misc fixes to LM

[light-and-matter.git] / lm / em / d.rbtex

<%
  require "../eruby_util.rb"
%>

<%
  chapter(
    '24',
    %q{Electromagnetism},
    'ch:em',
    '',
    {'opener'=>''}
  )
%>

<% begin_sec("The magnetic field",nil,'b-field') %>\index{magnetic field}\index{field!magnetic}

<% begin_sec("No magnetic monopoles") %>\index{monopoles!magnetic}

<% marg(10) %>
<%
  fig(
    'break-bar-magnet',
    %q{%
      Breaking a bar magnet in half doesn't create two monopoles, it creates two
      smaller dipoles.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'break-bar-magnet-atoms',
    %q{An explanation at the atomic level.}
  )
%>
<% end_marg %>
If you could play with a handful of electric dipoles and a
handful of bar magnets, they would appear very similar. For
instance, a pair of bar magnets wants to align themselves
head-to-tail, and a pair of electric dipoles does the same
thing. (It is unfortunately not that easy to make a
permanent electric dipole that can be handled like this,
since the charge tends to leak.)

You would eventually notice an important difference between
the two types of objects, however. The electric dipoles can
be broken apart to form isolated positive charges and
negative charges. The two-ended device can be broken into
parts that are not two-ended. But if you break a bar magnet
in half, \figref{break-bar-magnet}, you will find that you have simply made two
smaller two-ended objects.

The reason for this behavior is not hard to divine from our
microscopic picture of permanent iron magnets. An electric
dipole has extra positive ``stuff'' concentrated in one end
and extra negative in the other. The bar magnet, on the
other hand, gets its magnetic properties not from an
imbalance of magnetic ``stuff'' at the two ends but from the
orientation of the rotation of its electrons. One end is the
one from which we could look down the axis and see the
electrons rotating clockwise, and the other is the one from
which they would appear to go counterclockwise. There is no
difference between the ``stuff'' in one end of the
magnet and the other, \figref{break-bar-magnet-atoms}.

Nobody has ever succeeded in isolating a single magnetic
pole. In technical language, we say that magnetic \index{monopoles!magnetic}monopoles
do not seem to exist. Electric monopoles \emph{do} exist
--- that's what charges are.

Electric and magnetic forces seem similar in many ways. Both
act at a distance, both can be either attractive or
repulsive, and both are intimately related to the property
of matter called charge. (Recall that magnetism is an
interaction between moving charges.) Physicists's aesthetic
senses have been offended for a long time because this
seeming symmetry is broken by the existence of electric
monopoles and the absence of magnetic ones. Perhaps some
exotic form of matter exists, composed of particles that are
magnetic monopoles. If such particles could be found in
cosmic rays or moon rocks, it would be evidence that the
apparent asymmetry was only an asymmetry in the composition
of the universe, not in the laws of physics. For these
admittedly subjective reasons, there have been several
searches for magnetic monopoles. Experiments have been
performed, with negative results, to look for magnetic
monopoles embedded in ordinary matter. Soviet physicists in
the 1960's made exciting claims that they had created and
detected magnetic monopoles in particle accelerators, but
there was no success in attempts to reproduce the results
there or at other accelerators. The most recent search for
magnetic monopoles, done by reanalyzing data from the search
for the top quark at Fermilab, turned up no candidates,
which shows that either monopoles don't exist in nature or
they are extremely massive and thus hard to create in accelerators.

<% end_sec() %>
<% begin_sec("Definition of the magnetic field") %>

<% marg(100) %>
<%
  fig(
    'tesla',
    %q{The unit of magnetic field, the tesla, is named after Serbian-American inventor Nikola Tesla.},
    {'anonymous'=>true}
  )
%>
\spacebetweenfigs
<%
  fig(
    'current-loop-dipole',
    %q{%
      A standard dipole made from a square loop of wire shorting
      across a battery. It acts very much like a bar magnet, but its strength is more
      easily quantified.
    },
    {'anonymous'=>true}
  )
%>
\spacebetweenfigs
<%
  fig(
    'current-loop-aligns',
    %q{A dipole tends to align itself to the surrounding magnetic field.},
    {'anonymous'=>true}
  )
%>
<% end_marg %>

Since magnetic monopoles don't seem to exist, it would not
make much sense to define a magnetic field in terms of the
force on a test monopole. Instead, we follow the philosophy
of the alternative definition of the electric field, and
define the field in terms of the torque on a magnetic test
dipole. This is exactly what a magnetic compass does: the
needle is a little iron magnet which acts like a magnetic
dipole and shows us the direction of the earth's magnetic field.

To define the strength of a magnetic field, however, we need
some way of defining the strength of a test dipole, i.e., we
need a definition of the magnetic dipole moment. We could
use an iron permanent magnet constructed according to
certain specifications, but such an object is really an
extremely complex system consisting of many iron atoms, only
some of which are aligned. A more fundamental standard
dipole is a square current loop. This could be little
resistive circuit consisting of a square of wire shorting across a battery.

We will find that such a loop, when placed in a magnetic
field, experiences a torque that tends to align plane so
that its face points in a certain direction. (Since the loop
is symmetric, it doesn't care if we rotate it like a wheel
without changing the plane in which it lies.) It is this
preferred facing direction that we will end up defining as
the direction of the magnetic field.

Experiments show if the loop is out of alignment with the
field, the torque on it is proportional to the amount of
current, and also to the interior area of the loop. The
proportionality to current makes sense, since magnetic
forces are interactions between moving charges, and current
is a measure of the motion of charge. The proportionality to
the loop's area is also not hard to understand, because
increasing the length of the sides of the square increases
both the amount of charge contained in this circular
``river'' and the amount of leverage supplied for making
torque. Two separate physical reasons for a proportionality
to length result in an overall proportionality to length
squared, which is the same as the area of the loop. For
these reasons, we define the magnetic dipole moment of a
square current loop as
\begin{multline*}
                D_m         =    IA \qquad , \hfill\shoveright{\text{[definition of the magnetic}}\\
                                                            \text{ dipole moment of a square current loop]}
\end{multline*}
We now define the \index{magnetic field!defined}magnetic
field in a manner entirely analogous to the second
definition of the electric field:

\begin{lessimportant}[definition of the magnetic field]
The magnetic field vector, $\vc{B}$, at any location in space is
defined by observing the torque exerted on a magnetic test
dipole $D_{mt}$ consisting of a square current loop. The
field's magnitude is $|\vc{B}| =\tau/D_{mt} \sin \theta $, where
$\theta $ is the angle by which the loop is misaligned. The
direction of the field is perpendicular to the loop; of the
two perpendiculars, we choose the one such that if we look
along it, the loop's current is counterclockwise.
\end{lessimportant}

We find from this definition that the magnetic field has
units of $\zu{N}\unitdot\munit/\zu{A}\unitdot\munit^2=\zu{N}/\zu{A}\unitdot\munit$. This unwieldy combination of
units is abbreviated as the \index{tesla (unit)}tesla, 1
$\zu{T}=1\ \zu{N}/\zu{A}\unitdot\munit$. Refrain from memorizing the part about the
counterclockwise direction at the end; in section \ref{sec:em-waves} we'll
see how to understand this in terms of more basic principles.

<% marg(130) %>
<%
  fig(
    'iron-filings-around-magnet',
    %q{The magnetic field pattern of a bar magnet. This picture was made by putting iron filings on
      a piece of paper, and bringing a bar magnet up underneath it. Note how the field pattern
      passes across the body of the magnet, forming closed loops, as in figure \figref{electric-versus-magnetic-dipole}/2.
      There are no sources or sinks.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'electric-versus-magnetic-dipole',
    %q{Electric fields, 1, have sources and sinks, but magnetic fields, 2, don't.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'e-b-lorentz',
    %q{Transformation of the fields.}
  )
%>
<% end_marg %>

The nonexistence of magnetic monopoles means that unlike an
electric field, \figref{electric-versus-magnetic-dipole}/1, a magnetic one, \figref{electric-versus-magnetic-dipole}/2, can never have
sources or sinks. The magnetic field vectors lead in paths
that loop back on themselves, without ever converging or
diverging at a point.
<% end_sec() %>
<% begin_sec("Relativity") %>
The definition of the tesla as $1\ \zu{N}/\zu{A}\unitdot\munit$ looks messy, but relativity suggests
a simple explanation. We saw in __subsection_or_section(magnetic-interactions)
that a particular mixture of electric and magnetic fields appears to be a different mixture in a
different frame of reference. In a system of units designed for relativity, $\vc{E}$ and $\vc{B}$
have the same units. The SI, which predates relativity, wasn't designed this way, which is also why
$c$ is some number with units of $\munit/\sunit$ rather than simply equaling 1. The SI units of the
$\vc{E}$ and $\vc{B}$ fields are \emph{almost} the same: they differ only by a factor of $\munit/\sunit$.

