Introduction to Electrodynamics

David J. Griffiths, Reed College

Chapter 12

Electrodynamics and Relativity - all with Video Answers

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Chapter Questions

Problem 1

Use Galileo's velocity addition rule. Let $\mathcal{S}$ be an inertial reference system.
(a) Suppose that $\overline{\mathcal{S}}$ moves with constant velocity relative to $\mathcal{S}$. Show that $\overline{\mathcal{S}}$ is also an inertial reference system. [Hint: use the definition in footnote 1.
(b) Conversely, show that if $\overline{\mathcal{S}}$ is an inertial system, then it moves with respect to $\mathcal{S}$ at constant velocity.

Juliet Schive

Juliet Schive

Numerade Educator

Problem 2

As an illustration of the principle of relativity in classical mechanics, consider the following generic collision: In inertial frame $\mathcal{S},$ particle $A\left(\operatorname{mass} m_{A}, \text { velocity } \mathbf{u}_{A}\right)$ hits particle $B$ (mass $m_{B},$ velocity $\mathbf{u}_{B}$ ). In the course of the collision some mass rubs off $A$ and onto $B,$ and we are left with particles $C$ (mass $m_{C},$ velocity $\mathbf{u}_{C}$ ) and $D$ (mass $m_{D}$, velocity $\mathbf{u}_{D}$ ). Assume that momentum $(\mathbf{p} \equiv m \mathbf{u})$ is conserved in $\mathcal{S}$
(a) Prove that momentum is also conserved in inertial frame $\overline{\mathcal{S}}$, which moves with velocity $\mathbf{v}$ relative to $\mathcal{S}$. [Use Galileo's velocity addition rule - this is an entirely classical calculation. What must you assume about mass?]
(b) Suppose the collision is elastic in $\mathcal{S} ;$ show that it is also elastic in $\overline{\mathcal{S}}$

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 3

(a) What's the percent error introduced when you use Galileo's rule, instead of Einstein's, with $v_{A B}=5 \mathrm{mi} / \mathrm{h}$ and $v_{B C}=60 \mathrm{mi} / \mathrm{h} ?$
(b) Suppose you could run at half the speed of light down the corridor of a train going three quarters the speed of light. What would your speed be relative to the ground?
(c) Prove, using Eq. 12.3 , that if $v_{A B} < c$ and $v_{B C} < c$ then $v_{A C} < c .$ Interpret this result.

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 4

As the outlaws escape in their getaway car, which goes $\frac{3}{4} c,$ the police officer fires a bullet from the pursuit car, which only goes $\frac{1}{2} c$ (Fig. 12.3 ). The muzzle velocity of the bullet (relative to the gun) is $\frac{1}{3} c .$ Does the bullet reach its target (a) according to Galileo, (b) according to Einstein?

Juliet Schive

Numerade Educator

Problem 5

Synchronized clocks are stationed at regular intervals, a million km apart, along a straight line. When the clock next to you reads 12 noon:
(a) What time do you see on the 90 th clock down the line?
(b) What time do you observe on that clock?

Juliet Schive

Numerade Educator

Problem 6

Every 2 years, more or less, The New York Times publishes an article in which some astronomer claims to have found an object traveling faster than the speed of light. Many of these reports result from a failure to distinguish what is seen from what is observed-that is, from a failure to account for light travel time. Here's an example: A star is traveling with speed $v$ at an angle $\theta$ to the line of sight (Fig. 12.6 ). What is its apparent speed across the sky?

Juliet Schive

Numerade Educator

Problem 7

In a laboratory experiment a muon is observed to travel 800 m before disintegrating. A graduate student looks up the lifetime of a muon $\left(2 \times 10^{-6} \mathrm{s}\right)$ and concludes that its speed was
$$
v=\frac{800 \mathrm{m}}{2 \times 10^{-6} \mathrm{s}}=4 \times 10^{8} \mathrm{m} / \mathrm{s}
$$
Faster than light! Identify the student's error, and find the actual speed of this muon.

Anand Jangid

Numerade Educator

Problem 8

A rocket ship leaves earth at a speed of $\frac{3}{5} c .$ When a clock on the rocket says 1 hour has elapsed, the rocket ship sends a light signal back to earth.
(a) According to earth clocks, when was the signal sent?
(b) According to earth clocks, how long after the rocket left did the signal arrive back on earth?
(c) According to the rocket observer, how long after the rocket left did the signal arrive back on earth?

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 9

A Lincoln Continental is twice as long as a VW Beetle, when they are at rest. As the Continental overtakes the $V W$, going through a speed trap, a (stationary) policeman observes that they both have the same length. The $\mathrm{VW}$ is going at half the speed of light. How fast is the Lincoln going? (Leave your answer as a multiple of $c .$ )

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 10

A sailboat is manufactured so that the mast leans at an angle $\bar{\theta}$ with respect to the deck. An observer standing on a dock sees the boat go by at speed $v$ (Fig. 12.14 ). What angle does this observer say the mast makes?

