Direct current electrical conduction problems, such as electrolysis and computation of resistances of grounding plates, involve a steady current passing through a conductive medium. The current density J is related to the electric field E as follows:
where σ is the electric conductivity.
The electric field E is the gradient of the electric potential V:
Combining this definition with the homogeneous continuity equation
where ρ is the current density, yields this equation:
For DC conduction problems, Dirichlet boundary conditions specify the electric potential V on the boundary. The Neumann boundary conditions specify the surface current density, which is the value of the normal component of the current density on a face for a 3-D geometry or an edge for a 2-D geometry.