Introduction to Electrodynamics

David J. Griffiths, Reed College

Chapter 8

Conservation Laws - all with Video Answers

Educators

Chapter Questions

Problem 1

Calculate the power (energy per unit time) transported down the cables of Ex. 7.13 and Prob. 7.58 , assuming the two conductors are held at potential difference $V$, and carry current $I$ (down one and back up the other)

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Problem 2

Consider the charging capacitor in Prob. 7.31
(a) Find the electric and magnetic fields in the gap, as functions of the distance $s$ from the axis and the time $t . \text { (Assume the charge is zero at } t=0 .)$
(b) Find the energy density $u_{\mathrm{em}}$ and the Poynting vector $\mathbf{S}$ in the gap. Note especially the direction of S. Check that Eq. 8.14 is satisfied..
(c) Determine the total energy in the gap, as a function of time. Calculate the total power flowing into the gap, by integrating the Poynting vector over the appropriate surface. Check that the power input is equal to the rate of increase of energy in the gap (Eq. $8.9-$ in this case $W=0,$ because there is no charge in the gap). [If you're worried about the fringing fields, do it for a volume of radius $b<a$ well inside the gap.

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Problem 3

Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius $R$, angular velocity $\omega,$ and surface charge density $\sigma .[\text { This is the same as Prob. } 5.42,$ but this time use the Maxwell stress tensor and Eq. $8.22 .$

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Problem 4

(a) Consider two equal point charges $q$, separated by a distance $2 a$. Construct the plane equidistant from the two charges. By integrating Maxwell's stress tensor over this plane, determine the force of one charge on the other.
(b) Do the same for charges that are opposite in sign.

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Problem 5

Consider an infinite parallel-plate capacitor, with the lower plate (at $z=-d / 2$ ) carrying the charge density $-\sigma,$ and the upper plate (at $z=+d / 2$ ) carrying the charge density $+\sigma$
(a) Determine all nine elements of the stress tensor, in the region between the plates. Display your answer as a $3 \times 3$ matrix:
$$\left(\begin{array}{ccc}
T_{x x} & T_{x y} & T_{x z} \\
T_{y x} & T_{y y} & T_{y z} \\
T_{z x} & T_{z y} & T_{z z}
\end{array}\right)$$
(b) Use Eq. 8.22 to determine the force per unit area on the top plate. Compare Eq. 2.51
(c) What is the momentum per unit area, per unit time, crossing the $x y$ plane (or any other plane parallel to that one, between the plates)?
(d) At the plates this momentum is absorbed, and the plates recoil (unless there is some nonelectrical force holding them in position). Find the recoil force per unit area on the top
plate, and compare your answer to (b). [Note: This is not an additional force, but rather an alternative way of calculating the same force- $\operatorname{in}(\mathrm{b})$ we got it from the force law, and in $(\mathrm{d})$ we did it by conservation of momentum.

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Problem 6

A charged parallel-plate capacitor (with uniform electric field $\mathbf{E}=E \hat{\mathbf{z}}$ ) is placed in a uniform magnetic field $\mathbf{B}=B \hat{\mathbf{x}},$ as shown in Fig. $8.6 .^{3}$
(a) Find the electromagnetic momentum in the space between the plates.
(b) Now a resistive wire is connected between the plates, along the $z$ axis, so that the capacitor slowly discharges. The current through the wire will experience a magnetic force; what is the total impulse delivered to the system, during the discharge?
field. This will induce a Faraday electric field, which in turn exerts a force on the plates. Show that the total impulse is (again) equal to the momentum originally stored in the fields.

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Problem 7

In Ex. $8.4,$ suppose that instead of turning off the magnetic field (by reducing $I$ ) we turn off the electric field, by connecting a weakly $^{6}$ conducting radial spoke between the cylinders. (We'll have to cut a slot in the solenoid, so the cylinders can still rotate freely.) From the magnetic force on the current in the spoke, determine the total angular momentum delivered to the cylinders, as they discharge (they are now rigidly connected, so they rotate together). Compare the initial angular momentum stored in the fields (Eq. 8.35 ). (Notice that the mechanism by which angular momentum is transferred from the fields to the cylinders is entirely different in the two cases: in Ex. 8.4 it was Faraday's law, but here it is the Lorentz force law.

