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 λ moving along the z axis at speed υ (so that I = λυ), with a tiny gap of length ε , which reaches the origin at time t = 0. In the next instant (up to t =ε/υ) 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 z_{1}=v t-\epsilon \text { 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 (ε) goes to zero .^{35}