Figure \figref{e-b-lorentz} shows that something similar to the parallelogram diagrams developed in
ch.~\ref{ch:relativity} also works as a way of representing the transformation of the
$\vc{E}$ and $\vc{B}$ fields from one frame of reference to another. One frame is moving relative
to the other in the $x$ direction, and this mixes certain components of the fields. A dot on the
graph represents a particular set of fields, which are seen by each observer according to her
own coordinate grid.
<% end_sec() %> % Relativity
<% end_sec() %>
\vfill
<% begin_sec("Calculating magnetic fields and forces",4,'calculating-magnetism') %>

<% begin_sec("Magnetostatics") %>\index{magnetostatics}

Our study of the electric field built on our previous
understanding of electric forces, which was ultimately based
on Coulomb's law for the electric force between two point
charges. Since magnetism is ultimately an interaction
between currents, i.e., between moving charges, it is
reasonable to wish for a magnetic analog of Coulomb's law,
an equation that would tell us the magnetic force between
any two moving point charges.

Such a law, unfortunately, does not exist. Coulomb's law
describes the special case of electrostatics: if a set of
charges is sitting around and not moving, it tells us the
interactions among them. Coulomb's law fails if the charges
are in motion, since it does not incorporate any allowance
for the time delay in the outward propagation of a change in
the locations of the charges.

A pair of moving point charges will certainly exert magnetic
forces on one another, but their magnetic fields are like
the v-shaped bow waves left by boats. Each point charge
experiences a magnetic field that originated from the other
charge when it was at some previous position. There is no
way to construct a force law that tells us the force between
them based only on their current positions in space.

There is, however, a science of magnetostatics that covers a
great many important cases. Magnetostatics describes
magnetic forces among currents in the special case where the
currents are steady and continuous, leading to magnetic
fields throughout space that do not change over time.

The magnetic field of a long, straight wire is one example that we can
say something about without resorting to fancy mathematics. We saw in examples \ref{eg:line-charge-radial-dependence}
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
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.
In a frame of reference moving at velocity $v$ parallel to the line, this electric field will be observed
as a combination of electric and magnetic fields. It therefore follows that the magnetic
field of a long, straight, current-carrying wire must be proportional to $1/r$. We also expect that it will
be proportional to the Coulomb constant, which sets the strength of electric and magnetic
interactions, and to the current $I$ in the wire. The complete expression turns out to be
$B=(k/c^2)(2I/r)$. This is identical to the expression for $E$ except for replacement of
$q/L$ with $I$ and an additional factor of $1/c^2$.
The latter occurs because magnetism is a purely relativistic effect, and the relativistic
length contraction depends on $v^2/c^2$.

\pagebreak

Figure \figref{magnetic-field-equations} shows the equations for some of the more commonly
encountered configurations, with illustrations of their
field patterns. They all have a factor of $k/c^2$ in front, which shows that magnetism
is just electricity ($k$) seen through the lens of relativity ($1/c^2$). A convenient
feature of SI units is that $k/c^2$ has a numerical value of exactly $10^{-7}$, with units of $\nunit/\zu{A}^2$.
<% marg(-15) %>
<%
  fig(
    'magnetic-field-equations',
    %q{Some magnetic fields.}
  )
%>
<% end_marg %>

\begin{offsettopic}
\noindent\textit{Field created by a long, straight wire carrying current $I$:}
\begin{equation*}
 B = \frac{k}{c^2}\cdot\frac{2I}{r}
\end{equation*}
Here $r$ is the distance from the center of the wire. The field vectors trace
circles in planes perpendicular to the wire, going clockwise when viewed from along
the direction of the current.

\noindent\textit{Field created by a single circular loop of current:}\\
The field vectors form a dipole-like pattern, coming through the loop and back
around on the outside. Each oval path traced out by the field vectors appears clockwise
if viewed from along the direction the current is going when it punches through it.
There is no simple equation for a field at an arbitrary point in space, but
for a point lying \emph{along the central axis} perpendicular to the loop,
the field is
\begin{equation*}
  B = \frac{k}{c^2}\cdot 2\pi Ib^2\left(b^2+z^2\right)^{-3/2} \qquad ,
\end{equation*}
where $b$ is the radius of the loop and $z$ is the distance of the point
from the plane of the loop.

\noindent\textit{Field created by a solenoid (cylindrical coil):}\\
The field pattern is similar to that of a single loop, but for a long solenoid
the paths of the field vectors become very straight on the inside of the coil
and on the outside immediately next to the coil. For a sufficiently long solenoid,
the interior field also becomes very nearly uniform, with a magnitude of
\begin{equation*}
  B = \frac{k}{c^2}\cdot 4\pi I N/\ell \qquad ,
\end{equation*}
where $N$ is the number of turns of wire and $\ell$ is the length of the
solenoid. The field near the mouths or outside the coil is not constant, and
is more difficult to calculate. For a long solenoid, the exterior field is
much smaller than the interior field.
\end{offsettopic}

Don't memorize the equations! 

<% end_sec() %>
<% begin_sec("Force on a charge moving through a magnetic field",4) %>

We now know how to calculate magnetic fields in some typical
situations, but one might also like to be able to calculate
magnetic forces, such as the force of a solenoid on a moving
charged particle, or the force between two parallel
current-carrying wires.

We will restrict ourselves to the case of the force on a
charged particle moving through a magnetic field, which
allows us to calculate the force between two objects when
one is a moving charged particle and the other is one whose
magnetic field we know how to find. An example is the use of
solenoids inside a TV tube to guide the electron beam as
it paints a picture.

Experiments show that the magnetic force on a moving charged
particle has a magnitude given by
\begin{equation*}
 |\vc{F}|    = q|\vc{v}||\vc{B}|\sin\theta      \qquad   ,
\end{equation*}
where $\vc{v}$ is the velocity vector of the particle, and
$\theta $ is the angle between the $\vc{v}$ and $\vc{B}$ vectors.
Unlike electric and gravitational forces, magnetic forces do
not lie along the same line as the field vector. The force
is always \emph{perpendicular} to both $\vc{v}$ and $\vc{B}$. Given
two vectors, there is only one line perpendicular to both of
them, so the force vector points in one of the two possible
directions along this line. For a positively charged
particle, the direction of the force vector can be found as follows.
First, position the $\vc{v}$ and $\vc{B}$ vectors with their tails
together. The direction of $\vc{F}$ is such
that if you sight along it, the $\vc{B}$ vector is clockwise from
the $\vc{v}$ vector; for a negatively charged particle the
direction of the force is reversed. Note that since the
force is perpendicular to the particle's motion, the
magnetic field never does work on it.\label{geom-mag-force-on-charge}
<% marg(-36) %>
<%
  fig(
    'battery-wire-magnet',
    %q{%
      Example \ref{eg:battery-wire-magnet}.
    },
    {'anonymous'=>true}
  )
%>
<% end_marg %>

\enlargethispage{-2\baselineskip}

\begin{eg}{Magnetic levitation}\label{eg:battery-wire-magnet}
      In figure \figref{battery-wire-magnet}, a small, disk-shaped permanent magnet is stuck on the side of a
      battery, and a wire is clasped loosely around the battery, shorting it.
      A large current flows through the wire. The electrons moving through the
      wire feel a force from the magnetic field made by the permanent magnet,
      and this force levitates the wire.