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 11

A record turntable of radius $R$ rotates at angular velocity $\omega$ (Fig. 12.15). The circumference is presumably Lorentz-contracted, but the radius (being perpendicular to the velocity) is not. What's the ratio of the circumference to the diameter, in terms of $\omega$ and $R ?$ According to the rules of ordinary geometry, that has to be $\pi$. What's going on here? [This is known as Ehrenfest's paradox; for discussion and references see H. Arzelies, Relativistic Kinematics, Chap. IX (Elmsford, NY: Pergamon Press, 1966) and T. A. Weber, Am. J. Phys. $65,486(1997) .]$

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 12

Solve Eqs. 12.18 for $x, y, z, t$ in terms of $\bar{x}, \bar{y}, \bar{z}, \bar{t},$ and check that you recover Eqs. 12.19.

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 13

Sophie Zabar, clairvoyante, cried out in pain at precisely the instant her twin brother, $500 \mathrm{km}$ away, hit his thumb with a hammer. A skeptical scientist observed both events (brother's accident, Sophie's cry) from an airplane traveling at $\frac{12}{13} c$ to the right (see Fig. 12.19 ). Which event occurred first, according to the scientist? How much earlier was it, in seconds?

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 14

(a) In Ex. 12.6 we found how velocities in the $x$ direction transform when you go from $\mathcal{S}$ to
$\mathcal{S} .$ Derive the analogous formulas for velocities in the $y$ and $z$ directions.
(b) A spotlight is mounted on a boat so that its beam makes an angle $\bar{\theta}$ with the deck (Fig. 12.20 ). If this boat is then set in motion at speed $v,$ what angle $\theta$ does an observer on the dock say the beam makes with the deck? Compare Prob. 12.10 , and explain the difference.

Dominador Tan

Numerade Educator

Problem 15

You probably did Prob. 12.4 from the point of view of an observer on the ground. Now do it from the point of view of the police car, the outlaws, and the bullet. That is, fill in the gaps in the following table:
$$\begin{array}{|l||c|c|c|c||c|}
\hline \begin{array}{c}
\text { speed of } \rightarrow \\
\text { relative to } \downarrow
\end{array} & \text { Ground } & \text { Police } & \text { Outlaws } & \text { Bullet } & \text { Do they escape? } \\
\hline \text { Ground } & 0 & \frac{1}{2} c & \frac{3}{4} c & & \\
\hline \text { Police } & & & & \frac{1}{3} c & \\
\hline \text { Outlaws } & & & & & \\
\hline \text { Bullet } & & & & & \\
\hline
\end{array}$$

Dominador Tan

Numerade Educator

Problem 16

The twin paradox revisited. On their 21st birthday, one twin gets on a moving sidewalk, which carries her out to star $X$ at speed $\frac{4}{5} c ;$ her twin brother stays home. When the traveling twin gets to star $\mathrm{X}$, she immediately jumps onto the returning moving sidewalk and comes back to earth, again at speed $\frac{4}{5} c .$ She arrives on her 39 th birthday (as determined by her watch (a) How old is her twin brother (who stayed at home)?
(b) How far away is star X? (Give your answer in light years.)
Call the outbound sidewalk system $\overline{\mathcal{S}}$ and the inbound one $\tilde{\mathcal{S}}$ (the earth system is $\mathcal{S}$ ). All three systems set their master clocks, and choose their origins, so that $x=\bar{x}=\tilde{x}=0, t=\bar{t}=\tilde{t}=0$ at the moment of departure.
(c) What are the coordinates $(x, t)$ of the jump (from outbound to inbound sidewalk) in $\mathcal{S} ?$
(d) What are the coordinates $(\bar{x}, \bar{t})$ of the jump in $\overline{\mathcal{S}}$ ?
(e) What are the coordinates $(\tilde{x}, \tilde{t})$ of the jump in $\tilde{\mathcal{S}} ?$
(f) If the traveling twin wanted her watch to agree with the clock in $\tilde{\mathcal{S}},$ how would she have to reset it immediately after the jump? If she did this, what would her watch read when she got home? (This wouldn't change her $a g e$, of course - she's still 39-it would just make her watch agree with the standard synchronization in $\tilde{\mathcal{S}}$.)
(g) If the traveling twin is asked the question, "How old is your brother right now?", what is the correct reply (i) just before she makes the jump, (ii) just after she makes the jump? (Nothing dramatic happens to her brother during the split second between (i) and (ii), of course; what does change abruptly is his sister's notion of what "right now, back home" means.)
(h) How many earth years does the return trip take? Add this to (ii) from (g) to determine how old she expects him to be at their reunion. Compare your answer to (a).

