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.


BoundaryCond.jpg


Dominio2D.jpg


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}
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