University of Maryland, College Park

Spring 2011, Professor: Ted Jacobson

Homework

Problems from Griffiths, Introduction to Electrodynamics, 3rd edition

HW0 - due at beginning of class, Thursday 1/27/11

HW12 - due at beginning of class, Monday 5/9/11

11.9 (radiating charged ring)

11.15 (ultrarelativistic beaming of radiation) Modify the problem: in finding theta_max, consider only the

ultrarelativistic case, and find only the leading order form in the small quantity epsilon = 1- beta. To simplify

the calculation, anticipate that theta will be << 1 (see why?), and make the small angle approximations from the beginning.

11.22 (KRUD)

11.23 (pulsar spindown) For part (d), first assume the angle is 11°, then try 45°.

Add part (e): How long would it take the pulsar to lose all of it's rotational kinetic energy at this rate?

Treat the pulsar as a uniform sphere of radius 10 km and mass 1.4 M_sun.

(Hint: Let the spin axis be the z-direction, and think of the time dependent part of the dipole moment

as caused by a pair of magnetic dipoles in the x and y directions. The net power radiated is the sum of the

power from each of these, because they are 90 degrees out of phase, so the cross-terms in the Poynting vector

average to zero over a cycle.)

(FYI: While the spin-down rate slows of course as the spin rate slows, the time derivative of the square of the period

is constant in time. This yields a formula for the current age of a pulsar using the measured values of P and dP/dt,

assuming the initial period is much shorter than the currently observed one. Surprisingly (to me at least), this formula

is independent of the inclination, magnetic moment, and moment of inertia.

For an intro to pulsars see: http://www.cv.nrao.edu/course/astr534/Pulsars.html)

S12.1 (rotating dipole) Consider an electric dipole at the origin, rotating in the xy plane, with angular frequency omega.

Use (11.56) and (11.57) to write the electric and magnetic field (a) at a point on the z axis far from the dipole,

and (b) at a point on the x axis far from the dipole. (c) What is the polarization of the radiation in the two

cases (a) and (b)?

S12.2 As shown in class Tuesday 5/03, the vector potential for magnetic dipole radiation is given by

A = (µ

(b) the Poynting vector, and (c) the total radiated power. (Hint: This is almost the same as the electric dipole case.)

HW11 - due at beginning of class, Monday 5/2/11

9.30 (TM waveguide modes)

10.3 (V=0 gauge)

10.5 (gauge transformation for V=0 gauge)

10.12 (net effect of first order time dependence of current (see also Prob. 10.11))

11.3 (radiation resistance)

11.13 (Brehmsstrahlung)

(By contrast, if an 80 keV electron in an X-ray tube comes close enough to a

Tungsten nucleus to be deflected by a large angle, an X-ray carrying a sizable

fraction of the electron kinetic energy can be emitted.)

11.14 (decay of atomic electron orbit) Suggestion: P = -dE(r)/dt = dE/dr dr/dt.

Thus find and solve the differential equation for r(t), to see how long it takes

r to go from its initial value to zero.

HW10 - due at beginning of class, Monday 4/18/11

9.9 (plane electromagnetic waves) Add part (c): If the wave in part (a) reflects from a plane surface

perpendicular to the x direction, what average pressure is exerted on the surface?

9.14 (polarization of reflected and transmitted waves) Make the given problem part (a).

Add parts: Show that for the case of oblique incidence with incident polarization

(b) in the plane of incidence, or (c) perpendicular to the plane of incidence, the reflected

and transmitted waves are also polarized in the plane of incidence or perpendicular to

the plane of incidence, respectively. (Suggestion: Ignore Griffiths' hint. Instead, just show

in parts (a) and (b) that the y-components of the polarizations of the reflected and transmitted

waves must both vanish, and in part (c) that the x-components of these polarizations must vanish.)

9.18 (EM waves in conductors)

9.19 (skin depth) For part (c), compare the B/E ratio to its vacuum value.

9.21 (air to silver reflection coefficient)

9.37a,b,e,f (skip c&d) (evanescent waves)

S10.1 (a) Write the electric and magnetic fields for (i) right and (ii) left circularly polarized,

monochromatic plane waves of frequency omega and phase angle delta, traveling in the

z direction, as a superposition of waves with linear polarizations in the x and y directions.

