University of Maryland, College Park

Spring 2009, Professor: Ted Jacobson

Homework

Problems from Griffiths, Introduction to Electrodynamics, 3rd edition

HW0 - was due with HW1 - please hand in if you haven't yet done so.

HW11 - due at beginning of class, Monday 5/11/09

Note: These problems are all quite straightforward. If you're getting bogged down then you're doing something wrong.

10.3 (V = 0 gauge)

10.5 (gauge transformation from V = 0 gauge)

10.12 (net effect of first order time dependence of current) (See also problem 10.11.)

11.3 (radiation resistance)

11.13 (Brehmsstrahlung)

11.22 (KRUD)

11.23 (pulsar spindown) Add part (e): How long woud 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. (If you're ambitious, work out the angular velocity-dependent spin-down rate, and integrate over time to get the actual time dependence.) (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.)

HW10 - due at beginning of class, Monday 5/4/09

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)

S10.1 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.

HW9 - due at beginning of class, Thursday 4/16/09

9.14 (polarization of reflected and transmitted waves)

9.28 (waveguide example)

9.29 (group velocity and energy flux velocity of TE modes)

9.30 (TM modes)

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

HW8 - due at beginning of class, Thursday 4/09/09

7.37 (conduction and displacement currents in sea water)

8.2 (conservation of energy)

8.6 (conservation of linear momentum)

8.7 (conservation of angular momentum) Also read Problem 8.14 for your edification.

9.8 (circularly polarized string waves)

9.9 (plane electromagnetic waves)

S8.1 Show that for an isolated system of electromagnetic fields in a fluid with infinite conductivity (so-called "ideal MHD") , the magnetic helicity is a conserved quantity. (See problems S7.1,2, and make use of product rule (6) in Griffiths inside cover, among other things.)

HW7 - due at beginning of class, Thursday 4/02/09

7.1 (resistance between concentric shells)

7.12 (induced current in loop) Take the problem as written to be part (a), in which the self-inductance of the loop is neglected. Change notation so that the loop radius is r, and the solenoid radius is 2r. Add more parts: (b) If the self-inductance of the loop is L, what is the current in the loop as a function of time? (c) Suppose the current is carried uniformly within the loop, taken to be a ring of radius r with circular cross-section of radius a. Express the resistance R of the ring in terms of the resistivity rho, r, and a. (d) The self-inductance of the ring turns out to be approximately mu_0 r (ln(8r/a) - 7/4). How does the ratio L/R depend on r and a (neglecting the log)? (e) Assume a 60Hz AC current in the solenoid, r = 2 cm, and a = 0.5 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?

7.34 (spherical, time dependent electric field)

7.43 (spinning spherical conductor in magnetic field) [Compare with Ex. 7.4]

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)

S7.1 (magnetic helicity) The "magnetic helicity" is a measure of linking of magnetic field lines. The helicity of a field configuration can be defined as the integral of A·B over all space, assuming B falls to zero at the boundary. Show that, despite the fact that A itself depends on the gauge, the magnetic helicity is invariant under gauge transformations (addition of the gradient of a scalar to A).

S7.2 Show that in a conducting fluid with infinite conductivity, E·B=0.

HW6 - due at beginning of class, Thursday 3/26/09

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 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)

S6.1 (Aharonov-Bohm effect)

S6.2 (magnetic monopoles)

HW5 - due at beginning of class, Monday 3/09/09

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

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) Express the charge velocity in terms of the current I, charge carrier density n, and charge per carrier e, and use this in your answer to part (b) to find the relation between voltage, current and magnetic field.

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) Replace part (c) by the following: If the electron and proton were classical spheres with uniform charge to mass density, what would be the ratio of their magnetic moments? Express your answer in terms of their masses.

HW4 - due at beginning of class, Thursday 2/26/09

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)

4.21 (capacitance of dielectric coaxial cable)

4.23 (field in a dielectric sphere)

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)

S4.1 In class we charged a balloon and it stuck to the wall. 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 simple planar model: suppose the left half-space x < 0 is filled with a linear, isotropic dielectric material. The plane x = 0 is the "wall". (a) Find the electrostatic force per unit area on a planar "balloon", parallel to the wall, with a surface charge density sigma. Express your answer in terms of the dielectric constant epsilon_r of the material. Does the distance to the wall enter? Check your result using dimensional analysis. (b) Take the limit of your result as the dielectric constant goes to infinity, and show that the result is the same as if the dielectric were replaced by a conducting plane. (Hint: The force can be calculated using the electric field of the bound surface charge on the dielectric. To find this electric field, one approach is to first find the bound surface charge by a method like that of Ex. 4.8. A simpler approach is to express the total electric field both inside and outside the dielectric as a superposition of the electric field of the balloon E_b and the electric field E_d generated by the dielectric surface charge. Continuity of the normal component of the displacement then fixes E_d.)

HW3 - due by 5pm, Friday 2/20/09, turned in to Prof. Jacobson either in class, mailbox, 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 (multipole expansion for disk potential)

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)

S3.1 (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. Then

write the solution in the other half-space z < 0.

HW2 - due at the beginning of class, Thursday 2/12/09

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

2.36 (spherical cavities in a sphere)

2.48 (space charge) - To solve the differential equation that arises, try V(x) proportional to x^n and find the power n that works.

3.8 (point charge and conducting sphere) - For the last part, regarding the neutral (and ungrounded) sphere, I think the idea is

that to produce the field in the presence of a neutral sphere the total image charge inside must vanish - see Example 2.9.

3.10 (plates at right angle)

3.15 (cubical box)

HW1 - due at the beginning of class, Thursday 2/5/09

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

while 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)

2.18 (overlapping spherical charges)

2.21 (uniformly charged sphere) It seems simplest to get V using eqn. (2.21) and the E field found using Gauss' law.

2.24 (coaxial cable, potential difference)

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

2.33 (energy of uniformly charged sphere) Add part: use your result to estimate the electrostatic energy of a nucleus

with charge Z. Take the radius of the nucleus to be A^1/3 times 1.25 fermi, where A is the atomic mass number.

One fermi is E-15 meter. Express the result in units of MeV. [Also take a look at problem 2.32 for educational purposes.

If you want to practice and to test yourself, do 2.32 as well, but I think 2.33 is the simplest way to find the result.]