Problem Set 3

Problem Set 3

Due Friday, February 26, by 4pm in the homework locker

Problem 1 [10 points]
Suppose the electric and magnetic fields are:

E(r,t) = 1
4πε₀ q
r² θ(r-vt)e_r and B(r,t)=0

where e_r is the unit vector in the radial direction, and θ(x)=1 for x>0 and θ(x)=0 for x<0. Show that these fields satisfy all four of Maxwell's equations, and determine the charge density ρ and current density j. Describe the physical situation that gives rise to these fields.
Problem 2 [20 points]
Imagine the electon as a classical sphere of radius R, that has a uniform surface charge density with total charge -e, and is spinning about the z axis with angular velocity ω.
(a) Calculate the total energy contained in the electromagnetic fields (you of course have to first find, or look up, the E and B fields for this configuration).
(b) Calculate the total angular momentum contained in the electromagnetic fields.
(c) According to Einstein's formular E=mc², the energy contained in the electromagnetic fields should contribute to the mass of the electron. Suppose that all the rest mass of the electron is made of this electromagnetic energy. Similarly, imagine that the intrinsic angular momentum of the electon, hbar/2, is given by the total angular momentum of the electromagnetic fields. What would the radius R of the electron and its angular velocity ω have to be in order to get the observed electron rest mass and intrinsic angular momentum? Do these answers make physical sense?
Problem 3 [10 points]
(a) Find the electric and magnetic fields, and the charge and current densities, corresponding to the potentials

V(r,t)=0 and A(r,t) = -1
4πε₀ qt
r² e_r

(b) Use the gauge function λ(r,t) = -(1/4πε₀)(qt/r) to transform the potentials, and comment on the result.
Problem 4 [10 points]
Consider Maxwell's equations in the presence of charge and current sources ρ and j. Show that E and B now satisfy the inhomogeneous wave equation
∇²E − μ_oε_o∂²E/∂t² = g(r, t) and ∇²B − μ_oε_o∂²B/∂t² = h(r, t)
and express the functions g(r, t) and h(r, t) in terms of the charge density ρ(r, t) and the current density j(r, t) (i.e. the functions g and h should depend only on ρ and j and not on E and B).