The magnetic field of an infinite straight wire carrying a

Chapter 7, Problem 61P

(choose chapter or problem)

The magnetic field of an infinite straight wire carrying a steady current I can be obtained from the displacement current term in the Ampère/Maxwell law, as follows: Picture the current as consisting of a uniform line charge \(\lambda\) moving along the z axis at speed v (so that \(I=\lambda v\)), with a tiny gap of length \(\epsilon\), which reaches the origin at time t = 0. In the next instant (up to \(t=\epsilon / v\)) there is no real current passing through a circular Amperian loop in the xy plane, but there is a displacement current, due to the “missing” charge in the gap.

(a) Use Coulomb’s law to calculate the z component of the electric field, for points in the xy plane a distance s from the origin, due to a segment of wire with uniform density \(-\lambda\) extending from \(z_{1}=v t-\epsilon\) to \(z_{2}=v t\).

(b) Determine the flux of this electric field through a circle of radius a in the xy plane.

(c) Find the displacement current through this circle. Show that \(I_{d}\) is equal to I, in the limit as the gap width \((\epsilon)\) goes to \(\text { zero. }^{35}\)