DC Conduction

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:

$$J=\sigma E,$$

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

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

where ρ is the current density, yields this equation:

$$-\nabla \cdot \left(\sigma \nabla V\right)=0.$$

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