(You need not specify which is "right" and which is "left" circularly polarized; there are differing

conventions about this terminology.)

(b) Suppose each of the two circular polarized waves in part (a) separately pass through a "quarter wave plate"

that introduces, on top of the original phase relation, an extra phase lag of pi/2 for the y-polarization relative

to the x-polarization. What are the resulting polarizations of the two waves?

HW9 - due due at the beginning of class, Thursday 4/7/11

7.37 (conduction and displacement currents in sea water)

7.34 (spherical, time dependent electric field)

7.51 (self inductance and oscillating wire loop)

[Hint: First show that the total flux through the loop is constant in time.]

8.2 (local conservation of field energy in a charging capacitor)

S9.1 Assume that the electric and magnetic fields in vacuum are independent of the spatial coordinates

x and y, and that they both depend on z and t only via the combination z - vt for some constant v, i.e.,

they have the form E = E(z-vt) and B = B(z-vt). Find the most general fields of this form that satisfy all

of the vacuum Maxwell equations. Neglect fields that are constant in both space and time.

S9.2 Thompson's coil The primary coil has an AC current, and surrounds a ferromagnetic core.

The secondary is the jumping ring. The puzzle was to understand why this thing exerts a

constant nonzero average force. The answer is that the self inductance of the ring creates an

extra phase lag. I set up the problem in class, here you carry it out explicitly.

Let I_p be the current in the primary, and let M be the mutual inductance of the primary and

the ring when the ring is "floating", so the flux through the ring due to the primary current is

M I_p. Let L be the self-inductance of the ring, so the flux through the ring due to the ring's

current I is LI. Let R be the resistance of the ring.

(a) Write the differential equation satisfied by I.

(b) Solve for I in the steady state, using the method of complex exponentials. That is, assume

I_p = I_p0 e^iwt and I = I_0 e^iwt, and solve for the complex amplitude I_0 (w stands for the angular frequency "omega").

Show that if L=0 the ring current lags the primary by π/2, and if L is nonzero it lags by an extra phase, delta, and find delta.

(c) The repulsive force on the ring will be proportional to minus the product of the two currents,

F(t) ~ - Re(I_p)Re(I). Compute the average of this quantity over one cycle, and express the result in

terms of w, R, L, M, and I_p0. (Assume I_p0 is real.)

(d) Find the value of L for which the average force is maximized, with the other quantities held fixed.

(e) Suppose the ring is a loop or radius r and circular cross section of radius a. Express the resistance R

of the ring in terms of the resistivity rho, r, and a.

(f) The self inductance of such a ring at 60 Hz is given by L = µ_0 r [ln(8r/a) - 7/4].

Assume a 60Hz AC current in the solenoid, r = 2 cm, and a = 0.2 cm, and suppose the ring is made of copper

(see Table 7.1). Evaluate the ratio wL/R.

(g) Fixing r = 2cm, for what value of a is the inductive reactance equal to the resistance?

Bonus question (for 20 points extra credit, not required): Design a resistor that has exactly zero self inductance.

(It should be possible to actually fabricate this resistor.)

HW8 - due due at the beginning of class, Thursday 3/31/11

7.1 (resistance between concentric shells) [Suggestion for part c: to answer how you can "account for that",

think about adding resistors in series and in parallel.]

7.7 (sliding circuit)

7.10 (AC generator)

7.12 (induced current in loop) [Neglect any self-inductance of the loop, and use the quasistatic approximation.]

7.16 (induced electric field in coaxial cable) [Use the quasistatic approximation.]

7.49 (diamagnetic response of bound electron) [Note: The idea here is to show that the

mass times the change of centripetal acceleration at constant r matches the change of

centripetal force due to the field dB. Griffiths suggests that you compute the former via

the change in kinetic energy dT (using the work-energy theorem). You could also do it

just by computing the change in speed dv (using the tangential acceleration).]

HW7 - due due at the beginning of class, Thursday 3/10/11

5.34 (physical and pure magnetic dipole) (Note: "consistent ... when z >> R" means "agrees at leading order in 1/z",

which in this case is at order 1/z^3.)

