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5.7: Plane Waves at Oblique Incidence on a Planar Boundary- TM Case
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/05%3A_Wave_Reflection_and_Transmission/5.07%3A_Plane_Waves_at_Oblique_Incidence_on_a_Planar_Boundary-_TM_Case
Thus, continuity of the tangential component of the magnetic field across the boundary requires $\widetilde{\bf H}_1({\bf r}_0)=\widetilde{\bf H}_2({\bf r}_0)$ , where \({\bf r}_0\triangleq\hat{\bf x...Thus, continuity of the tangential component of the magnetic field across the boundary requires $\widetilde{\bf H}_1({\bf r}_0)=\widetilde{\bf H}_2({\bf r}_0)$ , where ${\bf r}_0\triangleq\hat{\bf x}x+\hat{\bf y}y$ since $z=0$ on the boundary. $\hat{\bf x}\cdot\widetilde{\bf E}^i({\bf r}_0) + \hat{\bf x}\cdot\widetilde{\bf E}^r({\bf r}_0) = \hat{\bf x}\cdot\widetilde{\bf E}^t({\bf r}_0) \nonumber$

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