DC Conduction Equations

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:

$J = σ E,$

where σ is the electric conductivity.

The electric field E is the gradient of the electric potential V:

$E = - \nabla V .$

Combining this definition with the homogeneous continuity equation

$\nabla \cdot J = - \frac{\partial ρ}{\partial t} = 0,$

where ρ is the current density, yields this equation:

$- \nabla \cdot (σ \nabla V) = 0.$

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 $(n \cdot (σ \nabla V))$ on a face for a 3-D geometry or an edge for a 2-D geometry.