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. \[\hat{\bf x}\cdot\widetilde{\bf E}^i({\bf r}_0) + \hat{\bf x}\cdot\widetilde{\bf E}^r({\bf r}_0) = \hat{\bf x}\cdot\widetilde{\bf E}^t({\bf r}_0) \nonumber \]
