Magnetic vector potential of a rotating uniformly charged shell.

Today we will solve the problem of finding magnetic vector potential of a rotating, uniformly charged spherical shell. We won’t discuss the general idea behind the vector potential (how it follows from Helmholtz theorem, and gauge freedom etc) and how its defined. That will be part of a conceptual lecture and will be available when the same would be created. The offline version is available, but the web version will call for a special priority to be assigned.

The problem is quite well defined. We just need to follow the straightforward method of implementing the basic definitions and carrying out the required steps. But we need to be mindful of the framework in which we need to accomplish these steps.

The framework I am talking about here is the coordinate system we need to set in order to solve the problem. Notice that the problem has been stated in the spherical coordinate system (which has been discussed couple of times in this website). But we need not worry about all the aspects of this coordinate system, we will only pick on those which are immediately applicable to our problem.

While this choice of the system where the polar axis (z-axis, wrt which the polar angle θ is measured in a r, θ, φ spherical coordinate system) coincides with the angular velocity vector ω is very natural, it isn’t the most convenient for carrying out the ensuing integral for the vector potential A.