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Problem 8

Imagine an iron sphere of radius $R$ that carries a charge $Q$ and a uniform magnetization $\mathbf{M}=M \hat{\mathbf{z}} .$ The sphere is initially at rest.
(a) Compute the angular momentum stored in the electromagnetic fields.
(b) Suppose the sphere is gradually (and uniformly) demagnetized (perhaps by heating it up past the Curie point). Use Faraday's law to determine the induced electric field, find the torque this field exerts on the sphere, and calculate the total angular momentum imparted to the sphere in the course of the demagnetization.
(c) Suppose instead of demagnetizing the sphere we discharge it, by connecting a grounding wire to the north pole. Assume the current flows over the surface in such a way that the charge density remains uniform. Use the Lorentz force law to determine the torque on the sphere, and cálculate the total angular momentum imparted to the sphere in the course of the discharge. (The magnetic field is discontinuous at the surface $\ldots$ does this matter?) [Answer:
$\left.\frac{2}{9} \mu_{0} M Q R^{2}\right]$

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Problem 9

A very long solenoid of radius $a$, with $n$ turns per unit length, carries a current $I_{s} .$ Coaxial with the solenoid, at radius $b \gg a,$ is a circular ring of wire, with resistance $R$ When the current in the solenoid is (gradually) decreased, a current $I_{r}$ is induced in the ring. (a) Calculate $I_{r},$ in terms of $d I_{s} / d t$
(b) The power $\left(I_{r}^{2} R\right)$ delivered to the ring must have come from the solenoid. Confirm this by calculating the Poynting vector just outside the solenoid (the electric field is due to the changing flux in the solenoid; the magnetic field is due to the current in the ring). Integrate over the entire surface of the solenoid, and check that you recover the correct total power.

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Problem 10

A sphere of radius $R$ carries a uniform polarization $\mathbf{P}$ and a uniform magnetization M (not necessarily in the same direction). Find the electromagnetic momentum of this configuration.

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Problem 11

Picture the electron as a uniformly charged spherical shell, with charge $e$ and radius $R,$ spinning at angular velocity $\omega$
(a) Calculate the total energy contained in the electromagnetic fields.
(b) Calculate the total angular momentum contained in the fields.
(c) According to the Einstein formula $\left(E=m c^{2}\right),$ the energy in the fields should contribute to the mass of the electron. Lorentz and others speculated that the entire mass of the electron might be accounted for in this way: $U_{\mathrm{em}}=m_{e} c^{2} .$ Suppose, moreover, that the electron's spin angular momentum is entirely attributable to the electromagnetic fields: $L_{\mathrm{em}}=\hbar / 2 .$ On these two assumptions, determine the radius and angular velocity of the electron. What is their product, $\omega R ?$ Does this classical model make sense?

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Problem 12

Suppose you had an electric charge $q_{e}$ and a magnetic monopole $q_{m} .$ The field of the electric charge is
$$\mathbf{E}=\frac{1}{4 \pi \epsilon_{0}} \frac{q_{e}}{r^{2}} \hat{\mathbf{r}}$$
of course, and the field of the magnetic monopole is
$$
\mathbf{B}=\frac{\mu_{0}}{4 \pi} \frac{q_{m}}{r^{2}} \hat{\mathbf{i}}
$$
Find the total angular momentum stored in the fields, if the two charges are separated by a distance $d .\left[\text

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Problem 13

Paul De Young, of Hope College, points out that because the cylinders in Ex. 8.4 are left rotating (at angular velocities $\omega_{a}$ and $\omega_{b},$ say), there is actually a residual magnetic field, and hence angular momentum in the fields, even after the current in the solenoid has been extinguished. If the cylinders are heavy, this correction will be negligible, but it is interesting to do the problem without making that assumption.
(a) Calculate (in terms of $\omega_{a}$ and $\omega_{b}$ ) the final angular momentum in the fields.
(b) As the cylinders begin to rotate, their changing magnetic field induces an extra azimuthal electric field, which, in turn, will make an additional contribution to the torques. Find the resulting extra angular momentum, and compare it to your result in (a)

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Problem 14

A point charge $q$ is a distance $a>R$ from the axis of an infinite solenoid (radius $R, n$ turns per unit length, current $I$ ). Find the linear momentum and the angular momentum in the fields. (Put $q$ on the $x$ axis, with the solenoid along $z$; treat the solenoid as a nonconductor, so you don't need to worry about induced charges on its surface.)

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Problem 15

(a) Carry through the argument in Sect. $8.1 .2,$ starting with Eq. $8.6,$ but using $\mathbf{J}_{f}$ in place of $\mathbf{J}$. Show that the Poynting vector becomes
$$
\mathbf{S}=\mathbf{E} \times \mathbf{H}
$$
and the rate of change of the energy density in the fields is
$$
\frac{\partial u_{\mathrm{em}}}{\partial t}=\mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t}+\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}
$$
For linear media, show that
$$
u_{\mathrm{em}}=\frac{1}{2}(\mathbf{E} \cdot \mathbf{D}+\mathbf{B} \cdot \mathbf{H})
$$
(b) In the same spirit, reproduce the argument in Sect. $8.2 .2,$ starting with Eq. $8.15,$ with $\rho_{f}$ and $\mathbf{J}_{f}$ in place of $\rho$ and $\mathbf{J} .$ Don't bother to construct the Maxwell stress tensor, but do show that the momentum density is
$$
\boldsymbol{\wp}=\mathbf{D} \times \mathbf{B}
$$
and the rate of change of the energy density in the fields is

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