## 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:

`$J=\sigma E,$`

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:

`$-\nabla \cdot \left(\sigma \text{\hspace{0.17em}}\nabla V\right)=0.$`

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 $\left(n\cdot \left(\sigma \text{\hspace{0.17em}}\nabla V\right)\right)$ on a face for a 3-D geometry or an edge for a 2-D geometry.