) Find the
1642 percentage drop in the field at the center of one coil,
1643 compared to the full strength at the center of the whole
1644 apparatus. (b) What value of $h$ (not equal to $b)$ would
1645 make this percentage difference equal to zero?
1650 <% begin_hw('b-from-photo') %>__incl(hw/b-from-photo)<% end_hw() %>
1654 <% begin_hw('corkscrew-beam',2) %>__incl(hw/corkscrew-beam)<% end_hw() %>
1658 <% begin_hw('deleted') %>This problem is now problem \ref{hw:cosmic-ray-lightning} on p.~\pageref{hw:cosmic-ray-lightning}<% end_hw() %>
1662 <% begin_hw('field-at-mouth-of-solenoid',2) %>__incl(hw/field-at-mouth-of-solenoid)<% end_hw() %>
1666 <% begin_hw('em-wave-energy-split') %>__incl(hw/em-wave-energy-split)<% end_hw() %>
1670 <% begin_hw('rel-vel-addition') %>__incl(../../share/relativity/hw/rel-vel-addition)<% end_hw() %>
1674 <% begin_hw('geroch-interval') %>__incl(../../share/relativity/hw/geroch-interval)<% end_hw() %>
1679 'hw-geroch-interval',
1681 Problem \ref{hw:geroch-interval}.
1687 <% begin_hw('pure-e-b-lorentz-graphical') %>__incl(hw/pure-e-b-lorentz-graphical)<% end_hw() %>
1691 <% begin_ex("Polarization","A") %>
1693 \noindent Apparatus:
1695 \begin{indentedblock}
1696 calcite (Iceland spar) crystal
1701 1. Lay the crystal on a piece of paper that has print on it.
1702 You will observe a double image. See what happens if
1703 you rotate the crystal.
1705 Evidently the crystal does something to the light that
1706 passes through it on the way from the page to your eye. One
1707 beam of light enters the crystal from underneath, but two
1708 emerge from the top; by conservation of energy the energy of
1709 the original beam must be shared between them. Consider the
1710 following three possible interpretations of what you have observed:
1712 (a) The two new beams differ from each other, and from the
1713 original beam, only in energy. Their other properties are the same.
1715 (b) The crystal adds to the light some mysterious new
1716 property (not energy), which comes in two flavors, X and
1717 Y. Ordinary light doesn't have any of either. One beam
1718 that emerges from the crystal has some X added to it, and
1719 the other beam has Y.
1721 (c) There is some mysterious new property that is possessed
1722 by all light. It comes in two flavors, X and Y, and most
1723 ordinary light sources make an equal mixture of type X and
1724 type Y light. The original beam is an even mixture of both
1725 types, and this mixture is then split up by the crystal into
1726 the two purified forms.
1728 In parts 2 and 3 you'll make observations that will allow
1729 you to figure out which of these is correct.
1731 2. Now place a polaroid film over the crystal and see what
1732 you observe. What happens when you rotate the film in the
1733 horizontal plane? Does this observation allow you to rule
1734 out any of the three interpretations?
1738 3. Now put the polaroid film under the crystal and try the
1739 same thing. Putting together all your observations, which
1740 interpretation do you think is correct?
1744 4. Look at an overhead light fixture through the polaroid,
1745 and try rotating it. What do you observe? What does this
1746 tell you about the light emitted by the lightbulb?
1750 5. Now position yourself with your head under a light
1751 fixture and directly over a shiny surface, such as a glossy
1752 tabletop. You'll see the lamp's reflection, and the light
1753 coming from the lamp to your eye will have undergone a
1754 reflection through roughly a 180-degree angle (i.e. it very
1755 nearly reversed its direction). Observe this reflection
1756 through the polaroid, and try rotating it. Finally, position
1757 yourself so that you are seeing glancing reflections, and
1758 try the same thing. Summarize what happens to light with
1759 properties X and Y when it is reflected. (This is the
1760 principle behind polarizing sunglasses.)
1763 <% begin_ex("Events and Spacetime","B") %>
1765 \includegraphics[width=168mm]{../share/relativity/figs/spacetime-ex-0.png}
1767 \includegraphics[width=168mm]{../share/relativity/figs/spacetime-ex-1.png}
1769 \includegraphics[width=168mm]{../share/relativity/figs/spacetime-ex-2.png}