Phys411 - Electricity and Magnetism
of Maryland, College Park
Spring 2011, Professor:
Problems from Griffiths,
Introduction to Electrodynamics, 3rd edition
HW0 - due at
class, Thursday 1/27/11
beginning of class, Monday
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.
11.23 (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
(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
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
For an intro to pulsars see:
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 = (µ0/(4π r c)) mdot x rhat,
the scalar potential is zero. Find (a) the electric and magnetic
(b) the Poynting vector, and (c) the total radiated power. (Hint: This is almost the same as
the electric dipole case.)
beginning of class, Monday
9.30 (TM waveguide
10.3 (V=0 gauge)
10.5 (gauge transformation for V=0 gauge)
10.12 (net effect of first order time dependence of
(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
fraction of the electron kinetic energy can be emitted.)
11.14 (decay of atomic electron orbit) Suggestion: P = -dE(r)/dt = dE/dr
Thus find and solve the differential equation for r(t), to see how long
r to go from its initial value to zero.
- due at
of class, Monday
9.9 (plane electromagnetic waves) Add part (c): If
from a plane surface
perpendicular to the x direction, what average pressure is exerted on
Make the given problem part (a).
Add parts: Show that for the case of oblique incidence with incident
(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
the plane of incidence, respectively. (Suggestion:
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.)
9.19 (skin depth) For part (c), compare the B/E ratio to
its vacuum value.
9.21 (air to silver reflection coefficient)
fields for (i) right and (ii) left
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
HW9 - due due
7.37 (conduction and displacement currents in sea
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).
that satisfy all
of the vacuum Maxwell equations. Neglect fields that are constant in
both space and time.
coil The primary coil has an AC current, and surrounds a
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
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
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
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
HW7 - due due
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
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
in magnitude to about 0.68 times the proton magnetic moment. (The
neutron mass differs from the proton
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
greater than the uniform estimate by a factor of 2 plus a small QED
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
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
5.3 (J.J. Thompson's e/m measurement)
5.12 (speeding to escape the Force)
5.14 (slab current) (Suggestion:
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 = µ0J (or use the integral form, with
amperian loops). Note that a uniform magnetic field
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
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
to find the relation
voltage, current, magnetic field, charge carrier density n (assumed
dimensions of the bar.
HW5 - due due
3.32 (field of three charges) - Modify this problem
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
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?)
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.)
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
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
to compute the net force on the balloon, treating
(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
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
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
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
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
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
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
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
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
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:
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 xn and find the power n that works.)
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/2n
for any integer n. I don't know if there are other cases but it seems
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
(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 A1/3 times 1.25 fermi.
is E-15 meter. Express the result in units of MeV (mega electron volts).
(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
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:
``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/(r3 + a3),
where r is the radial vector and a is a constant length.
(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.)
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
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
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.