The magnetic field of an infinite straight wire carrying a steady current I can be obtained from the displacement current term in the Ampere/Maxwell law, as follows: Picture the current as consisting of a uniform line charge λ moving along the z axis at speed v (so that I = λv), with a tiny gap of length ε, which reaches the origin at time t = 0. In the next instant (up to t = ε/v) there is no real current passing through a circular AmperJan 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 – λ extending from z1 = vt – ε to z2 = vt.

(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 Id is equal to I, in the limit as the gap width (ε) goes to zero.