In an Ohmic conductor, the current density is related to the electric field as follows:
<math> \mathbf{J(r)} = \sigma \mathbf{E(r)}= -\sigma \nabla \Phi(\mathbf{r}) </math>
where σ is the electric conductivity. In addition, the continuity equation for a stationary current in a closed region requires that
These equations lead to the Laplace equation inside an Ohmic conductor medium:
<math>\nabla^2 \Phi(\mathbf{r}) = 0</math>
In addition, the boundary condition at a conductor-dielectric interface requires a vanishing normal derivative of the electric potential:
<math> \frac{\partial \Phi}{\partial n} = 0 = 0</math>
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At the interface between two contiguous conductors, the normal component of the current density must be continuous.
== Magnetostatics Analysis==