The magnetic vector potential contributed by a length with current running through it is
What is the magnetic vector potential a distance from a long straight current element?
Assume the wire is placed on the z-axis. Hence the distance from a differential element to a point in space is
Now it is necessary to integrate over the length of the rod. Since the rod is arbitrarily said to be on the z-axis, it can be said, for simplicity, to extend from to This simply requires multiplying by 2. Also, use the substitution
The magnetic field is the curl of the vector potential.
Find the magnetic field in a region with magnetic vector potential
Since is in spherical coordinates, use the spherical definition of the curl.
The only part that will survive for the given is
The partial derivative of the magnetic vector potential contributes partially to the induced electric field according to faraday's law.
Recall Faraday's law: