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When a uniform plane electromagnetic wave propagates in free space and strikes a boundary with a different material, the behavior of the wave can be analyzed using Maxwell’s equations and the boundary conditions at the interface. Let us denote the free space with permittivity (\varepsilon_0), permeability (\mu_0), and assume the material has permittivity (\varepsilon), permeability (\mu), and conductivity (\sigma). The wave is assumed to strike the boundary normally, so the analysis simplifies to one-dimensional propagation along the (z)-axis. The incident electric field vector is given as (\mathbf{E}_i = \mathbf{E}_0 e^{-j k_0 z} \hat{x}), where (\mathbf{E}_0) is the amplitude, (k_0 = \omega \sqrt{\mu_0 \varepsilon_0}) is the wave number in free space, (\omega) is the angular frequency, and (\hat{x}) indicates polarization along the x-direction. The corresponding incident magnetic field can be obtained from the __________ ________ __________ ______ ____.
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