Problem 1:

Consider the magnetic vector potentials A₁ = (-By, 0, 0) and A₂ = -½(By, -Bx, 0).
(a)  Show that both vector potentials are associated with the same magnetic field B.
(b)  Construct a gauge transformation ψ(r) which connects the two representations of the vector potential.

Problem 2:

The Aharanov-Bohm experiment is illustrated in the figure below.

It is a two-slit electron scattering experiment where a solenoid is placed in the region behind the screen and between the two classical paths that electrons passing through the slits would follow to reach a point on the screen.  The long and thin solenoid confines the magnetic field to regions that the electrons should not pass through.  In terms of the cylindrical coordinates in Fig (b), the magnetic field may be assumed given by

inside solenoid:              B_r = 0,  B_φ = 0,  B_z = B.
outside solenoid:            B_r = 0,  B_φ = 0,  B_z = 0.

(a)  Show that a vector potential given in the cylindrical coordinates by
inside the solenoid:        A_r = A_z = 0,   A_φ = Br/2.
outside the solenoid:     A_r = A_z = 0,  A = BR²/(2r),
leads to the magnetic field components inside and outside the solenoid given above.

Thus, in the Aharanov-Bohm experiment the electrons never experience a finite magnetic field but they may encounter a non-zero vector potential outside the solenoid.

(b)  What does this result, and that in the Aharanov-Bohm experiment the interference pattern is observed to be shifted when current is flowing in the solenoid, say about the relative importance of the magnetic field and the vector potential in classical and quantum mechanics?

Problem 3:

The concentric cylindrical shells of a cylindrical capacitor have radii a and b > a,  respectively, and height h >> b.  The capacitor charge is Q, with +Q on the inner shell of radius a, and -Q on the outer shell of radius b (see figure).  The whole capacitor rotates about its axis with angular velocity ω = 2π/T.  Neglect edge effects.

(a)  Find the capacitance of the capacitor. 
(b)  Evaluate the magnetic field B generated by the rotating capacitor over all space.
(c)  Find the direction and magnitude of the Poynting vector.

Problem 4:

Assume magnetic charges exist and Maxwell's equations are of the form
∇∙E = ρ/ε_0,   ∇∙B = μ₀ρ_m,   -∇×E = μ₀j_m + ∂B/∂t,   ∇×B = μ₀j + μ₀ε₀∂E/∂t.
Assume a magnetic monopole of magnetic charge q_m is located at the origin, and an electric charge q_e is placed on the z-axis at a distance R from it.
(a)  Write down expressions for the electric field E(r) and the magnetic field B(r).  Make a sketch.
(b)  Write down expressions for the momentum density g(r) and angular momentum density ℒ(r) of the electromagnetic field.
(c)  Show that there is electromagnetic angular momentum L_z about the z-axis and derive an expression for it.  Show that L_z is independent of R.
Useful vector identity:  (a∙∇)n = (1/r)[a - n(a∙n)].   Here n is r/r is the unit radial vector.