Dominador Tan

Numerade Educator

Problem 17

Check Eq. 12.29, using Eq. 12.27. [This only proves the invariance of the scalar product for transformations along the $x$ direction. But the scalar product is also invariant under rotations, since the first term is not affected at all, and the last three constitute the three dimensional dot product a $\cdot \mathbf{b}$. By a suitable rotation, the $x$ direction can be aimed any way you please, so the four-dimensional scalar product is actually invariant under arbitrary Lorentz transformations.

Dominador Tan

Numerade Educator

Problem 18

(a) Write out the matrix that describes a Galilean transformation (Eq. 12.12).
(b) Write out the matrix describing a Lorentz transformation along the $y$ axis.
(c) Find the matrix describing a Lorentz transformation with velocity $v$ along the $x$ axis followed by a Lorentz transformation with velocity $\bar{v}$ along the $y$ axis. Does it matter in what order the transformations are carried out?

Dominador Tan

Numerade Educator

Problem 19

The parallel between rotations and Lorentz transformations is even more striking if we introduce the rapidity: $$\theta \equiv \tanh ^{-1}(v / c)$$
(a) Express the Lorentz transformation matrix $\Lambda$ (Eq. 12.24 ) in terms of $\theta$, and compare it to the rotation matrix (Eq. 1.29 ).

In some respects rapidity is a more natural way to describe motion than velocity. [See E.
F. Taylor and J. A. Wheeler, Spacetime Physics (San Francisco: W. H. Freeman, 1966).] For one thing, it ranges from $-\infty$ to $+\infty,$ instead of $-c$ to $+c .$ More significantly, rapidities add. whereas velocities do not.
(b) Express the Einstein velocity addition law in terms of rapidity.

Sam Stansfield

Numerade Educator

Problem 20

(a) Event $A$ happens at point $\left(x_{A}=5, y_{A}=3, z_{A}=0\right)$ and at time $t_{A}$ given by $c t_{A}=15$ event $B$ occurs at (10,8,0) and $c t_{B}=5,$ both in system $\mathcal{S}$
(i) What is the invariant interval between $A$ and $B ?$
(ii) Is there an inertial system in which they occur simultaneously? If so, find its velocity (magnitude and direction) relative to $\mathcal{S}$
(iii) Is there an inertial system in which they occur at the same point? If so, find its velocity relative to $\mathcal{S}$
(b) Repeat part (a) for $A=(2,0,0), c t=1 ;$ and $B=(5,0,0), c t=3$

Dominador Tan

Numerade Educator

Problem 21

The coordinates of event $A$ are $\left(x_{A}, 0,0\right), t_{A},$ and the coordinates of event $B$ are $\left(x_{B}, 0,0\right), t_{B} .$ Assuming the interval between them is space like, find the velocity of the system in which they are simultaneous.

Juliet Schive

Numerade Educator

Problem 22

(a) Draw a space-time diagram representing a game of catch (or a conversation) between two people at rest, $10 \mathrm{ft}$ apart. How is it possible for them to communicate, given that their separation is spacelike?
(b) There's an old limerick that runs as follows:
There once was a girl named Ms. Bright, Who could travel much faster than light. She departed one day, The Einsteinian way, And returned on the previous night.
What do you think? Even if she could travel faster than the speed of light, could she return
before she set out? Could she arrive at some intermediate destination before she set out? Draw a space-time diagram representing this trip.

Dominador Tan

Numerade Educator

Problem 23

Inertial system $\overline{\mathcal{S}}$ moves in the $x$ direction at speed $\frac{3}{5} c$ relative to system $\mathcal{S}$ (The $\bar{x}$ axis slides long the $x$ axis, and the origins coincide at $t=\bar{t}=0,$ as usual.)
(a) On graph paper set up a Cartesian coordinate system with axes $c t$ and $x$. Carefully draw in lines representing $\bar{x}=-3,-2,-1,0,1,2,$ and $3 .$ Also draw in the lines corresponding to $c \bar{t}=-3,-2,-1,0,1,2,$ and $3 .$ Label your lines clearly.
(b) In $\overline{\mathcal{S}},$ a free particle is observed to travel from the point $\bar{x}=-2$ at time $c \bar{t}=-2$ to the point $\bar{x}=2$ at $c \bar{t}=+3 .$ Indicate this displacement on your graph. From the slope of this line. determine the particle's speed in $\mathcal{S}$
(c) Use the velocity addition rule to determine the velocity in $\mathcal{S}$ algebraically, and check that your answer is consistent with the graphical solution in (b).

