3.2.4.7. Voltage Equation

The electric potential or voltage $V$ is frequently used in determining the electric field, $\vector{E} = -\grad V$ . While equation (3.1) cannot be applied to the voltage, the equation governing the voltage – Gauss’ law from Maxwell’s equations – has a similar form. Writing the electric displacement $\vector{D}$ as $\vector{D} = \elecperm\vector{E}$ , where $\elecperm$ is the electric permittivity, Gauss’ law is

(3.28) $- \div\elecperm\grad V = \density_e$

where the permittivity is taken to be a constant and $\density_e$ is the volumetric free charge density.

Using equation (3.3), the G/FEM residual form is

(3.29) $\symRes_V^i = \int\limits_\Vol \left(-\density_e \phi^i + \grad\phi^i\bcdot\elecperm\grad V\right)\dV + \int\limits_\Surf q_n\phi^i\dS = 0$

In Aria, each term in (3.29) is specified separately as identified in equation (3.30).

(3.30) $\symRes_V^i = - \underbrace{\int\limits_\Vol \density_e \phi^i\dV}_\mathrm{SRC} + \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\elecperm\grad V\dV}_\mathrm{DIFF} + \int\limits_\Surf q_n\phi^i\dS = 0$