      From the photo, it's possible to find the direction of the magnetic field made by the permanent magnet.
      The electrons in the copper wire are negatively charged, so they flow from the negative (flat) terminal
      of the battery to the positive terminal (the one with the bump, in front). As the electrons pass by the
      permanent magnet, we can imagine that they would experience a field either toward the magnet, or away from
      it, depending on which way the magnet was flipped when it was stuck onto the battery.
      Imagine sighting along the upward force vector, which you could do if
      you were a tiny bug lying on your back underneath the wire. Since the electrons are negatively charged,
      the $\vc{B}$ vector must be counterclockwise from the $\vc{v}$ vector, which means toward the magnet.
\end{eg}

\vspace{10mm}

\begin{eg}{A circular orbit}\label{eg:circular-orbit}
      Magnetic forces cause a beam of electrons to move in a circle.
              The beam is created in a vacuum tube, in which a small amount of
              hydrogen gas has been left. A few of the electrons strike hydrogen
              molecules, creating light and letting us see the beam. A magnetic
              field is produced by passing a current (meter) through the circular
              coils of wire in front of and behind the tube. In the bottom figure,
              with the magnetic field turned on, the force perpendicular to the
              electrons' direction of motion causes them to move in a circle.
\end{eg}
<% marg(42) %>
<%
  fig(
    'circular-orbit',
    %q{%
      Example \ref{eg:circular-orbit}.
    },
    {'anonymous'=>true}
  )
%>
<% end_marg %>

\enlargethispage{-2\baselineskip}

\vspace{10mm}

\begin{eg}{Nervous-system effects during an MRI scan}\index{MRI scan}
During an MRI scan of the head, the patient's nervous system
is exposed to intense magnetic fields, and there are ions moving
around in the nerves. The resulting forces on the ions can cause
symptoms such as vertigo.
\end{eg}

\vspace{10mm}

<% end_sec() %> % Force on a charge moving through a magnetic field
<% begin_sec("Energy in the magnetic field") %>\label{sec:b-energy}\index{energy!stored in magnetic field}
On p.~\pageref{sec:energy-in-fields} I gave equations for the energy stored in the gravitational and electric fields.
Since a magnetic field is essentially an electric field seen in a different frame of reference, we expect the magnetic-field
equation to be closely analogous to the electric version, and it is:
\begin{align*}
(\text{energy stored in the gravitational field per $\munit^3$}) &= -\frac{1}{8\pi G}|\vc{g}|^2\\
(\text{energy stored in the electric field per $\munit^3$}) &= \frac{1}{8\pi k}|\vc{E}|^2\\
(\text{energy stored in the magnetic field per $\munit^3$}) &= \frac{c^2}{8\pi k}|\vc{B}|^2
\end{align*}
The idea here is that $k/c^2$ is the magnetic version of the electric quantity $k$, the $1/c^2$ representing the fact
that magnetism is a relativistic effect.

\pagebreak

\begin{eg}{Getting killed by a solenoid}
 Solenoids are very common electrical devices, but they can
be a hazard to someone who is working on them. Imagine a
solenoid that initially has a DC current passing through it.
The current creates a magnetic field inside and around it,
which contains energy. Now suppose that we break the
circuit. Since there is no longer a complete circuit,
current will quickly stop flowing, and the magnetic field
will collapse very quickly. The field had energy stored in
it, and even a small amount of energy can create a dangerous
power surge if released over a short enough time interval.
It is prudent not to fiddle with a solenoid that has current
flowing through it, since breaking the circuit could be
hazardous to your health.

 As a typical numerical estimate, let's assume a 40 cm $\times$
40 cm $\times$ 40 cm solenoid with an interior magnetic field of
1.0 T (quite a strong field). For the sake of this rough
estimate, we ignore the exterior field, which is weak, and
assume that the solenoid is cubical in shape. The energy
stored in the field is

\begin{align*}
 (\text{energy per unit volume})(\text{volume})
 &= \frac{c^2}{8\pi k}|\vc{B}|^2V\\
 &= 3\times10 ^4\ \junit
\end{align*}
That's a lot of energy!
\end{eg}

<% end_sec() %> % Energy in the magnetic field
<% end_sec() %>
<% begin_sec("The universal speed $c$") %>\label{sec:universal-speed-c}
Let's think a little more about the role of the 45-degree diagonal in the Lorentz transformation.
Slopes on these graphs are interpreted as velocities.
This line has a slope of 1 in relativistic units, but that slope corresponds to $c$ in ordinary metric units.
We already know that the relativistic distance unit must
be extremely large compared to the relativistic time unit, so $c$ must be extremely large.
Now note what happens when we perform a Lorentz transformation: this particular line gets stretched, but the new version
of the line lies right on top of the old one, and its slope stays the
same. In other words, if one observer says  that something has a velocity equal to $c$, every other observer will agree
on that velocity as well. (The same thing happens with $-c$.) 

<% marg(80) %>
<%
  fig(
    'ring-laser-gyro',
    %q{If you've flown in a jet plane, you can thank relativity for helping you to avoid
crashing into a mountain or an ocean. The figure shows a standard
piece of navigational equipment called a ring laser gyroscope. A beam of light is
split into two parts, sent around the perimeter of the device, and reunited. Since
light travels at the universal speed $c$, which is constant, we expect the two parts to come back together at the
same time. If they don't, it's evidence that the device has been rotating. The plane's
computer senses this and notes how much rotation has accumulated.}
  )
%>
<% end_marg %>

<% begin_sec("Velocities don't simply add and subtract.") %>\label{relativistic-combination-of-vel}\index{velocity!addition of!relativistic}
This is counterintuitive, since we expect velocities to
add and subtract in relative motion. If a dog is running away from me at
5 m/s relative to the sidewalk, and I run after it at 3 m/s,
the dog's velocity in my frame of reference is 2 m/s.
According to everything we have learned about motion (p.~\pageref{vel-addition-newtonian}), the
dog must have different speeds in the two frames: 5 m/s in
the sidewalk's frame and 2 m/s in mine. But velocities are measured by dividing a distance by a time, and
both distance and time are distorted by relativistic effects, so we actually shouldn't expect the ordinary
arithmetic addition of velocities to hold in relativity; it's an approximation that's valid at velocities
that are small compared to $c$.
<% end_sec %>

<% begin_sec("A universal speed limit") %>\label{c-as-speed-limit}
For example, suppose Janet takes a trip in a
spaceship, and accelerates until she is moving at $0.6c$ relative to the
earth. She then launches a space probe in the forward
direction at a speed relative to her ship of $0.6c$. We might think that the
probe was then moving at a velocity of $1.2c$, but in fact the answer is still less 
than $c$ (problem \ref{hw:six-tenths-c-twice}, page \pageref{hw:six-tenths-c-twice}).
This is an example of a more general fact about relativity, which is that $c$ represents
a universal speed limit. This is required by causality, as shown in figure \figref{speed-limit}.
<% end_sec %>
<% marg(100) %>
<%
  fig(
    'speed-limit',
    %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.
       In the new frame, shown by the parallelogram, A happens after $t=0$, but B happens before $t=0$; that is,
       B happens before A. The time ordering of the two events has been reversed. This can only happen because
       events A and B are very close together in time and fairly far apart in space. The line segment
       connecting A and B has a slope greater than 1, meaning that if we wanted to be present at both events, we would
       have to travel at a speed greater than $c$ (which equals 1 in the units used on this graph). You will find that if
       you pick any two points for which the slope of the line segment connecting them is less than 1, you can never get them
       to straddle the new $t=0$ line in this funny, time-reversed way. Since different observers disagree on the time order
       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
       we would have time-travel paradoxes. The conclusion is that $c$ is the maximum speed of cause and effect in relativity.
    }
  )
%>
<% end_marg %>

<% begin_sec("Light travels at $c$.") %>\label{sec:light-travels-at-c}
Now consider a beam of light. We're used to talking casually about the ``speed of light,'' but what does that really
mean? Motion is relative, so normally if we want to talk about a velocity, we have to specify what it's measured
relative to. A sound wave has a certain speed relative to the air, and a water wave has its own speed relative to the
water. If we want to measure the speed of an ocean wave, for example, we should make sure to measure it in a frame
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,
as when rays of sunlight travel to earth. This seems like a paradox: light is supposed to have a specific speed,
but there is no way to decide what frame of reference to measure it in. The way out of the paradox is that light
must travel at a velocity equal to $c$. Since all observers agree on a velocity of $c$, regardless of their frame
of reference, everything is consistent.
<% end_sec %>

<% begin_sec("The Michelson-Morley experiment") %>
The constancy of the speed of light had in fact already been
observed when Einstein was an 8-year-old boy, but because nobody could
figure out how to interpret it, the result was largely ignored.
In 1887 Michelson and Morley set up a clever apparatus to
measure any difference in the speed of light beams traveling
east-west and north-south. The motion of the earth around
the sun at 110,000 km/hour (about 0.01\% of the speed of
light) is to our west during the day. Michelson and Morley
believed that light was a vibration of a mysterious medium called the ether, so they expected that the
speed of light would be a fixed value relative to the ether.
As the earth moved through the ether, they thought they
would observe an effect on the velocity of light along an
east-west line. For instance, if they released a beam of
light in a westward direction during the day, they expected
that it would move away from them at less than the normal
speed because the earth was chasing it through the ether.
They were surprised when they found that the expected 0.01\%
change in the speed of light did not occur.\index{Michelson-Morley experiment}\index{ether}
<%
  fig(
    'michelson',
    %q{The Michelson-Morley experiment, shown in photographs, and drawings from the original 1887 paper.
       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
       mirror $\zu{h}_1$. The two half-intensity parts of the beam are reflected by the mirrors at a and b, reunited,
       and observed in the telescope, t. If the earth's surface was supposed to be moving through the ether,
       then the times taken by the two light waves to pass through the moving ether would be unequal, and
       the resulting time lag would be detectable by observing the interference between the waves when they were reunited.
       2. In the real apparatus, the light beams were reflected multiple times. The effective length of each arm was
       increased to 11 meters, which greatly improved its sensitivity to the small expected difference in the speed of light.
       3. In an earlier version of the experiment, they had run into problems with its ``extreme sensitiveness to vibration,''
       which was ``so great that it was impossible to see the interference fringes except at brief intervals \ldots even at
       two o'clock in the morning.'' They therefore mounted the whole thing on a massive stone floating in a pool of mercury,
       which also made it possible to rotate it easily. 4. A photo of the apparatus.
    },
    {
      'width'=>'wide',
      'sidecaption'=>false,
      'sidepos'=>'t',
      'anonymous'=>true
    }
  )
%>