Dominador Tan

Numerade Educator

Problem 24

(a) Equation 12.40 defines proper velocity in terms of ordinary velocity. Invert that equation to get the formula for $\mathbf{u}$ in terms of $\eta$
(b) What is the relation between proper velocity and rapidity (Eq. 12.34 )? Assume the velocity is along the $x$ direction, and find $\eta$ as a function of $\theta$

Dominador Tan

Numerade Educator

Problem 25

A car is traveling along the $45^{\circ}$ line in $\mathcal{S}$ (Fig. 12.25 ), at (ordinary) speed $(2 / \sqrt{5}) c$ (a) Find the components $u_{x}$ and $u_{y}$ of the (ordinary) velocity.
(b) Find the components $\eta_{x}$ and $\eta_{y}$ of the proper velocity.
(c) Find the zeroth component of the 4-velocity, $\eta^{0}$ System $\overline{\mathcal{S}}$ is moving in the $x$ direction with (ordinary) speed $\sqrt{2 / 5} c,$ relative to $\mathcal{S} .$ By using the appropriate transformation laws:
(d) Find the (ordinary) velocity components $\bar{u}_{x}$ and $\bar{u}_{y}$ in $\overline{\mathcal{S}}$

Dominador Tan

Numerade Educator

Problem 26

Find the invariant product of the 4 -velocity with itself, $\eta^{\mu} \eta_{\mu}$

Dominador Tan

Numerade Educator

Problem 27

Consider a particle in hyperbolic motion.
$$x(t)=\sqrt{b^{2}+(c t)^{2}}, \quad y=z=0$$
(a) Find the proper time $\tau$ as a function of $t,$ assuming the clocks are set so that $\tau=0$ when $t=0 .[\text {Hint: Integrate Eq. } 12.37 .]$
(b) Find $x$ and $v$ (ordinary velocity) as functions of $\tau$
(c) Find $\eta^{\mu}$ (proper velocity) as a function of $t$

Dominador Tan

Numerade Educator

Problem 28

(a) Repeat Prob. 12.2 using the (incorrect) definition $\mathbf{p}=m \mathbf{u},$ but with the (correct) Einstein velocity addition rule. Notice that if momentum (so defined) is conserved in $\mathcal{S}$, it is not conserved in $\overline{\mathcal{S}}$. Assume all motion is along the $x$ axis.
(b) Now do the same using the correct definition, $\mathbf{p}=m \eta$. Notice that if momentum (so defined) is conserved in $\mathcal{S}$ it is automatically also conserved in $\overline{\mathcal{S}}$. [Hint: Use Eq. 12.43 to transform the proper velocity.] What must you assume about relativistic energy?

Dominador Tan

Numerade Educator

Problem 29

If a particle's kinetic energy is $n$ times its rest energy, what is its speed?

Juliet Schive

Numerade Educator

Problem 30

Suppose you have a collection of particles, all moving in the $x$ direction, with energies $E_{1}, E_{2}, E_{3}, \ldots$ and momenta $p_{1}, p_{2}, p_{3}, \ldots .$ Find the velocity of the center of momentum frame, in which the total momentum is zero.

Narayan Hari

Numerade Educator

Problem 31

Find the velocity of the muon in Ex. 12.8

Juliet Schive

Numerade Educator

Problem 32

A particle of mass $m$ whose total energy is twice its rest energy collides with an identical particle at rest. If they stick together, what is the mass of the resulting composite particle? What is its velocity?

Penny Riley

Numerade Educator

Problem 33

A neutral pion of (rest) mass $m$ and (relativistic) momentum $p=\frac{3}{4} m c$ decays into two photons. One of the photons is emitted in the same direction as the original pion, and the other in the opposite direction. Find the (relativistic) energy of each photon.

Juliet Schive

Numerade Educator

Problem 34

In the past, most experiments in particle physics involved stationary targets:
one particle (usually a proton or an electron) was accelerated to a high energy $E,$ and collided with a target particle at rest (Fig. 12.29 a). Far higher relative energies are obtainable (with the same accelerator) if you accelerate both particles to energy $E,$ and fire them at each other (Fig. $12.29 \mathrm{b}$ ). Classically, the energy $\bar{E}$ of one particle, relative to the other, is just $4 \mathrm{E}(\text { why? })-$ not much of a gain (only a factor of 4). But relativistically the gain can be enormous. Assuming the two particles have the same mass, $m$, show that
$$\bar{E}=\frac{2 E^{2}}{m c^{2}}-m c^{2}$$
Suppose you use protons $ (m c^{2}=1 \mathrm{GeV})$ with $E=30 \mathrm{GeV} .$ What $E$ do you get? What multiple of $E$ does this amount to? $(1 \mathrm{GeV}=10^{9}$ electron volts.) [Because of this relativistic enhancement, most modern elementary particle experiments involve colliding beams, instead of fixed targets.

Penny Riley

Numerade Educator

Problem 35

In a pair annihilation experiment, an electron (mass $m$ ) with momentum $p_{e}$ hits a positron (same mass, but opposite charge) at rest. They annihilate, producing two photons. (Why couldn't they produce just one photon?) If one of the photons emerges at $60^{\circ}$ to the incident electron direction, what is its energy?

