The magnetic field of an infinite straight wire carrying a steady current I can be obtained from the displacement current term in the Amp\`ere/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 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 −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.