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.. admonition:: Work in Progress
   :class: warning

   This document is still under development and may change frequently.

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Surface Impedance Boundary Condition
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   :caption: Surface Impedance Boundary Condition

A surface impedance boundary condition is applied at the boundaries of well-conducting materials, whose conductivity, although high, cannot be considered infinite. In such cases, it is assumed that the electromagnetic field penetrates the boundary material to a non-negligible skin depth. The amplitude of electric field at the boundary with such a material must satisfy the following equation:

.. math::
   \hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = -i \frac{\omega}{Z_s} \left[\hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}} \times \hat{\mathbf{n}}\right)\right],

where :math:`Z_s = \sqrt{\mu_s/(\varepsilon_s + i\sigma_s/\omega)}` is the surface impedance with :math:`\mu_s`, :math:`\varepsilon_s`, and :math:`\sigma_s` being the permeability, permittivity, and conductivity of the boundary material.