Penny Riley

Numerade Educator

Problem 36

In classical mechanics Newton's law can be written in the more familiar form $\mathbf{F}=m \mathbf{a} .$ The relativistic equation, $\mathbf{F}=d \mathbf{p} / d t,$ cannot be so simply expressed. Show, rather that
$$\mathbf{F}=\frac{m}{\sqrt{1-u^{2} / c^{2}}}\left[\mathbf{a}+\frac{\mathbf{u}(\mathbf{u} \cdot \mathbf{a})}{c^{2}-u^{2}}\right]$$
where $\mathbf{a} \equiv d \mathbf{u} / d t$ is the ordinary acceleration.

Dominador Tan

Numerade Educator

Problem 37

Show that it is possible to outrun a light ray, if you're given a sufficient head start, and your feet generate a constant force.

Narayan Hari

Numerade Educator

Problem 38

Define proper acceleration in the obvious way:
$$\alpha^{\mu} \equiv \frac{d \eta^{\mu}}{d \tau}=\frac{d^{2} x^{\mu}}{d \tau^{2}}$$
(a) Find $\alpha^{0}$ and $\alpha$ in terms of $\mathbf{u}$ and $\mathbf{a}$ (the ordinary acceleration).
(b) Express $\alpha_{\mu} \alpha^{\mu}$ in terms of $\mathbf{u}$ and $\mathbf{a}$
(c) Show that $\eta^{\mu} \alpha_{\mu}=0$
(d) Write the Minkowski version of Newton's second law, Eq. 12.70 , in terms of $\alpha^{\mu}$. Evaluate the invariant product $K^{\mu} \eta_{\mu}$

Dominador Tan

Numerade Educator

Problem 39

Show that
$$
K_{\mu} K^{\mu}=\frac{1-\left(u^{2} / c^{2}\right) \cos ^{2} \theta}{1-u^{2} / c^{2}} F^{2}
$$

Dominador Tan

Numerade Educator

Problem 40

Show that the (ordinary) acceleration of a particle of mass $m$ and charge $q$ moving at velocity u under the influence of electromagnetic fields $\mathbf{E}$ and $\mathbf{B}$, is given by $$\mathbf{a}=\frac{q}{m} \sqrt{1-u^{2} / c^{2}}\left[\mathbf{E}+\mathbf{u} \times \mathbf{B}-\frac{1}{c^{2}} \mathbf{u}(\mathbf{u} \cdot \mathbf{E})\right]$$

Dominador Tan

Numerade Educator

Problem 41

Why can't the electric field in Fig. 12.35b have a z component? After all, the magnetic field does.

Juliet Schive

Numerade Educator

Problem 42

A parallel-plate capacitor, at rest in $\mathcal{S}_{0}$ and tilted at a 45 $^{\circ}$ angle to the $x_{0}$ axis, carries charge densities $\pm \sigma_{0}$ on the two plates (Fig. 12.41 ). System $\mathcal{S}$ is moving to the right at speed $v$ relative to $\mathcal{S}_{0}$
(a) Find $\mathbf{E}_{0},$ the field in $\mathcal{S}_{0}$
(b) Find $\mathbf{E},$ the field in $\mathcal{S}$
(c) What angle do the plates make with the $x$ axis?
(d) Is the field perpendicular to the plates in $\mathcal{S}$ ?

Dominador Tan

Numerade Educator

Problem 43

(a) Check that Gauss's law, $\int \mathbf{E} \cdot d \mathbf{a}=\left(1 / \epsilon_{0}\right) Q_{\text {enc }}$, is obeyed by the field of a point charge in uniform motion, by integrating over a sphere of radius $R$ centered on the charge.
(b) Find the Poynting vector for a point charge in uniform motion. (Say the charge is going in the $z$ direction at speed $v,$ and calculate $\mathbf{S}$ at the instant $q$ passes the origin.)

Dominador Tan

Numerade Educator

Problem 44

(a) Charge $q_{A}$ is at rest at the origin in system $\mathcal{S} ;$ charge $q_{B}$ flies by at speed $v$ on a trajectory parallel to the $x$ axis, but at $y=d .$ What is the electromagnetic force on $q_{B}$ as it crosses the $y$ axis?
(b) Now study the same problem from system $\overline{\mathcal{S}}$, which moves to the right with speed $v$. What is the force on $q_{B}$ when $q_{A}$ passes the $\bar{y}$ axis? [Do it two ways: (i) by using your answer to (a) and transforming the force; (ii) by computing the fields in $\overline{\mathcal{S}}$ and using the Lorentz force law.]

