Question

a) If we are not using the "Coulomb Gauge", it is not necessarily true that ▽A-0. We argued that we can always find a gauge transformation that will make this true. Find this transformation explicitly: starting from some general choice of A, show that there always exists a gauge transformation such that ▽ . Anew = 0. b) Assume that ▽ . A asymptotes to zero at infinity. Can you write down a general integral equation one could use to calculate the appropriate for a given initial choice of ?

Answer 1

The electric E^bar and magnetic B^bar field is produced by the scalar and vector potential say V and A respectively. Now suppose field E^bar and B^bar are also produced by another potentials and both potential are related by A'^bar = A^bar + alpha^bar and V' = V + beta Then B' = B nable^bar times A'^bar = nable^bar times A^bar nable^bar times A^bar + nable^bar times alpha^bar = nable^bar times A^bar nabla^bar times alpha^bar = 0 alpha^bar = nabla^bar lambda \r\n We have A_new = A^bar + nabla^bar lambda We assume that nabla^bar middot A^bar = 0 at infinity. So nabla^bar A^bar_new = nabla^bar middot A^bar + nabla^2 lambda nabla^2 lambda = nabla^bar A^bar_new This equation looks like a Poisson equation so its solution is given by lambda = 1/4 pi integral nabla^bar A^bar_new/r d rho where d rho = r^2 sin dr d theta d phi