University of Maryland, College Park

Spring 2010, Professor: Ted Jacobson

Homework

Problems from Griffiths, Introduction to Electrodynamics, 3rd edition

HW0 - due at beginning of class, Monday 1/28/10

HW11 - due at beginning of class, Thursday 5/06/10

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.

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. You may assume that the net power radiated

is the sum of that from each of these, which it is since the cross-terms in the Poynting vector average to zero over a cycle.)

(Optional: Show that while the spin down rate slows as the spin rate slows, the time derivative of the square of the

period is constant in time, and derive from this 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.

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

S11. (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)?

HW10 - due at beginning of class, Thursday 4/29/10

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))

10.13 (potential for circulating charge)

10.20 (fields for circulating charge) Modify the problem: (a) Find the fields for any point on the axis.,

not just at the center, and (b) find the leading order contribution to the Poynting vector on the axis at large

distance from the charge. Skip the part about the steady current loop.

Suggestion: This problem is easier to deal with if you use vector notation, taking advantage of vector

orthogonality, etc. To this end, let z be the field point on the axis, and let w(t) = R [cos(wt) xhat + sin(wt) yhat]

be the position of the particle on the circle at time t (where w in the argument of the sin and cos is omega, not double-u).

Note that z is orthogonal to w, v, and a; and v is orthogonal to w and a.

HW9 - due at beginning of class, Thursday 4/15/10

9.14 (polarization of reflected and transmitted waves)

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)

9.25 (group and phase velocity with resonant dispersion)

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

S9. Materials with negative permittivity and permeability (for a certain range of frequencies) were first

discussed in 1967 by Veselago, and in recent years so-called "metamaterials" with these properties have

been fabricated. They exhibit a negative index of refraction. See Reversing Light With Negative Refraction,

by John B. Pendry and David R. Smith. That article attributes this phenomenon to the need to take the negative

square root in computing the index, due to analytic properties of the solutions for the fields. This may be so,

but it is at least somewhat obscure, and there is a simpler way to view this phenomenon. Consider then a

linear medium (the metamaterial) with negative dielectric permittivity epsilon and magnetic permeability mu,

and take a look at Maxwell's equations in this medium, in the absence of free charge density or current density.

The equations are identical to those with permittivity |epsilon| and permeability |mu|, so it looks at first as if

there is no difference at all. However, the Poynting vector in a medium is given by (1/mu)ExB, whose sign flips

with the sign of mu. Hence the energy flow is anti-parallel to the wave vector in a plane wave. When electromagnetic

radiation strikes a plane interface from, say, vacuum to such a material, Snell's law must hold as usual in terms

of the tangential component of the wave vector (cf. Sec. 9.3.3 of Griffiths), but the wave vector is antiparallel

to the energy flux. (a) Use this to derive the version of Snell's law that relates the energy flux directions, rather

than the wave vector directions. (b) Explain the functioning of the planar slab lens in Fig. 4(c) of the article linked above.

HW8 - due at beginning of class, Thursday 4/08/10

7.37 (conduction and displacement currents in sea water)

8.2 (conservation of energy)

8.7 (conservation of angular momentum)

9.8 (circularly polarized string waves)

9.9 (plane electromagnetic waves)

HW7 - due at beginning of class, Thursday 4/01/10

7.1 (resistance between concentric shells)

7.12 (induced current in loop) [Neglect any self-inductance of the loop.]

7.34 (spherical, time dependent electric field)

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).]

7.51 (self inductance and oscillating wire loop)

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

S7. 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. 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.

(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 small cylinder of radius r and height r. Assuming the current is carried

uniformly within the ring, and (crudely) treating the magnetic field as uniform inside, find the self-inductance of the ring.

(f) Express the resistance R of the ring in terms of the resistivity rho, r, and the thickness of the ring wall a<<r.

(g) How does the ratio L/R depend on r and a?

(h) 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 omega L/R, where omega is the angular frequency. For what values of

a and r is the inductive reactance equal to the resistance?