Dominador Tan

Numerade Educator

Problem 45

Two charges $\pm q$, are on parallel trajectories a distance $d$ apart, moving with equal speeds $v$ in opposite directions. We're interested in the force on $+q$ due to $-q$ at the instant they cross (Fig. 12.42 ). Fill in the following table, doing all the consistency checks you can think of as you go along.
$$\begin{array}{|l||l|l|l|}
\hline & \begin{array}{c}
\text { System } A \\
\text { (Fig. }12.42)
\end{array} & \begin{array}{c}
\text { System } B \\
(+q \text { at rest })
\end{array} & \begin{array}{c}
\text { System } C \\
(-q \text { at rest })
\end{array} \\
\hline \mathbf{E} \text { at }+q \text { due to }-q: & & & \\
\hline \mathbf{B} \text { at }+q \text { due to }-q: & & & \\
\hline \mathbf{F} \text { on }+q \text { due to }-q: & & & \\
\hline
\end{array}$$

Dominador Tan

Numerade Educator

Problem 46

(a) Show that $(\mathbf{E} \cdot \mathbf{B})$ is relativistically invariant.
(b) Show that $\left(E^{2}-c^{2} B^{2}\right)$ is relativistically invariant.
(c) Suppose that in one inertial system $\mathbf{B}=0$ but $\mathbf{E} \neq 0$ (at some point $P$ ). Is it possible to find another system in which the electric field is zero at $P ?$

Sam Stansfield

Numerade Educator

Problem 47

An electromagnetic plane wave of (angular) frequency $\omega$ is traveling in the $x$ direction through the vacuum. It is polarized in the $y$ direction, and the amplitude of the electric field is $E_{0}$
(a) Write down the electric and magnetic fields, $\mathbf{E}(x, y, z, t)$ and $\mathbf{B}(x, y, z, t) .$ [Be sure to define any auxiliary quantities you introduce, in terms of $\omega, E_{0},$ and the constants of nature.
(b) This same wave is observed from an inertial system $\overline{\mathcal{S}}$ moving in the $x$ direction with speed $v$ relative to the original system $\mathcal{S}$. Find the electric and magnetic fields in $\overline{\mathcal{S}}$, and express them in terms of the $\overline{\mathcal{S}}$ coordinates: $\overline{\mathbf{E}}(\bar{x}, \bar{y}, \bar{z}, \bar{t})$ and $\overline{\mathbf{B}}(\bar{x}, \bar{y}, \bar{z}, \bar{t}) .$ [Again, be sure to define
any auxiliary quantities you introduce.
(c) What is the frequency $\bar{\omega}$ of the wave in $\overline{\mathcal{S}} ?$ Interpret this result. What is the wavelength $\bar{\lambda}$ of the wave in $\overline{\mathcal{S}} ?$ From $\bar{\omega}$ and $\bar{\lambda},$ determine the speed of the waves in $\overline{\mathcal{S}} .$ Is it what you expected?

Dominador Tan

Numerade Educator

Problem 48

Work out the remaining five parts to Eq. 12.117

Sam Stansfield

Numerade Educator

Problem 49

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if $t^{\mu v}$ is symmetric, show that $\bar{t}^{\mu \nu}$ is also symmetric, and likewise for antisymmetric)

Dominador Tan

Numerade Educator

Problem 50

Recall that a covariant 4-vector is obtained from a contravariant one by changing the sign of the zeroth component. The same goes for tensors: When you "lower an index" to make it covariant, you change the sign if that index is zero. Compute the tensor invariants
$$F^{\mu \nu} F_{\mu \nu}, \quad G^{\mu \nu} G_{\mu \nu}, \text { and } F^{\mu \nu} G_{\mu \nu}$$
in terms of $\mathbf{E}$ and $\mathbf{B}$. Compare Prob. 12.46

Dominador Tan

Numerade Educator

Problem 51

A straight wire along the $z$ axis carries a charge density $\lambda$ traveling in the $+z$ direction at speed $v .$ Construct the field tensor and the dual tensor at the point $(x, 0,0)$

Narayan Hari

Numerade Educator

Problem 52

Obtain the continuity equation (12.125) directly from Maxwell's equations (12.126)

Penny Riley

Numerade Educator

Problem 53

Show that the second equation in (12.126) can be expressed in terms of the field tensor $F^{\mu v}$ as follows:
$$\frac{\partial F_{\mu v}}{\partial x^{\lambda}}+\frac{\partial F_{v \lambda}}{\partial x^{\mu}}+\frac{\partial F_{\lambda \mu}}{\partial x^{v}}=0$$

Dominador Tan

Numerade Educator

Problem 54

Work out, and interpret physically, the $\mu=0$ component of the electromagnetic force law, Eq. 12.127

Dominador Tan

Numerade Educator

Problem 55

You may have noticed that the four-dimensional gradient operator $\partial / \partial x^{\mu}$ functions like a covariant 4-vector- in fact, it is often written $\partial_{\mu},$ for short. For instance, the continuity equation, $\partial_{\mu} J^{\mu}=0,$ has the form of an invariant product of two vectors. The corresponding contravariant gradient would be $\partial^{\mu} \equiv \partial x_{\mu} .$ Prove that $\partial^{\mu} \phi$ is a (contravariant) 4-vector, if $\phi$ is a scalar function, by working out its transformation law, using the chain rule.