<% end_sec %> % The Michelson-Morley experiment

<% end_sec %> % Universality of $c$

\pagebreak

\startdqs

\vspace{15mm}

\begin{dq}
The figure shows a famous thought experiment devised by Einstein. A train is moving at
constant velocity to the right when bolts of lightning strike the ground near its front and
back. Alice, standing on the dirt at the midpoint of the flashes, observes that the light
from the two flashes arrives simultaneously, so she says the two strikes must have occurred
simultaneously. Bob, meanwhile, is sitting aboard the train, at its middle. 
He passes by Alice at the moment when Alice later figures out that the flashes happened.
Later, he receives flash 2, and then flash 1. He infers that since both flashes traveled
half the length of the train, flash 2 must have occurred first.
How can this be reconciled with Alice's belief that the flashes were simultaneous?
Explain using a graph.
\end{dq}

<%
 fig(
   'einstein-train',
   '',
   {
     'width'=>'wide',
     'sidecaption'=>true,
     'anonymous'=>true
   }
 )
%>

\vspace{40mm}

\begin{dq}
Use a graph to resolve the following relativity paradox.
Relativity says that in one frame of reference, event A could happen before event B, but in someone else's
frame B would come before A. How can this be? Obviously the two people could meet up at A and talk
as they cruised past each other. Wouldn't they have to
agree on whether B had already happened?
\end{dq}

\pagebreak

\begin{dq}\label{dq:machine-gun-ftl}
The machine-gunner in the figure sends out a spray of bullets. Suppose that the bullets are being shot into
       outer space, and that the distances traveled are trillions of miles (so that the human figure
       in the diagram is not to scale). After a long time, the bullets reach the points shown with dots
       which are all equally far from the gun. Their arrivals at those points are events A through E,
       which happen at different times. The chain of impacts extends across space at a speed greater
       than $c$. Does this violate special relativity?
\end{dq}

<%
  fig(
    'machine-gun-ftl',
    %q{Discussion question \ref{dq:machine-gun-ftl}.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true,
      'anonymous'=>true
    }
  )
%>


<% begin_sec("Induction",4,'induction') %>\index{induction}

<% begin_sec("The principle of induction") %>
Physicists of Michelson and Morley's generation thought that light was
a mechanical vibration of the ether, but we now know that it is a ripple
in the electric and magnetic fields. With hindsight, relativity essentially
requires this:
\begin{enumerate}
\item Relativity requires that changes in any field propagate as waves at a finite speed (p.~\pageref{subsec:time-delays}).
\item Relativity says that if a wave has a fixed speed but is not a mechanical disturbance in a physical medium, then it
must travel at the universal velocity $c$ (p.~\pageref{sec:light-travels-at-c}).
\end{enumerate}
What is less obvious is that there are not two separate kinds of waves, electric and magnetic.
In fact an electric wave can't exist without a magnetic one, or a magnetic one without an electric one.
This new fact follows from the principle of induction, which was discovered experimentally
by Faraday\index{Faraday, Michael} in 1831, seventy-five years before Einstein. Let's state Faraday's idea first, and
then see how something like it must follow 
inevitably from relativity:
<% marg(0) %>
<%
  fig(
    'induced-field-geometry',
    %q{%
      The geometry of induced fields. The induced field tends to
              form a whirlpool pattern around the change in the vector producing it.
              Note how they circulate in opposite directions.
    }
  )
%>
<% end_marg %>

\begin{lessimportant}[the principle of induction]
Any electric field that changes over time will produce a
magnetic field in the space around it.

\noindent Any magnetic field that changes over time will produce an
electric field in the space around it.
\end{lessimportant}

The induced field tends to have a whirlpool pattern, as
shown in figure \figref{induced-field-geometry}, but the whirlpool image is not to be
taken too literally; the principle of induction really just
requires a field pattern such that, if one inserted a
paddlewheel in it, the paddlewheel would spin. All of the
field patterns shown in figure \figref{curl-concept} are ones that
could be created by induction; all have a counterclockwise ``curl'' to them.

<%
  fig(
    'curl-concept',
    %q{Three fields with counterclockwise ``curls.''},
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

<%
  fig(
    'relativity',
    %q{%
        Observer 1 is at rest with respect to the bar magnet, and
        observes magnetic fields that have different strengths at different
        distances from the magnet. Observer 2, hanging out in the region to the left of the
        magnet, sees the magnet moving toward her, and detects that the
        magnetic field in that region is getting stronger as time passes. 
    },
    {
      'width'=>'wide',
      'sidecaption'=>true,
      'narrowfigwidecaption'=>true,
      'float'=>false,
      'anonymous'=>false
    }
  )
%>


Figure \figref{relativity} shows an example of the fundamental
reason why a changing $\vc{B}$ field must create an $\vc{E}$ field.
In section \ref{sec:magnetic-interactions} we established that according to
relativity, what one observer describes as a purely magnetic field,
an observer in a different state of motion describes as a mixture of magnetic and electric fields.
This is why there must be both an $\vc{E}$ and a $\vc{B}$ in observer 2's frame.
Observer 2 cannot explain the electric field as coming from any
charges. In frame 2, the $\vc{E}$ can only be explained as an effect
caused by the changing $\vc{B}$.

Observer 1 says, ``2 feels a changing $\vc{B}$ field because he's moving through
a static field.'' Observer 2 says, ``I feel a changing $\vc{B}$ because the
magnet is getting closer.'' 

Although this argument doesn't prove the ``whirlpool'' geometry, we can
verify that the fields I've drawn in figure \figref{relativity} are consistent
with it.
The $\Delta\vc{B}$ vector is upward,
        and the electric field has a curliness to it: a paddlewheel inserted
        in the electric field would spin clockwise as seen from above, since
        the clockwise torque made by the strong electric field on the right is
        greater than the counterclockwise torque made by the weaker electric
        field on the left.

\begin{eg}{The generator}\index{generator}
A generator, \figref{generator}, consists of a permanent magnet that
rotates within a coil of wire. The magnet is turned by a
motor or crank, (not shown). As it spins, the nearby
magnetic field changes. According to the principle of
induction, this changing magnetic field results in an
electric field, which has a whirlpool pattern. This electric
field pattern creates a current that whips around the coils
of wire, and we can tap this current to light the lightbulb.
\end{eg}
<% marg(33) %>
<%
  fig(
    'generator',
    %q{A generator.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'transformer',
    %q{A transformer.}
  )
%>
<% end_marg %>

<% self_check('alternator',<<-'SELF_CHECK'
When you're driving a car, the engine recharges the
battery continuously using a device called an alternator,
which is really just a generator like the one shown on the
previous page, except that the coil rotates while the
permanent magnet is fixed in place. Why can't you use the
alternator to start the engine if your car's battery is dead?
  SELF_CHECK
  ) %>

\begin{eg}{The transformer}
In example \ref{hvtransmission} on p.~\pageref{hvtransmission} we discussed the advantages of transmitting
power over electrical lines using high voltages and low
currents. However, we don't want our wall sockets to operate
at 10000 volts! For this reason, the electric company uses a
device called a \index{transformer}transformer, \figref{transformer}, to
convert to lower voltages and higher currents inside your
house. The coil on the input side creates a magnetic field.
Transformers work with alternating current, so the magnetic
field surrounding the input coil is always changing. This
induces an electric field, which drives a current around the output coil.

                If both coils were the same, the arrangement would be
symmetric, and the output would be the same as the input,
but an output coil with a smaller number of coils gives the
electric forces a smaller distance through which to push the
electrons. Less mechanical work per unit charge means a
lower voltage. Conservation of energy, however, guarantees
that the amount of power on the output side must equal the
amount  put in originally, $I_{in}V_{in} = I_{out}V_{out}$,
so this reduced voltage must be accompanied by an increased current.

