boundary conditions for D and H

From Physics wiki

Let us consider the boundary conditions for $\mathbf{D}\,$ and $\mathbf{H}\,$ across the interface between two materials when fields are constant in time. Integrating Maxwell's equations over cylinders and loops straddling the interface gives

$\boldsymbol{\nabla} \cdot \mathbf{D} = \rho_f$ ,		implies		$\hat\mathbf{n} \cdot \left( \mathbf{D}_{out} - \mathbf{D}_{in} \right) = \sigma_f\,$ ,
$\boldsymbol{\nabla} \times \mathbf{E} = 0\,$		implies		$\hat\mathbf{n} \times \left( \mathbf{E}_{out} - \mathbf{E}_{in} \right) = 0\,$ ,

and

$\boldsymbol{\nabla} \cdot \mathbf{B} = 0$ ,		implies		$\hat\mathbf{n} \cdot \left( \mathbf{B}_{out} - \mathbf{B}_{in} \right) = 0\,$ ,
$\boldsymbol{\nabla} \times \mathbf{H} = \mathbf{J}$		implies		$\hat\mathbf{n} \times \left( \mathbf{H}_{out} - \mathbf{H}_{in} \right) = \mathbf{K}_f \,$ ,

back to Maxwell's equations in matter

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boundary conditions for D and H

From Physics wiki

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