5.36 (dipole moment of a spinning charged shell) (Compute the dipole moment either by considering the sphere as a

stack of loops and using (5.84) for each loop, or using the more general formula for magnetic moment that I gave in

class, the integral of 1/2 r x J. (If you're curious why this integral gives the magnetic moment, one method of showing it is

suggested in Problem 5.60c.) Then use (5.66) and (5.83) to show that the field outside is that of a pure dipole with

precisely the dipole moment that you computed.)

5.56 (gyromagnetic ratio) For (b) assume only that the spinning body has axisymmetric charge and mass density,

and the ratio of charge to mass density is uniform. Replace part (c) by the following: If the electron and proton

were classical spheres with uniform charge to mass density ratio, and angular momentum hbar/2 (as befits a

spin-1/2 particle), what would be the ratio of their magnetic moments?* Express your answer in terms of their masses.

Add part (d): The neutron also has spin hbar/2, and has a magnetic moment opposite to the spin direction and equal

in magnitude to about 0.68 times the proton magnetic moment. (The neutron mass differs from the proton mass by

less than 0.2%, so the mass difference is nearly irrelevant.) How must the charge be distributed in a neutron in order

to produce a magnetic moment opposite to the spin?

* Point of information: As mentioned in the problem in the book, the electron magnetic moment is actually

greater than the uniform estimate by a factor of 2 plus a small QED correction. The magnetic moment of a proton,

on the other hand, is larger than the above estimate by a factor of about 5.58. The difference from the electron

is because the proton is a composite, extended object. Thus the true ratio of electron to proton magnetic moments is

2/5.58 times the mass ratio, i.e. ~ 658.

6.12 (non-uniformly magnetized cylinder)

6.19 (estimate of diamagnetic [and paramagnetic] susceptibility)

Call the given problem part (a), and add parts. For (a) take the size of the orbit to be one Angstrom, the volume

per atom to be (2 Angstroms)^3, and consider the contribution from just one electron per atom.

(Note: This problem concerns only very crude estimates.)

(b) (i) Estimate the electron spin contribution to the paramagnetic susceptibility of liquid oxygen at 73K, coming from one electron

per atom and the same volume per atom as in (a), and assuming that the average spin alignment is weighted by the thermal ratio mB/3kT,

where m is the magnetic moment of the electron, the "Bohr magneton", (see problem 5.56), and is equal to about 5.8 E-5 eV/Tesla.

(This is valid in the limit that mB/kT is small compared to one. (ii) How large is this ratio in this case if B is 1 Tesla?)

(iii) How does your paramagnetic susceptibility estimate compare with the value for liquid oxygen at 73K in Table 6.1?

(It should be the right order of magnitude, as oxygen molecules have unpaired electrons. By contrast, liquid nitrogen

has no unpaired electrons and a much smaller susceptibility. The metals in the table, like aluminum, also have all spins paired.)

6.21 (energy of magnetic dipole interactions)

HW6 - due due at the beginning of class, Thursday 3/3/11

5.3 (J.J. Thompson's e/m measurement)

5.12 (speeding to escape the Force)

5.14 (slab current) (Suggestion: First write down the most general form magnetic field that is compatible with

the planar symmetry of the problem (assume the slab is infinite in the x and y directions), then impose

div B = 0 and curl B = µ

in any direction can always be added to a solution to obtain another solution. Resolve this ambiguity by

imposing symmetry under 180 rotation about the x axis, which is a symmetry of the current density.)

5.24 (vector potential for uniform magnetic field) Let part (a) be the problem as stated.

Add (b) If the uniform field points in the z direction, find a vector potential that depends only upon x and satisfies

div A = 0. (c) Explain why the difference between the potentials in (a) and (b) must be the gradient of a scalar function

that satisfies Laplace's equation, and verify explicitly that this is the case in this example.

5.25 (vector potential of an infinite line current) (Note: This should say find "a" vector potential, not "the" vector

potential, and it should specify that you should find one that satisfies div A = 0.) (Hint: There are several ways to solve this.

Perhaps the easiest way is to first find the magnetic field, then write the simplest form of A consistent with the symmetry

and satisfying div A = 0, and set its curl equal to the magnetic field you previously found. Then solve for A.