\end{eg}

<% end_sec() %>
<% end_sec() %>

<% begin_sec("Electromagnetic waves",0,'em-waves') %>\index{electromagnetic waves}\index{waves!electromagnetic}

The most important consequence of induction is the existence
of electromagnetic waves. Whereas a gravitational wave would
consist of nothing more than a rippling of gravitational
fields, the principle of induction tells us that there can
be no purely electrical or purely magnetic waves. Instead,
we have waves in which there are both electric and magnetic
fields, such as the sinusoidal one shown in the figure.
Maxwell proved that such waves were a direct consequence of
his equations, and derived their properties mathematically.
The derivation would be beyond the mathematical level of
this book, so we will just state the results.

\enlargethispage{-\baselineskip}

<%
  fig(
    'em-wave',
    %q{An electromagnetic wave.},
    {
      'width'=>'wide',
      'sidecaption'=>true,
      'anonymous'=>true
    }
  )
%>

A sinusoidal electromagnetic wave has the geometry shown above.
The $\vc{E}$ and $\vc{B}$ fields are perpendicular
to the direction of motion, and are also perpendicular to
each other. If you look along the direction of motion of the
wave, the $\vc{B}$ vector is always 90 degrees clockwise from the
$\vc{E}$ vector. In a plane wave, the magnitudes of the two fields are related by
$|\vc{E}|=c|\vc{B}|$.

How is an electromagnetic wave created? It could be emitted,
for example, by an electron orbiting an atom or currents
going back and forth in a transmitting antenna. In general
any accelerating charge will create an electromagnetic wave,
although only a current that varies sinusoidally with time
will create a sinusoidal wave. Once created, the wave
spreads out through space without any need for charges or
currents along the way to keep it going. As the electric
field oscillates back and forth, it induces the magnetic
field, and the oscillating magnetic field in turn creates
the electric field. The whole wave pattern propagates
through empty space at the velocity $c$.

\begin{eg}{Einstein's motorcycle}\label{eg:einstein-motorcycle}
As a teenage physics student, Einstein imagined the following paradox. 
(See p.~\pageref{qualitative-doppler-light}.) What if
he could get on a motorcycle and ride at speed $c$, alongside a beam of light?
In his frame of reference, he observes constant electric and magnetic fields.
But only a \emph{changing} electric field can induce a magnetic field, and
only a \emph{changing} magnetic field can induce an electric field. The laws
of physics are violated in his frame, and this seems
to violate the principle that all frames of reference are equally valid.

The resolution of the paradox is that $c$ is a universal speed limit, so the
motorcycle can't be accelerated to $c$. Observers can never be at rest relative
to a light wave, so no observer can have a frame of reference in which a light
wave is observed to be at rest.
\end{eg}

\begin{eg}{Reflection}\label{eg:em-wave-refl-conductor}
The wave in figure \figref{em-wave-refl-conductor} hits a silvered mirror.
The metal is a good conductor, so it has constant voltage throughout, and
the electric field equals zero inside it: the wave
doesn't penetrate and is 100\%
reflected. If the electric field is to be zero at the surface as well, the reflected wave must have
its electric field inverted (p.~\pageref{wave-inversion}), so that the incident and reflected
fields cancel there.

But the magnetic field of the reflected wave is \emph{not} inverted. This is because the reflected
wave, when viewed along its leftward direction of propagation, needs to have its $\vc{B}$ vector
90 degrees clockwise from its $\vc{E}$ vector.
\end{eg}

<% begin_sec("Polarization") %>
\index{polarization}

Two electromagnetic waves traveling in the same direction
through space can differ by having their electric and
magnetic fields in different directions, a property of the
wave called its polarization.

<% end_sec() %>

\enlargethispage{-\baselineskip}

<% begin_sec("Light is an electromagnetic wave") %>

Once Maxwell had derived the existence of electromagnetic
waves, he became certain that they were the same phenomenon
as light. Both are transverse waves (i.e., the vibration is
perpendicular to the direction the wave is moving), and the
velocity is the same.
<% marg(0) %>
<%
  fig(
    'em-wave-refl-conductor',
    %q{Example \ref{eg:em-wave-refl-conductor}. The incident and reflected waves are drawn offset from
       each other for clarity, but are actually on top of each other so that their fields superpose.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'hertz',
    %q{Heinrich Hertz (1857-1894).},
    {'anonymous'=>true}
  )
%>
<% end_marg %>

Heinrich \index{Hertz, Heinrich}Hertz (for whom the unit of
frequency is named) verified Maxwell's ideas experimentally.
Hertz was the first to succeed in producing, detecting, and
studying electromagnetic waves in detail using antennas and
electric circuits. To produce the waves, he had to make
electric currents oscillate very rapidly in a circuit. In
fact, there was really no hope of making the current reverse
directions at the frequencies of $10^{15}$ Hz possessed by
visible light. The fastest electrical oscillations he could
produce were $10^9$ Hz, which would give a wavelength of
about 30 cm. He succeeded in showing that, just like light,
the waves he produced were polarizable, and could be
reflected and refracted (i.e., bent, as by a lens), and he
built devices such as parabolic mirrors that worked
according to the same optical principles as those employing
light. Hertz's results were convincing evidence that light
and electromagnetic waves were one and the same.

<% end_sec() %>
<% begin_sec("The electromagnetic spectrum") %>\index{spectrum!electromagnetic}\index{electromagnetic spectrum}

Today, electromagnetic waves with frequencies in the range
employed by Hertz are known as radio waves. Any remaining
doubts that the ``Hertzian waves,'' as they were then
called, were the same type of wave as light waves were soon
dispelled by experiments in the whole range of frequencies
in between, as well as the frequencies outside that range.
In analogy to the spectrum of visible light, we speak of the
entire electromagnetic spectrum, of which the visible
spectrum is one segment.

<%
 fig(
   'em-spectrum',
   '',
   {'width'=>'fullpage'}
 )
%>

The terminology for the various parts of the spectrum is
worth memorizing, and is most easily learned by recognizing
the logical relationships between the wavelengths and the
properties of the waves with which you are already familiar.
 Radio waves have wavelengths that are comparable to the
size of a radio antenna, i.e., meters to tens of meters.
Microwaves were named that because they have much shorter
wavelengths than radio waves; when food heats unevenly in a
microwave oven, the small distances between neighboring hot
and cold spots is half of one wavelength of the standing
wave the oven creates. The infrared, visible, and
ultraviolet obviously have much shorter wavelengths, because
otherwise the wave nature of light would have been as
obvious to humans as the wave nature of ocean waves. To
remember that ultraviolet, x-rays, and gamma rays all lie on
the short-wavelength side of visible, recall that all three
of these can cause cancer. (As we'll discuss later in the
course, there is a basic physical reason why the cancer-causing
disruption of DNA can only be caused by very short-wavelength
electromagnetic waves. Contrary to popular belief,
microwaves cannot cause cancer, which is why we have
microwave ovens and not x-ray ovens!)

\vspace{20mm}

\begin{eg}{Switching frames of reference}\label{eg:null-field}
If we switch to a different frame of
reference, a legal light wave should still be legal. Consider the requirement $\vc{E}=c\vc{B}$,
in the case where observer 1 says observer 2 is trying to run away from the wave.
In figure \figref{e-b-lorentz} on p.~\pageref{fig:e-b-lorentz}, we saw that the familiar parallelogram
graphs described the transformation of electric and magnetic fields from one frame of
reference to another. These pictures are intended to be used in units where $c=1$, so
the requirement for the fields becomes $\vc{E}=\vc{B}$, and such a combination of fields is represented
by a dot on the diagonal, which is the same line in both frames.
\end{eg}

\vspace{20mm}


<% marg(60) %>
<%
  fig(
    'null-field',
    %q{%
      Example \ref{eg:null-field}: an electromagnetic wave that is legal in one frame of reference is legal in another.
      As in figure \figref{e-b-lorentz} on p.~\pageref{fig:e-b-lorentz}, the each frame of reference is
      in motion relative to the other along the $x$ axis.
      If the wave's electric field
      is aligned with the $y$ axis, and its magnetic field with $z$, then $x$ is also the direction in which the
      wave is moving, as required for our example. 
    },
    {'anonymous'=>true}
  )
%>
<% end_marg %>


\begin{eg}{Why the sky is blue}
When sunlight enters the upper atmosphere, a particular air mole\-cule
finds itself being washed over by an electromagnetic wave of frequency $f$.
The molecule's charged
particles (nuclei and electrons) act like oscillators being driven by
an oscillating force, and respond by vibrating at the same frequency $f$.
Energy is sucked out of the incoming beam of sunlight and converted into the
kinetic energy of the oscillating particles. However, these particles are
accelerating, so they act like little radio antennas that put the energy back
out as spherical waves of light that spread out in all directions.
An object oscillating at a frequency $f$ has an acceleration proportional to
$f^2$, and an accelerating charged particle creates an electromagnetic wave
whose fields are proportional to its acceleration, so the field of the
reradiated spherical wave is proportional to $f^2$. The energy of a field
is proportional to the square of the field, so the energy of the reradiated wave
is proportional to $f^4$. Since blue light has about twice the frequency of
red light, this process is about $2^4=16$ times as strong for blue as for
red, and that's why the sky is blue.
\end{eg}

