boundary conditions for E and B

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Let us consider the boundary conditions for \mathbf{E}\, and \mathbf{B}\, across the surface of a conductor when all fields are constant in time. Integrating Maxwell's equations over cylinders and loops straddling the interface gives

\boldsymbol{\nabla} \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}, implies \hat\mathbf{n} \cdot \left( \mathbf{E}_{out} - \mathbf{E}_{in} \right) = \frac{\sigma}{\epsilon_0}\,,
\boldsymbol{\nabla} \times \mathbf{E} = 0\, implies \hat\mathbf{n}\times \left( \mathbf{E}_{out} - \mathbf{E}_{in} \right) = 0\,,

and

\boldsymbol{\nabla} \cdot \mathbf{B} = 0, implies \hat\mathbf{n} \cdot\left( \mathbf{B}_{out} - \mathbf{B}_{in} \right) = 0\,,
\boldsymbol{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} implies \hat\mathbf{n}\times\left( \mathbf{B}_{out} - \mathbf{B}_{in} \right) = \mu_0 \mathbf{K} \,,
back to Maxwell's equations
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