Problem Set 6

Due Thursday, November 14, uploaded to Blackboard by 11:59pm (grader Elise)

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 flowing from wire 1 to wire 2 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 [20 points]
Consider, as a classical model of an electron, a spherical shell of radius R with charge e distributed uniformly over its surface, and 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) 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.
d) 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.
e) Are your results in (c) and (d) physically reasonable for the electron?