\vfill

<% end_sec() %> % The electromagnetic spectrum


<%
  fig(
    'maxwellian-momentum-of-light',
    %q{An electromagnetic wave strikes an ohmic surface. The wave's electric field causes currents to flow up and down.
            The wave's magnetic field then acts on these currents, producing a force in the direction of the wave's propagation.
            This is a pre-relativistic argument that light must possess inertia.},
    {
      'width'=>'wide',
      'narrowfigwidecaption'=>true,
      'float'=>false
    }
  )
%>

\spacebetweenfigs

<%
  fig(
    'nichols-radiometer',
    %q{A simplified drawing of the 1903 experiment by Nichols and Hull that verified the predicted momentum
       of light waves. Two circular mirrors were hung from a fine quartz fiber, inside an evacuated
       bell jar. A 150 mW beam of light was shone on one of the mirrors for 6 s, producing a tiny rotation,
       which was measurable by an optical lever (not shown). The force was within 0.6\%
       of the theoretically predicted value
       (problem \ref{hw:ultrarelativistic} on p.~\pageref{hw:ultrarelativistic})
       of $0.001\ \mu\nunit$.
       For comparison, a short clipping of a single human hair weighs $\sim 1\ \mu\nunit$.},
    {
      'width'=>'wide',
      'narrowfigwidecaption'=>true,
      'float'=>false
    }
  )
%>

<% begin_sec("Momentum of light") %>\index{light!momentum of}\index{electromagnetic wave!momentum of}\index{momentum!of light}

Newton defined momentum as $mv$, which would imply that light, which has no mass,
should have no momentum. But Newton's laws only work at speeds small compared to
the speed of light, and light travels \emph{at} the speed of light.
In fact, it's straightforward to show that electromagnetic waves have momentum. If a light wave strikes
an ohmic surface, as in figure \figref{maxwellian-momentum-of-light}, the wave's electric field causes charges to vibrate back and forth
in the surface. These currents then
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
the resulting force is in the direction of propagation of the wave. A light wave
has momentum and inertia. This is explored further in 
problem \ref{hw:ultrarelativistic} on p.~\pageref{hw:ultrarelativistic}. Figure
\figref{nichols-radiometer} shows an experimental confirmation.

<% end_sec() %> % Momentum of light
<% end_sec() %> % Electromagnetic waves
<% begin_sec("Symmetry and handedness",nil,'',{'optional'=>true}) %>\index{symmetry}\index{handedness}

Imagine that you establish
radio contact with an alien on another planet. Neither of
you even knows where the other one's planet is, and you
aren't able to establish any landmarks that you both
recognize. You manage to learn quite a bit of each other's
languages, but you're stumped when you try to establish the
definitions of left and right (or, equivalently, clockwise
and counterclockwise). Is there any way to do it?

<% marg(0) %>
<% fig(
     '../../../share/mechanics/figs/swan-lake-symmetry',
     %q{In this scene from Swan Lake, the choreography has a symmetry with respect to left and right.}
   )
%>
\label{fig:swan-lake-symmetry}
<% end_marg %>

If there was any way to do it without reference to external
landmarks, then it would imply that the laws of physics
themselves were asymmetric, which would be strange. Why
should they distinguish left from right? The gravitational
field pattern surrounding a star or planet looks the same in
a mirror, and the same goes for electric fields.
The field patterns shown in section \ref{sec:calculating-magnetism} seem to violate this
principle, but do they really? Could you use these patterns
to explain left and right to the alien? In fact, the answer
is no. If you look back at the definition of the magnetic
field in section \ref{sec:b-field}, it also contains a reference to
handedness: the counterclockwise direction of the loop's
current as viewed along the magnetic field. The aliens might
have reversed their definition of the magnetic field, in
which case their drawings of field patterns would look like
mirror images of ours.

<% marg(80) %>
<% fig(
     '../../../share/mechanics/figs/c-s-wu-with-beamline',
     %q{C.S.~Wu},
     {'anonymous'=>true}
   )
%>
\label{fig:c-s-wu-with-beamline}
\spacebetweenfigs
<%
  fig(
    'six-tenths-c',
    %q{%
      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
         short one is half its former length, and the area is the same as before.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    '../../../lm/vw/figs/doppler',
    %q{%
      The pattern of waves made by a point source moving to the right
      across the water. Note the shorter wavelength of the forward-emitted waves and the longer
      wavelength of the backward-going ones.
    },
    {'suffix'=>'2'} # doesn't work, gets labeled by subsec number ... why?
  )
%>
<% end_marg %>

Until the middle of the twentieth century, physicists
assumed that any reasonable set of physical laws would have
to have this kind of symmetry between left and right. An
asymmetry would be grotesque. Whatever their aesthetic
feelings, they had to change their opinions about reality
when experiments by C.S.~Wu et al.~showed that the \index{weak nuclear
force}weak \index{nuclear forces}nuclear force (section \ref{sec:energy-in-fields})
violates right-left symmetry! It is still a mystery why
right-left symmetry is observed so scrupulously in general,
but is violated by one particular type of physical process.

<% end_sec() %>
<% begin_sec("Doppler shifts and clock time",nil,'doppler-and-clock',{'optional'=>true}) %>

Figure \figref{six-tenths-c} shows our now-familiar method of visualizing a Lorentz transformation,
in a case where the numbers come out to be particularly simple. This diagram has two geometrical features that we have referred to
before without digging into their physical significance: the \emph{stretch factor} of the diagonals, and the \emph{area}. In this section
we'll see that the former can be related to the Doppler effect, and the latter to clock time.


<% begin_sec("Doppler shifts of light",nil,'doppler-light') %>\index{light!Doppler effect for}\index{Doppler effect!for light}
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
different objects: the source, the receiver, and the medium. But light waves don't have a medium. Therefore Doppler shifts of
light can only depend on the relative motion of the source and observer.

One simple case is the one in which the relative motion of the source and the receiver is perpendicular to the line connecting them.
That is, the motion is transverse.
Nonrelativistic Doppler shifts happen because the distance between the source and receiver is changing, so in nonrelativistic
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.
For example, the photo
% Can't get it to label correctly, so can't do figref.
shows shortened and lengthened wavelengths to the right and left, along the source's line of motion,
but an observer above or below the source measures just the normal, unshifted wavelength and frequency. But relativistically, we have
a time dilation effect, so for light waves emitted transversely, there is a Doppler shift of $1/\mygamma$ in frequency (or $\mygamma$ in wavelength).

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
receding from one another. For example, distant galaxies are receding from our galaxy due to the expansion of the universe, and this expansion was
originally detected because Doppler shifts toward the red (low-frequency) end of the spectrum were observed. 

Nonrelativistically, we would expect
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
objects: the source, the wave's medium, and the receiver. Relativistically, things get simpler, because light isn't a vibration of a physical medium,
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
increasing.

The square in figure \figref{doppler-geometry} is the ``graph paper'' used by someone who considers the source to be at rest, while
the parallelogram plays a similar role for the receiver. The figure is drawn for the case where $v=3/5$ (in units where $c=1$),
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
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
shift is a factor of 1/2 in frequency. That is, the squish factor of the short diagonal is interpreted as the Doppler shift.
To get this as a general equation for velocities other than 3/5, one can show by straightforward fiddling with the result of
part c of problem \ref{hw:gamma-derivation} on p.~\pageref{hw:gamma-derivation} that
the Doppler shift is
\begin{equation*}
  D(v) = \sqrt{\frac{1-v}{1+v}} \qquad .
\end{equation*}
Here $v>0$ is the case where the source and receiver are getting farther apart, $v<0$ the case where they are approaching.
(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
use positive values of $v$ in the case of cosmological red-shifts, which are the most important application.)
<% marg(100) %>
<%
  fig(
    'doppler-geometry',
    %q{%
      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.
      At P, the source emits another wave crest, and at Q the receiver receives it.
    }
  )
%>
<% end_marg %>

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.
When I first learned relativity, the thing that caused me the most pain was understanding how each observer could say that the
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,
then naturally I would see everybody \emph{else} as being \emph{slow}. Shouldn't the same apply to relativity? But suppose Alice and Betty
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
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
over the radio two seconds apart, but the situation is exactly symmetric, and Alice hears the same thing if Betty claps.
Each twin analyzes the situation using a diagram identical to \figref{doppler-geometry}, and attributes her sister's
observations to a complicated combination of time distortion, the time taken by the radio signals to propagate, and
the motion of her twin relative to her.

