Problem Set 6

Due Wednesday, November 22, by noon in the homework locker

Problem 1 [10 points]
Two infinite parallel wires carrying currents I₁ and I₂ are separated by a distance d. Compute the flux of electromagnetic momentum -∫daT⋅n passing through an infinite plane half way between the wires; the normal n to the plane is in the direction d. Consider both the cases where the currents are parallel and anti-parallel. Interpret your answer.
Problem 2 [10 points]
Consider a spherical conducting shell of radius R that has a total charge Q. Compute the total force on the northern hemisphere of the shell.
Problem 3 [20 points]
Consider, as a classical model of an electron, a uniformly charged spherical shell with charge e and radius R, spinning with angular velocity ω.
a) Compute the total energy contained in the electromagnetic fields.
b) Compute the total angular momentum contained in the electromagnetic fields. If Π is the electromagnetic momentum density, then r×Π is the angular momentum density.
c) According to Einstein, the rest energy of a particle is related to its rest mass by E=mc². If one assumes that all the rest mass m is due to the energy of the electron's electromagnetic field computed in (a), compute the radius R of the electron.
d) Assuming that the total angular momentum computed in (b) is equal to the intrinsic angular momentum of the electron, ℏ/2, compute the angular velocity ω of the electron.
e) Are your results in (c) and (d) physically reasonable for the electron?