5.29 (field inside a spinning charged sphere) (Note: "sphere" means "solid sphere")

5.39 (Hall effect) Add part (d) Eliminate the charge velocity from your answer to part (b) to find the relation

between voltage, current, magnetic field, charge carrier density n (assumed uniform), charge per carrier e,

and the dimensions of the bar.

HW5 - due due at the beginning of class, Thursday 2/24/11

3.32 (field of three charges) - Modify this problem as follows:

(a) Evaluate the monopole and dipole moments.

(b) Write down the monopole and dipole parts of the potential using (a).

(c) Write down the electric field using (3.104) for the dipole part. Write it in two ways:

(i) using the unit vectors in the radial and z directions, (ii) using the unit vectors in the radial and theta directions.

4.4 (force between charge and polarized atom) Compute (a) the force on the charge,

and (b) the force on the dipole using Eq. (4.5). They should be equal and opposite!

4.17 (electret sketches) Be careful to indicate where field lines are kinked, if anywhere.

4.21 (capacitance of dielectric coaxial cable)

4.32 (charge in a dielectric sphere) Note: use Eqn. (1.99) to deal with divergence at the origin.

4.33 (kink in electric field lines at dielectric interface)

S5.1 (Quadrupole from three charges?) (a) Find a configuration of three point charges that has zero monopole and

dipole moments, but nonzero quadrupole moment, and calculate the quadrupole part of the potential (3.95).

(b) Find the quadrupole moment tensor Q_ij for your configuration (see Problem 3.45). Orient your axes

conveniently, and give all nine independent components of Q_ij. (Make sure that your Q_ij is symmetric and

has zero trace.)

S5.2 Charged balloon on wood door

In class I charged a balloon and it stuck to a wooden door. The induced dipoles provided a normal force,

which produced a static friction force that may be what held the balloon up against the force of gravity.

Let's investigate this situation with a simple model: suppose the left half-space x < 0 is filled with a

linear, isotropic dielectric material. The plane x = 0 is the surface of the "door". (a) Suppose the balloon

is a sphere of radius R, uniformly charged at an electrostatic potential V. Use the result of Example 4.8

to compute the net force on the balloon, treating all the charge on the balloon as if it were at the center.

(b) Assuming R = 10 cm, and a dielectric constant of 4 for the wood, and a mass of 1 gram for the balloon,

for what voltage V is the electrostatic force equal in magnitude to the weight of the balloon? (If the

coefficient of static friction is unity, a balloon with this voltage would be supported.)

HW4 - due due at the beginning of class, Thursday 2/17/11

3.15 (cubical box)

3.20 (charged conducting sphere in electric field) - You can obtain the result by a simple modification

of the result in Example 3.8, allowing for the charge Q on the sphere. Be sure to justify why your

modification yields the correct potential. Alternatively, you can revisit the analysis ab initio. If you

proceed this way, note that the potential can no longer vanish everywhere in the equatorial plane.

You may choose where it is to vanish, e.g. on the sphere itself, or in the equatorial plane at infinity.

Apply Gauss' law to fix the value of the 1/r term.

3.21 (spherical expansion for disk potential)

(Hint: Match a Taylor expansion in R/r to the spherical expansion on the axis.)

3.23 (separation of variables in cylindrical coordinates) - (Hints: Follow steps similar to those of section 3.3.2.

Don't forget to impose periodicity under addition of 2 pi to the angle. Also, to expand on Griffiths' tip,

in the case when the separation constants are zero, you should still find two independent solutions to the

differential equation for the radial function. Probably the method you used yields just one solution, the

constant one. To have a complete set of solutions you'll need to find a second solution.)

3.24 (conducting pipe in an electric field)

S4. (Faraday cage leakage) In a class demo it was shown that a wire mesh strainer acts like a good Faraday cage,

shielding electric fields. If we compare the wire mesh to a continuous conductor, the difference is that the charge

on the surface is redistributed in a pattern with a spatial periodicity on the length scale of the distance between the

mesh wires. It is clear that far from the mesh the effects of this redistribution will be small, but how small?

To explore the effect of this pattern of charge density let's consider the simpler problem in which on the z=0 plane

the potential is given by some function V(x,y,0) that is periodic in the x and y directions with period L, given by

a double Fourier series,

V(x,y,0) = V_0(x,y) = Sum_mn V_mn sin(2pi m x/L)sin(2pi n y/L).