<% self_check('doppler-approaching',<<-'SELF_CHECK'
Turn your book upside-down and reinterpret figure \figref{doppler-geometry}.
  SELF_CHECK
  ) %>

\begin{eg}{A symmetry property of the Doppler effect}\label{eg:doppler-abc}
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
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
this product must equal 1, so we must have $D(-v)D(v)=1$, which can be verified directly from the equation.
\end{eg}

\begin{eg}{The Ives-Stilwell experiment}\index{Ives-Stilwell experiment}
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.
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 $c$.
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,
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
have been detectable in their experiment. In 2003, Saathoff et al.~carried out an extremely precise version of the Ives-Stilwell technique
with $\zu{Li}^+$ ions moving at 6.4\% of $c$. The frequencies observed, in units of MHz, were:

\noindent\begin{tabular}{rl}
$f_\zu{o}$ &=      $546466918.8  \pm 0.4$ \\
           & \hfill (unshifted frequency)\\
$f_\zu{o}D(-v)$ & =          $582490203.44 \pm .09$ \\
           & \hfill (shifted frequency, forward)\\
$f_\zu{o} D(v)$ & =          $512671442.9  \pm 0.5$ \\
           & \hfill (shifted frequency, backward)\\
$\sqrt{f_\zu{o}D(-v)\cdot f_\zu{o} D(v)}$ &= $546466918.6  \pm 0.3$ 
\end{tabular}

\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
relativistically because $D(v)D(-v)$ is supposed to equal 1. The agreement extends to 9 significant figures, whereas
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.
The spectacular agreement with theory has made this experiment a lightning rod for
anti-relativity kooks.
\end{eg}

We saw on p.~\pageref{relativistic-combination-of-vel} that relativistic velocities should not be expected to be exactly additive,
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
at $0.6c$ and B relative to C at $0.6c$ --- the result \emph{not} being $1.2c$.\index{velocity!addition of!relativistic}
The relativistic Doppler shift provides a simple way of deriving a general equation for the relativistic combination of velocities;
problem \ref{hw:rel-vel-addition} on p.~\pageref{hw:rel-vel-addition} guides you through the steps of this derivation.

<% end_sec() %>
<% begin_sec("Clock time",nil,'proper-time',{'optional'=>true}) %>
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.
We used this only as a stepping stone toward the Lorentz transformation, but it is natural to wonder whether
this kind of area has any physical interest of its own.

The equal-area result is not relativistic, since the proof never appeals to property 5 on page \pageref{spacetime-properties}. Cases
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
figure \figref{galilean-boost} on p.~\pageref{fig:galilean-boost}, where the distortion of the rectangle could be accomplished by cutting it into
vertical slices and then displacing the slices upward without changing their areas.

But the area does have a nice interpretation in the relativistic
case. Suppose that we have events A (Charles VII is restored to the throne) and B (Joan of Arc is executed). Now imagine
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
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
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
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 $\pm 1$.
We call this type of rectangle a light-rectangle, because its sides could represent the motion of rays of light.

<%
  fig('light-rectangles',
    %q{1.~The gray light-rectangle represents the set of all events such as P that could be visited after A and before B.\\\\
       2.~The rectangle becomes a square in the frame in which A and B occur at the same location in space.\\\\
       3.~The area of the dashed square is $\tau^2$, so the area of the gray square is $\tau^2/2$.},
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

The area of this rectangle will be the same regardless of one's frame of reference. In particular, we could choose a
special frame of reference, panel 2 of the figure, such that A and B occur
in the same place. (They do not occur at the same place, for example, in the sun's frame,
because the earth is spinning and going around the sun.) Since the speed $c$, which equals 1 in our units,
is the same in all frames of reference, and the sides of
the rectangle had slopes $\pm 1$ in frame 1, they must still have slopes $\pm 1$ in frame 2.
The rectangle becomes a square with its diagonals parallel to the $x$ and $t$ axes, and the length of these diagonals equals the
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,
meeting up with the planet earth at the appointed time.
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.
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
on units, the expression must have the form $(\ldots)t^2+(\ldots)tx+(\ldots)x^2$, where each $(\ldots)$ represents a unitless constant.
The $tx$ coefficient must be zero by property 2 on p.~\pageref{spacetime-properties}. For consistency with figure \subfigref{light-rectangles}{3},
the $t^2$ coefficient must equal 1. Since the area vanishes for $x=t$, the $x^2$ coefficient must equal $-1$.}

When $|x|$ is greater than $|t|$, events A and B are so far apart in space and so close together in time that
it would be impossible to have a cause and effect relationship between them, since $c=1$ is the maximum speed of cause and effect.
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
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}

No matter what, $t^2-x^2$ is the same as measured in all frames of reference. Geometrically, it plays the same role
in the $x$-$t$ plane that ruler measurements play in the Euclidean plane. In Euclidean
geometry, the ruler-distance between any two points stays the same regardless of rotation, i.e., regardless of 
the angle from which we view the scene; according to the Pythagorean theorem, the square of this distance is $x^2+y^2$.
In the $x$-$t$ plane, $t^2-x^2$ stays the same regardless of the
frame of reference.

To avoid overloading the reader with terms to memorize, some commonly used terminology is
relegated to problem \ref{hw:geroch-interval} on p.~\pageref{hw:geroch-interval}.
<% end_sec() %> % Clock time
<% end_sec() %> % Doppler shifts and clock time
\begin{summary}

\begin{vocab}

\vocabitem{magnetic field}{a field of force, defined in terms of the
torque exerted on a test dipole}

\vocabitem{magnetic dipole}{an object, such as a current loop, an atom,
or a bar magnet, that experiences torques due to magnetic
forces; the strength of magnetic dipoles is measured by
comparison with a standard dipole consisting of a square
loop of wire of a given size and carrying a given amount of current}

\vocabitem{induction}{the production of an electric field by a changing
magnetic field, or vice-versa}

\end{vocab}

\begin{notation}
\notationitem{$\vc{B}$}{the magnetic field}

\notationitem{$D_m$}{magnetic dipole moment}

\end{notation}

\begin{summarytext}

The magnetic field is defined in terms of
the torque on a magnetic test dipole. It has no sources or
sinks; magnetic field patterns never converge on or
diverge from a point.

Relativity dictates a maximum speed limit $c$ for cause and effect. This
speed is the same in all frames of reference.

Relativity requires that the magnetic and electric fields be intimately related. The
principle of induction states that any changing electric
field produces a magnetic field in the surrounding space,
and vice-versa. These induced fields tend to form whirlpool patterns.

The most important consequence of the principle of induction
is that there are no purely magnetic or purely electric
waves. Electromagnetic disturbances
propagate outward at $c$ as combined magnetic and electric waves,
with a well-defined relationship between the magnitudes
and directions of the electric and magnetic fields. These electromagnetic waves are what light
is made of, but other forms of electromagnetic waves exist
besides visible light, including radio waves, x-rays, and gamma rays.
\end{summarytext}
\end{summary}

 %%===============================================================================

<% begin_hw_sec %>

<% begin_hw('six-tenths-c-twice') %>
The figure illustrates a Lorentz transformation using the conventions employed in section \ref{sec:x-t-distortion}.
For simplicity, the transformation chosen is one that lengthens one diagonal by a factor of 2. Since Lorentz transformations
preserve area, the other diagonal is shortened by a factor of 2. Let the original frame of reference, depicted with the square,
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 $0.6c$.
(b) Print out a copy of the page. With a ruler, draw a third parallelogram that represents a second successive Lorentz transformation, one
that lengthens the long diagonal by another factor of 2. Call this third frame C. Use measurements with a ruler to determine
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
out the way it does.\answercheck
% v=.6 exactly; combined v = tanh(2 atanh(.6))=.8824; verified graphically using inkscape that it's .8824
<% end_hw() %>

\vfill

\enlargethispage{3\baselineskip}

<%
  fig(   
    'hw-six-tenths-c-twice',
    '',
    {
      'width'=>'fullpage'
      #'sidecaption'=>true
    }
  )
%>

\pagebreak

\pagebreak

<% begin_hw('side-by-side-lasers') %>__incl(../../share/relativity/hw/side-by-side-lasers)<% end_hw() %>

\vfill

<% begin_hw('nestedsolenoids') %>
Consider two solenoids, one of which is smaller so that
it can be put inside the other. Assume they are long enough
so that each one only contributes significantly to the field
inside itself, and the interior fields are nearly uniform.
Consider the configuration where the small one is inside the
big one with their currents circulating in the same
direction, and a second configuration in which the currents
circulate in opposite directions. Compare the energies of
these configurations with the energy when the solenoids are
far apart. Based on this reasoning, which configuration is
stable, and in which configuration will the little solenoid
tend to get twisted around or spit out? [Hint: A stable
system has low energy; energy would have to be added to
change its configuration.]
<% end_hw() %>