HW6 - due at beginning of class, Monday 3/29/10

5.25 (vector potential of an infinite line current)

5.29 (field of a spinning charged sphere)

5.36 (dipole moment of a spinning charged shell) (Compute the dipole moment from integrating (5.84),

and then use the results in (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.)

6.12 (non-uniformly magnetized cylinder)

6.19 (estimate of diamagnetic susceptibility) (For the size of the orbit take one Angstrom, and for the

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

This is supposed to be only a rough, order of magnitude estimate. Besides its inherent roughness

regarding diamagnetism, it neglects any competing residual paramagnetic effects.)

6.21 (energy of magnetic dipole interactions)

6.24 (slick trick for finding potentials for uniform polarization or magnetization)

S6. (Aharonov-Bohm effect)

HW5 - due by 5pm, Friday 3/05/10, turned in either in class before that or at Dr. J's office,

Room 4115 (under the door if he is not there).

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

5.12 (speeding to escape the Force)

5.14 (slab current)

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.

5.34 (physical and pure magnetic dipole)

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, charge per carrier e, and the dimensions of the bar.

5.55 (dipole in a uniform field) (Suggestion: Find the value of r for which the dot product of the total field with the

radial unit vector vanishes. Use (5.87) to write the dipole part.)

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.

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

greater than the uniform estimate by almost exactly a factor of 2. 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 comes from

the fact that the proton is a composite object, with finite sized substructure. Thus the true ratio of electron

to proton magnetic moments is 2/5.58 times the mass ratio, i.e. ~ 658. By the way, 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.) Evidently there is a separation of positive and negative charge inside the

neutron, with the positive charge concentrated toward the center.

HW4 - due at beginning of class, Monday 3/01/09

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)

S4. Charged balloon interaction with wall

In class we charged a balloon and it stuck to the wooden wall (door). I guess the induced dipoles provided a normal force,

as well as a vertical force acting on the curved balloon surface, which together with static friction held the balloon up

against the force of gravity. Let's investigate the dipole force in this situation with a (ridiculously) simple planar 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 "wall".

The "balloon" is an infinite plane, parallel to the wall, with a surface charge density sigma.

(a) Find the electrostatic force per unit area on the balloon. Express your answer in terms of the dielectric constant epsilon_r

of the wood. Does the distance to the wall enter? Check your result using dimensional analysis.

(Note that if the wall were replaced by a slab of finite thickness, the net force would be zero. To simplify the more complicated

problem of a finite sized balloon, I have pushed the back side of the door off to infinity, thus ignoring its contribution to the force.)

(b) Take the limits of your result as the dielectric constant goes to unity and to infinity,

and show that the result is the same as if the dielectric were replaced by vacuum or a conducting plane, respectively.

(c) Determine the surface charge density of the balloon by assuming the balloon is an isolated sphere of radius 10 cm at an

electrostatic potential of 1000 V (it could be several times this after I rubbed it with the fur.) Assume the dielectric constant of the

wood is 4, and calculate the force per unit area in the planar model with this surface charge density. Find the force on

100 square centimeters, and compare this to the weight of the balloon, assumed to have a mass of 1g.

(Hint: The force can be calculated using the electric field of the bound surface charge on the wall.

To find the bound surface charge I suggest you follow the method of Ex. 4.8.)

HW3 - due by 5pm, Friday 2/19/10, turned in to Prof. Jacobson either in class, or at his office (under the door if he is not there).

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) - Follow steps similar to those of section 3.3.2.

Don't forget to impose periodicity under addition of 2 pi to the angle.

3.24 (conducting pipe in an electric field)

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.

3.38 (multipole expansion of charged line segment)

S3. (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.

HW2 - due at the beginning of class, Thursday 2/11/10

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

2.36 (spherical cavities in a sphere)

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...

3.15 (cubical box)

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.

HW1 - due at the beginning of class, Thursday 2/4/10

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 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.)

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.