(A generic periodic function would include also cosine terms in the Fourier expansion, but the sine series is

sufficiently general to illustrate the point.) Solve Laplace's equation in the half space z > 0 using separation

of variables, with the above boundary condition, together with the condition V(x,y,z) -> 0 as z -> + infinity.

Show that the field falls off exponentially as z grows, and find the slowest possible rate of falloff under these

circumstances. Write the solution in the other half-space z < 0, assuming the potential vanishes as z -> - infinity.

HW3 - due at the beginning of class, Thursday 2/10/11

2.30(b) (cylindrical tube) - do only part(b)

2.36 (spherical cavities in a sphere) (Hint: To answer these questions one can invoke the second uniqueness theorem of Chapter 3.

I think this problem seems out of place in Chapter 2 in that regard. Did you havea way to solve it that doesn't use the uniqueness theorem?)

2.48 (space charge) - (Hint: To solve the differential equation that arises, try V(x) proportional to x

3.8 (point charge and non-grounded conducting sphere)

3.10 (plates at right angle) - You need not answer the questions about plates at angles other than π/2. I think it can be done for π/2

for any integer n. I don't know if there are other cases but it seems unlikely...

S2. Nuclear fission (a) Find the energy of a uniformly charged sphere. (Use any method you like. See problems 2.32 and 2.33 for suggested methods.)

(b) Use your result to determine how the electrostatic energy of a nucleus depends on the charge Z and atomic mass number A.

Take the radius of the nucleus to be A

(c) If a nucleus (A,Z) splits into two nuclei (A/2, Z/2), how much electrostatic energy is liberated? Give your answer both as a fraction

of the initial energy and in MeV. (d) When Uranium-235 absorbs a neutron it can fission into Barium-141+ Krypton-92 + 3 neutrons.

Estimate the electrostatic energy released in such a reaction.

HW2 - due at the beginning of class, Thursday 2/3/11

2.6 (disk) When taking limits, show that for z << R the field approaches the constant value for an infinite

planar charge density, and for z >> R it approaches, at leading order in the small ratio R/z, the field of a point charge.

2.9 (radial r^3 field) Do part (b) by (i) integrating the charge density, and (ii) using Gauss' law.

2.16 (coaxial cable) (Hint: Use symmetry and Gaussian surfaces.)

2.18 (overlapping spherical charges) (Hint: First find the field inside a single sphere, then change the sign and

``shift" this to find the field of the second sphere, then add these fields.)

2.21 (uniformly charged sphere)

2.24 (coaxial cable, potential difference)

2.25(c) (potential of disk) (Do only part (c).)

S1. Dirac delta function and field of a point charge

(a) Compute the divergence of the vector field r/(r

(b) Show that the result is a spike at the origin that becomes infinitely high and infinitely narrow as a goes to zero.

(c) Show that the integral of this spike over all of space is 4π, independent of a. (Note: You may of course use the divergence theorem...)

HW1 - due by 5pm, Friday 1/28/11 (turn in in class Thursday, or to the envelope on Dr. J's office door, Room 4115 by Friday 5pm.)

Reading:

a. Read and familiarize yourself with the vector derivatives, identities, and integral theorems on

the inside of the front cover of Griffiths (a.k.a. the "GIC"), and the spherical and cylindrical coordinates

on the inside of the back cover.

b. Read the Preface and Advertisement, pp. ix to xv.

c. Look through Chapter 1 and read the parts you don't already know about.

Problems:

S1. List the topics in Chapter 1 that you do not feel you already know and understand reasonably well. For each of

these, state if they have been treated to some extent in any math or physics courses you've taken, and if so which courses.

1.48 (3d delta function and integration by parts)

1.57 (Stokes' theorem)

1.61 (vector area)

1.62 (divergence and curl of rhat/r = r/r

In parts (a) and (b) use the spherical coordinate expressions for divergence and curl.

Add parts (c) and (d): compute the divergence and curl using the product rules (5) and (7) from the

GIC. To this end, it's easier to work with the non-unit radial vector r, rather than rhat. First compute

grad r, div r and curl r.