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}


References

  1. Sci-Tech Encyclopedia: Electrostatics
  2. wikipedia Electrostatics
  3. wikipedia Static electricity
  4. wikipedia Electric charge
  5. Electric force
  6. Electric force animation
  7. Electric flux
  8. Electric displacement field
  9. Poisson's Equation
  10. Electrostatics Summary
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