Dominador Tan

Numerade Educator

Problem 56

Show that the potential representation (Eq. 12.132) automatically satisfies $\partial G^{\mu \nu} / \partial x^{\nu}=0 .[\text {Suggestion: Use Prob. } 12.53 .]$

Dominador Tan

Numerade Educator

Problem 57

Inertial system $\overline{\mathcal{S}}$ moves at constant velocity $\mathbf{v}=\beta c(\cos \phi \hat{\mathbf{x}}+\sin \phi \hat{\mathbf{y}})$ with
respect to $\mathcal{S}$. Their axes are parallel to one another, and their origins coincide at $t=\bar{t}=0,$ as usual. Find the Lorentz transformation matrix $\Lambda$ (Eq. 12.25 ).

Dominador Tan

Numerade Educator

Problem 58

Calculate the threshold (minimum) momentum the pion must have in order for the process $\pi+p \rightarrow K+\Sigma$ to occur. The proton $p$ is initially at rest. Use $m_{\pi} c^{2}=$ $\left.150, m_{K} c^{2}=500, m_{p} c^{2}=900, m_{\Sigma} c^{2}=1200 \text { (all in } \mathrm{MeV}\right) .$ [Hint: To formulate the
threshold condition, examine the collision in the center-of-momentum frame (Prob. 12.30 ).

Sam Stansfield

Numerade Educator

Problem 59

A particle of mass $m$ collides elastically with an identical particle at rest. Classically, the outgoing trajectories always make an angle of $90^{\circ} .$ Calculate this angle relativistically, in terms of $\phi$, the scattering angle, and $v$, the speed, in the center-of-momentum frame.

Penny Riley

Numerade Educator

Problem 60

Problem 12.60 Find $x$ as a function of $t$ for motion starting from rest at the origin under the influence of a constant Minkowski force in the $x$ direction. Leave your answer in implicit form ( $t$ as a function of $x$ )

Dominador Tan

Numerade Educator

Problem 61

An electric dipole consists of two point charges $(\pm q),$ each of mass $m,$ fixed to the ends of a (mass less) rod of length $d .$ (Do not assume $d$ is small.)
(a) Find the net self-force on the dipole when it undergoes hyperbolic motion (Eq. 12.62 ) along a line perpendicular to its axis. [Hint: Start by appropriately modifying Eq. $11.90 .$
(b) Notice that this self-force is constant ( $t$ drops out), and points in the direction of motion $-$ just right to produce hyperbolic motion. Thus it is possible for the dipole to undergo self sustaining accelerated motion with no external force at all! $1^{18}$ [Where do you suppose the
energy comes from?] Determine the self-sustaining force, $F$, in terms of $m, q,$ and $d$.

Dominador Tan

Numerade Educator

Problem 62

An ideal magnetic dipole moment $m$ is located at the origin of an inertial system $\overline{\mathcal{S}}$ that moves with speed $v$ in the $x$ direction with respect to inertial system $\mathcal{S} .$ In $\overline{\mathcal{S}}$ the vector potential is
$$\overline{\mathbf{A}}=\frac{\mu_{0}}{4 \pi} \frac{\overline{\mathbf{m}} \times \overline{\hat{\mathbf{r}}}}{\bar{r}^{2}}$$
(Eq. 5.83 ), and the electric potential $\bar{V}$ is zero.
(a) Find the scalar potential $V$ in $\mathcal{S}$. [Answer: $\left(1 / 4 \pi \epsilon_{0}\right)\left(\hat{\mathbf{R}} \cdot(\mathbf{v} \times \mathbf{m}) / c^{2} R^{2}\right)\left(1-v^{2} / c^{2}\right) /(1-$
$\left.\left(v^{2} / c^{2}\right) \sin ^{2} \theta\right)^{3 / 2}$
(b) In the nonrelativistic limit, show that the scalar potential in $\mathcal{S}$ is that of an ideal electric dipole of magnitude
$$
\mathbf{p}=\frac{\mathbf{v} \times \mathbf{m}}{c^{2}}
$$
located at $\overline{\mathcal{O}}$

Dominador Tan

Numerade Educator

Problem 63

A stationary magnetic dipole, $m=m \hat{\mathbf{z}},$ is situated above an infinite uniform surface current, $\mathbf{K}=K \hat{\mathbf{x}}(\text { Fig. } 12.44)$
(a) Find the torque on the dipole, using Eq. 6.1
(b) Suppose that the surface current consists of a uniform surface charge $\sigma,$ moving at velocity $\mathbf{v}=v \hat{\mathbf{x}},$ so that $\mathbf{K}=\sigma \mathbf{v},$ and the magnetic dipole consists of a uniform line charge $\lambda$ circulating at speed $v$ (same $v$ ) around a square loop of side $l$, as shown, so that $m=\lambda v l^{2}$ Examine the same configuration from the point of view of system $\overline{\mathcal{S}},$ moving in the $x$ direction at speed $v .$ In $\overline{\mathcal{S}}$ the surface charge is at rest, so it generates no magnetic field. Show that in this frame the current loop carries an electric dipole moment, and calculate the resulting torque, using Eq. 4.4.

