Lorenz gauged formulation of Maxwell’s equations

mathematics
1

I’m trying to solve an EM problem and finding the E and B fields in a system that only has potential (voltage) sources defined as time-harmonic values in phasor form. I use the scalar potential \Phi and the vector potential A so that E field and B field can be defined as

\begin{align} E =& -j\omega A - \nabla\Phi \\ B =& \nabla \times A \end{align}

By using the Lorenz gauge (\nabla\cdot A + \mu\hat{\sigma}\Phi = 0) the following Helmholtz equation can be derived that can be solved to get the vector potential A.

\nabla^2 A - \mu\epsilon\omega^2 A = -\mu J

Similarly one can also derive a Helmholtz equation for the scaler potential \Phi.

\nabla^2 \Phi - \mu\epsilon\omega^2 \Phi = \frac{\rho}{\epsilon}

However, my problem is that I only have potential sources (voltages) in my problem and there are no currents applied. The medium is conductive so there will be current flowing due to the potentials but how can I incorporate these conditions into the Helmholtz equations?

I can use Dolfinx solve the equation for the scalar potential by incorporating the potential sources as Dirichlet boundary conditions but I cannot calculate E without knowing A and I cannot solve for A without setting the proper boundary conditions that should somehow include the voltage potential sources.

According to the textbook I only need to solve for A as \Phi can be calculated using the Lorenz gauge so this implies I need to set the boundary conditions for the vector potential equation (A) to include the presence of the voltage sources but as A does not represent a charge how can I do this?