The Ampere's Law with some assumptions can be written as: curl (H) = grad(phi) - (1)

The current conservation equation can be written as div.(grad(phi)) - (2)

(2) can also be derived by taking a divergence of (1) since divergence of a curl of any vector should be zero.

So do (2) & (1) represent a complete set of equations or do we need another equation. The other equation could be a gauge condition which is my second question. Is gauge condition absoloutely necessary or is it just needed to simplify a coupled set of PDEs.

I assume you want to simulate a MHD flow... so you need to model eitheir the electric potential (phi) or the induced magnetic field (H)...you have to choose....

If I did not answer you correctly, please detail what you want to model.