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Ph. D. Qualifying Exam

Electrodynamics October 3, 2012

Directions: Work six of the following eight problems.

1. A plane electromagnetic wave traveling in the z direction is circularlypolarized with positive helicity. The three-vector potential in the Coulomb

gauge (such that the electric field E = At ) is given by

A =1

2A0 (x + iy) e

i(kzt),

where A0 is a (possibly complex) constant, x and y are unit vectors in thex and y directions, respectively, k is the wave number, and = kc, wherec is the speed of light.

(a) A point particle of charge q and mass m is accelerated by this wave.Assume the fields are small enough that the motion is non-relativisticsuch that the particles speed v is much less than c. Show that themagnetic force on the particle in this case is much less than theelectric force.

(b) Solve for x(t), the particles position as a function of time. You mayassume the particle is at the origin (x = y = z = 0) at time t = 0.

2. A point charge with charge q and mass m is attached to the end of amassless, rigid rod of length R. The other end of the rod is attached toa fixed point about which the charge can rotate freely. As a consequence,the charge moves in a circle of fixed radius R in the x y plane. Thecharge is initially moving with speed v0

c. The charge slows down

due to emission of radiation. Compute the speed v as a function of time.Assume that the charge loses only a small amount of radiation per orbit sothat 2, where is the angular frequency of the charge in its orbit.

3. An insulating circular ring of radius R lies in the XY plane, centered atthe origin. It carries a linear charge density 0 = 0 cos , where 0 is aconstant and is the angle in the x y plane measured from the positivex axis.

(a) Make a sketch of this charge distribution using several + and signsto indicate where the charge is most positive or negative. Is this anelectric dipole, a magnetic dipole, or something else? Give an answerand calculate the x,y, and z components of the multipole moment

vector.(b) At time t = 0 the ring is set spinning about the z axis with constant

angular velocity . Is the outgoing wave in the far field region alongeach of the coordinate axes linearly polarized or circularly polarized?In the case of linear polarized waves, please give the direction of theE vector.

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Figure 3: Figure for Problem 6.

6. A beam of white light is incident at an angle on two parallel glass platesseparated by an air film a distance t thick, as shown in figure 3. Thereflected light is admitted into a prism spectrometer. At what wavelengthsin the spectrum are dark lines seen? Take the index of refraction of air tobe n = 1.

7. A laboratory frame has static fields E and B. Find a Lorentz frame inwhich both fields vanish or prove that such a frame does not exist.

8. The charge density of a hydrogen atom in the ground state is given by

(r) =q

a3e2r/a

where a is the Bohr radius and q is electrons charge.

(a) Calculate the electric field of the electron cloud Ee(r). It may helpto know that

0

exx2dx =2

3

1 e

2

2e 1

2e,

where and are positive, real constants.

(b) Calculate the atomic polarizability of the hydrogen atom.