Show wave equations for E & B consistent with potentials

[insynC]

Homework Statement

There are a couple of introductory questions that aren't relevant to the last parts then it says:

(d) If the current density is time independent and divergence free, show that the Maxwell Equations separate into independent equations for E and B.

(e) Express E in terms of the electrostatic potential φ, and B in terms of the vector potential A. Show that the equations in question (d) are satisfied when (in Coulomb Gauge) φ and A satisfy Poisson equations. Write down those equations.

The Attempt at a Solution

I have (with some help) figured out (d) and I got a homogeneous wave equation for B and an inhomogeneous wave equation for E (the inhomogeneity created by a ∇(ρ/ε) term).

However subbing in A and φ for B and E leads only to mess, I have no idea how to simplify it. I looked through a list of identities, but they were no help. I need to find a way to rearrange it so that I can make use of properties like the Coulomb Gauge and Poisson's equation, but I'm not sure how.

Any help would be great, thanks.

 

[gabbagabbahey]

If you show me the mess you are getting, I'll be better able to help you.