References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Problem 10.9.
One more of calculating the retarded potential. We have a loop of wire in the following shape. It extends along the axis from to , then in a semicircular loop of radius clockwise around to , then along the axis from to , then in a semicircular loop of radius back to . A linearly increasing current
flows through the loop in the direction given above. Assuming the wire is electrically neutral, so our job is to find .
Calculating in general is a complex task, so we’ll look only at the value of at the origin. Consider first the inner loop of radius . All points on this loop are at the same distance from the origin, so the retarded time is the same for all points on the loop. Since the current goes clockwise around the semicircle, the contribution to is
We get a similar expression for the loop around the outer semicircle except this time the current flows counterclockwise so the sign is reversed:
Adding these two together we get
The contributions from each of the two horizontal segments are equal, so for these two segments we have
The total potential is then
Because we have the potential at only a single point in space, we can’t calculate any of its derivatives, so we can’t calculate . However we can calculate :
The electric field is constant in time at the origin. An electrically neutral wire can produce an electric field since the changing current induces a changing magnetic field which in turn produces an electric field.