Dominador Tan

Numerade Educator

Problem 64

In a certain inertial frame $\mathcal{S},$ the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ are neither parallel nor perpendicular, at a particular space-time point. Show that in a different inertial system $\overline{\mathcal{S}},$ moving relative to $\mathcal{S}$ with velocity $\mathbf{v}$ given by
$$\frac{\mathbf{v}}{1+v^{2} / c^{2}}=\frac{\mathbf{E} \times \mathbf{B}}{B^{2}+E^{2} / c^{2}}$$
the fields $\overline{\mathbf{E}}$ and $\overline{\mathbf{B}}$ are parallel at that point. Is there a frame in which the two are perpendicular?

Dominador Tan

Numerade Educator

Problem 65

Two charges $\pm q$ approach the origin at constant velocity from opposite directions along the $x$ axis. They collide and stick together, forming a neutral particle at rest. Sketch the electric field before and shortly after the collision (remember that electromagnetic "news" travels at the speed of light). How would you interpret the field after the collision, physically? 19

Dominador Tan

Numerade Educator

Problem 66

"Derive" the Lorentz force law, as follows: Let charge $q$ be at rest in $\overline{\mathcal{S}},$ so $\overline{\mathbf{F}}=q \overline{\mathbf{E}},$ and let $\overline{\mathcal{S}}$ move with velocity $\mathbf{v}=v \hat{\mathbf{x}}$ with respect to $\mathcal{S}$. Use the transformation rules (Eqs. 12.68 and 12.108 ) to rewrite $\overline{\mathbf{F}}$ in terms of $\mathbf{F}$, and $\overline{\mathbf{E}}$ in terms of $\mathbf{E}$ and $\mathbf{B}$. From these deduce the formula for $\mathbf{F}$ in terms of $\mathbf{E}$ and $\mathbf{B}$.

Dominador Tan

Numerade Educator

Problem 67

A charge $q$ is released from rest at the origin, in the presence of a uniform electric field $\mathbf{E}=E_{0} \hat{\mathbf{z}}$ and a uniform magnetic field $\mathbf{B}=B_{0} \hat{\mathbf{x}} .$ Determine the trajectory of the particle by transforming to a system in which $\mathbf{E}=0,$ finding the path in that system and then transforming back to the original system. Assume $E_{0}<c B_{0}$. Compare your result with Ex. 5.2.

Dominador Tan

Numerade Educator

Problem 68

(a) Construct a tensor $D^{\mu \nu}$ (analogous to $F^{\mu \nu}$ ), out of $\mathbf{D}$ and $\mathbf{H}$. Use it to express Maxwell's equations inside matter in terms of the free current density $J_{f}^{\mu} .$
(b) Construct the dual tensor $H^{\mu \nu}$ (analogous to $G^{\mu \nu}$ ).
(c) Minkowski proposed the relativistic constitutive relations for linear media:
$$
D^{\mu \nu} \eta_{\nu}=c^{2} \epsilon F^{\mu \nu} \eta_{\nu} \text { and } H^{\mu \nu} \eta_{\nu}=\frac{1}{\mu} G^{\mu \nu} \eta_{\nu}
$$
where $\epsilon$ is the proper $^{20}$ permittivity, $\mu$ is the proper permeability, and $\eta^{\mu}$ is the 4-velocity of the material. Show that Minkowski's formulas reproduce Eqs. 4.32 and 6.31 , when the material is at rest.
(d) Work out the formulas relating $\mathbf{D}$ and $\mathbf{H}$ to $\mathbf{E}$ and $\mathbf{B}$ for a medium moving with (ordinary) velocity u.

Dominador Tan

Numerade Educator

Problem 69

Use the Larmor formula (Eq. 11.70 ) and special relativity to derive the Liénard formula (Eq. 11.73 ).

Prabhat Tyagi

Numerade Educator

Problem 70

The natural relativistic generalization of the Abraham-Lorentz formula(Eq. 11.80) would seem to be
$$
K_{\mathrm{rad}}^{\mu}=\frac{\mu_{0} q^{2}}{6 \pi c} \frac{d \alpha^{\mu}}{d \tau}
$$
This is certainly a 4 -vector, and it reduces to the Abraham-Lorentz formula in the nonrelativistic limit $v \ll c$ (a) Show, nevertheless, that this is not a possible Minkowski force. [Hint: See Prob. 12.38d.]
(b) Find a correction term that, when added to the right side, removes the objection you raised in (a), without affecting the 4-vector character of the formula or its nonrelativistic limit.21

Prabhat Tyagi

Numerade Educator

Problem 71

Generalize the laws of relativistic electrodynamics (Eqs. 12.126 and 12.127) to include magnetic charge. [Refer to Sect. 7.3.4.]

Prabhat Tyagi

Numerade Educator