Assignment 2

Problem 1:

A thin ring of radius R made from a dielectric carries a total charge Q uniformly distributed over its circumference.  It lies in the xy-plane centered at the origin and is rotating about the z-axis with angular velocity ω k.
(a)  Write down a lowest-order expression for E(r) for r >> R.
(b)  Write down a lowest-order expression for B(r) for r >> R.
(c)  Does the ring produce electromagnetic radiation?  Why or why not?
(d)  Is the Pointing vector zero or nonzero for r >> R?

Problem 2:

A finite spherically symmetrical charge distribution disperses under the influence of mutually repulsive forces. 
Suppose that the charge density ρ(r, t), as a function of the distance r from the center of symmetry and of time, is known. 

(a)  Prove that the curl of the magnetic field is zero at any point.
(b)  Use this to show that B is zero at any point.

Problem 3:

Assume magnetic charges exist and Maxwell's equations are of the form
E = ρ/ε0,   B = μ0ρm,   -E = μ0jm + ∂B/∂t,   ∇B = μ0j + μ0ε0E/∂t.
Assume a magnetic monopole of magnetic charge qm is located at the origin, and an electric charge qe 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 (r) of the electromagnetic field.
(c)  Show that there is electromagnetic angular momentum Lz about the z-axis and derive an expression for it.  Show that Lz is independent of R.
Useful vector identity:  (a)n = (1/r)[a - n(an)].   Here n is r/r is the unit radial vector.

Problem 4:

Given are Maxwell's equations in macroscopic form,
D = ρfE = -∂B/∂t,  B = 0,  H = jf + ∂D/∂t,
and the constitutive relations for a linear, isotropic medium
D = εEB = μH.

Let jf = σcE + jb, where σc is the conductivity of the medium and jb represents the density of forced currents, such as currents inside a battery.

(a)  What is the relationship between the field quantities E and B and the potentials A and V?
(b)  E and B do not specify A and V uniquely.  Let A = A' - ψ,  V = V' + ∂ψ/∂t.  Show that Maxwell's equations are satisfied by A' and V' if they are satisfied by A and V.
(c)  Briefly discuss the gauge transformation represented in part b and its implications.
(d)  Write the relationship between the fields given in answers to part (a) in terms of H, D, A, and V.
(e)  Find the wave equation for A in terms of V, μ , ε , jb, σc , and  v = (μ0ε0)- in the Lorentz gauge.
(f)  Starting with Maxwell's equations, derive the wave equation for V in the Lorentz gauge.

Problem 5:

A thin wire of radius b is used to form a circular wire loop of radius a (a >> b) and total resistance R.
The loop is rotating about the z-axis with constant angular velocity ωk in a region with constant magnetic field B = B0i
At t = 0 the loop lies in the y-z plane and the point A at the center of the wire crosses the y-axis.

Let the (θ/θ) direction be tangential to the loop and be equal to the positive z direction at point A.
Let the (φ/φ) direction be tangential to the wire and be equal to the direction indicated in the figure.
(a)  Find the current flowing in the loop.  Neglect the self-inductance of the loop.  What is current density J as a function of time?
(b)  Find the thermal energy generated per unit time, averaged over one revolution.
(c)  Write down an expression for the the pointing vector S on the surface of the wire.
(d)  Use S to find the field energy per unit time flowing into the wire, averaged over one revolution.