Use Maxwell's equations,
to derive expressions for and in terms of and .
Now suppose that there exists a scalar potential such that , and as . If is spherically symmetric, calculate using Gauss's flux method, i.e. by integrating a suitable equation inside a sphere centred at the origin. Use your result to find and in the case when for and otherwise.
For each integer , let be the sphere of radius centred at the point . Suppose that vanishes outside , and has the constant value in the volume between and for . Calculate and at the point .