Electrostatic Boundary Conditions

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Boundary conditions for electrostatic fields at the interface between two different media are:

$\left . \hat n \times (\vec{E}_1-\vec{E}_2) \right |_{\Gamma_{1 \to 2}} = \left . \left ( E_{t_1}-E_{t_2} \right ) \right |_{\Gamma_{1 \to 2}} = 0$

$\left . \hat n \cdot (\vec{D}_1-\vec{D}_2) \right |_{\Gamma_{1 \to 2}} = \left . \left ( D_{n_1}-D_{n_2} \right ) \right |_{\Gamma_{1 \to 2}} = \rho_C$

with $\hat n \,$ the normal vector to the interface between the two media (see picture) and $\rho_C \,$ the charge density in the boundary.

For a domain ${\Omega} \,$ , we should consider three different boundary conditions:

Dirichlet boundary condition:

$\left . V - \bar V = 0 \right |_{\Gamma_{V}}$

Neumann boundary condition:

$\left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}}$

Infinit condition (when no physical boundary are presents -free space-):

$\left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r}$

Electrostatic Boundary Conditions

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