\vfill

<% marg(0) %>
<%
  fig(
    'hw-nested-wire-loops',
    %q{Problem \ref{hw:nested-wire-loops}.}
  )
%>
<% end_marg %>
<% begin_hw('nested-wire-loops') %>__incl(hw/nested-wire-loops)<% end_hw() %>

\pagebreak

<% begin_hw('atom') %>
One model of the hydrogen atom has the electron circling
around the proton at a speed of $2.2\times10^6$ m/s, in an
orbit with a radius of 0.05 nm. (Although the electron and
proton really orbit around their common center of mass, the
center of mass is very close to the proton, since it is 2000
times more massive. For this problem, assume the proton is
stationary.) In homework problem \ref{hw:lpcurrent} on page \pageref{hw:lpcurrent}, you
calculated the electric current created.\hwendpart
(a) Now estimate the magnetic field created at the center 
of the atom by the electron. We are treating the circling
electron as a current loop, even though it's only a single particle.\answercheck\hwendpart
(b) Does the proton experience a nonzero force from the
electron's magnetic field? Explain.\hwendpart
(c) Does the electron experience a magnetic field from
the proton? Explain.\hwendpart
(d) Does the electron experience a magnetic field created by
its own current? Explain.\hwendpart
(e) Is there an electric force acting between the proton 
and electron? If so, calculate it.\answercheck\hwendpart
(f) Is there a gravitational force acting between the proton
and electron? If so, calculate it.\hwendpart
(g) An inward force is required to keep the electron in its
orbit -- otherwise it would obey Newton's first law and go
straight, leaving the atom. Based on your answers to the
previous parts, which force or forces (electric, magnetic
and gravitational) contributes significantly to this inward force?\hwendpart
[Based on a problem by Arnold Arons.]
<% end_hw() %>

\vfill

<% begin_hw('velocityfilter') %>__incl(hw/velocityfilter)<% end_hw() %>

\vfill

<% begin_hw('solenoid-u-quadruple-i') %>__incl(hw/solenoid-u-quadruple-i)<% end_hw() %>

\vfill

<% begin_hw('mystery-magnet',2) %>__incl(hw/mystery-magnet)<% end_hw() %>

\pagebreak

<% begin_hw('solenoid-sex') %>
Consider two solenoids, one of which is smaller so that
it can be put inside the other. Assume they are long enough
to act like ideal solenoids, so that each one only
contributes significantly to the field inside itself, and
the interior fields are nearly uniform. Consider the
configuration where the small one is partly inside and
partly hanging out of the big one, with their currents
circulating in the same direction. Their axes are constrained to coincide.

(a) Find the magnetic potential energy as a function of the
length $x$ of the part of the small solenoid that is inside
the big one. (Your equation will include other relevant
variables describing the two solenoids.)

(b) Based on your answer to part (a), find the force acting
between the solenoids.
<% end_hw() %>

<%
  fig(
    'hw-wire-box',
    %q{Problem \ref{hw:wire-box}.},
    {
      'width'=>'wide'
    }
  )
%>

\vfill

<% begin_hw('wire-box') %>__incl(hw/wire-box)<% end_hw() %>

\vfill

<% begin_hw('force-on-wire-in-b') %>__incl(hw/force-on-wire-in-b)<% end_hw() %>

\vfill

<% begin_hw('force-between-wires') %>__incl(hw/force-between-wires)<% end_hw() %>

\vfill

<% begin_hw('cyclic-vbf') %>__incl(hw/cyclic-vbf)<% end_hw() %>

\vfill

<% begin_hw('dipoleshape',2) %>__incl(hw/dipoleshape)<% end_hw() %>

\vfill

<% marg(0) %>
<%
  fig(
    'hw-helmholtz-coil',
    %q{Problem \ref{hw:helmholtzcoil}.}
  )
%>
<% end_marg %>
<% begin_hw('helmholtzcoil') %>
A Helmholtz coil is defined as a pair of identical
circular coils separated by a distance, $h$, equal to their
radius, $b$. (Each coil may have more than one turn of
wire.) Current circulates in the same direction in each
coil, so the fields tend to reinforce each other in the
interior region. This configuration has the advantage of
being fairly open, so that other apparatus can be easily
placed inside and subjected to the field while remaining
visible from the outside. The choice of $h=b$ results in the
most uniform possible field near the center. (a) Find the
percentage drop in the field at the center of one coil,
compared to the full strength at the center of the whole
apparatus. (b) What value of $h$ (not equal to $b)$ would
make this percentage difference equal to zero?
<% end_hw() %>

\vfill

<% begin_hw('b-from-photo') %>__incl(hw/b-from-photo)<% end_hw() %>

\vfill

<% begin_hw('corkscrew-beam',2) %>__incl(hw/corkscrew-beam)<% end_hw() %>

\pagebreak

<% begin_hw('deleted') %>This problem is now problem \ref{hw:cosmic-ray-lightning} on p.~\pageref{hw:cosmic-ray-lightning}<% end_hw() %>

\vfill

<% begin_hw('field-at-mouth-of-solenoid',2) %>__incl(hw/field-at-mouth-of-solenoid)<% end_hw() %>

\vfill

<% begin_hw('em-wave-energy-split') %>__incl(hw/em-wave-energy-split)<% end_hw() %>

\vfill

<% begin_hw('rel-vel-addition') %>__incl(../../share/relativity/hw/rel-vel-addition)<% end_hw() %>

\pagebreak

<% begin_hw('geroch-interval') %>__incl(../../share/relativity/hw/geroch-interval)<% end_hw() %>

<% marg(200) %>
<%
  fig(
    'hw-geroch-interval',
    %q{%
      Problem \ref{hw:geroch-interval}.
    }
  )
%>
<% end_marg %>

<% begin_hw('pure-e-b-lorentz-graphical') %>__incl(hw/pure-e-b-lorentz-graphical)<% end_hw() %>

<% end_hw_sec %>

<% begin_ex("Polarization","A") %>

\noindent Apparatus:

\begin{indentedblock}
calcite (Iceland spar) crystal

polaroid film
\end{indentedblock}

1. Lay the crystal on a piece of paper that has print on it.
You will observe a double image. See what happens if
you rotate the crystal.

        Evidently the crystal does something to the light that
passes through it on the way from the page to your eye. One
beam of light enters the crystal from underneath, but two
emerge from the top; by conservation of energy the energy of
the original beam must be shared between them. Consider the
following three possible interpretations of what you have observed:

(a) The two new beams differ from each other, and from the
original beam, only in energy. Their other properties are the same.

(b) The crystal adds to the light some mysterious new
property (not energy), which comes in two flavors, X and
Y. Ordinary light doesn't have any of either. One beam
that emerges from the crystal has some X added to it, and
the other beam has Y.

(c) There is some mysterious new property that is possessed
by all light. It comes in two flavors, X and Y, and most
ordinary light sources make an equal mixture of type X and
type Y light. The original beam is an even mixture of both
types, and this mixture is then split up by the crystal into
the two purified forms.

In parts 2 and 3 you'll make observations that will allow
you to figure out which of these is correct.

2. Now place a polaroid film over the crystal and see what
you observe. What happens when you rotate the film in the
horizontal plane? Does this observation allow you to rule
out any of the three interpretations?

\vspace{20mm}

3. Now put the polaroid film under the crystal and try the
same thing. Putting together all your observations, which
interpretation do you think is correct?

\vspace{20mm}

4. Look at an overhead light fixture through the polaroid,
and try rotating it. What do you observe? What does this
tell you about the light emitted by the lightbulb?

\vspace{20mm}

5. Now position yourself with your head under a light
fixture and directly over a shiny surface, such as a glossy
tabletop. You'll see the lamp's reflection, and the light
coming from the lamp to your eye will have undergone a
reflection through roughly a 180-degree angle (i.e. it very
nearly reversed its direction). Observe this reflection
through the polaroid, and try rotating it. Finally, position
yourself so that you are seeing glancing reflections, and
try the same thing. Summarize what happens to light with
properties X and Y when it is reflected. (This is the
principle behind polarizing sunglasses.)
<% end_ex %>

<% begin_ex("Events and Spacetime","B") %>

\includegraphics[width=168mm]{../share/relativity/figs/spacetime-ex-0.png}

\includegraphics[width=168mm]{../share/relativity/figs/spacetime-ex-1.png}

\includegraphics[width=168mm]{../share/relativity/figs/spacetime-ex-2.png}
<% end_ex %>

<% end_chapter() %>