### Perfect Conductor Inside of the conductor: - $\phi=\text{Constant}$ - $\vec{E}=0$ - $\vec{B} = 0$ (6.2210 assumption; strictly only $\frac{ \partial \vec{B} }{ \partial t }=0$) At the boundary: - For EQS systems: - $\hat{n}\cdot \bigl[ \lvert \epsilon_{0}\vec{E} \rvert \bigr]=\rho_{s}$ - $\hat{n}\cdot \bigl[ \lvert \vec{J} \rvert \bigr]=-\frac{ \partial \rho_{s} }{ \partial t }$ - $\hat{n}\times \bigl[ \lvert \vec{E} \rvert \bigr]=0$ - $\vec{E}$ terminates perpendicular to boundary - For MQS systems: - $\hat{n}\cdot \bigl[ \lvert \vec{B} \rvert \bigr]=0$ - $\hat{n}\cdot \bigl[ \lvert \vec{J} \rvert \bigr]=0$ - $\hat{n}\times \bigl[ \lvert \vec{E} \rvert \bigr]=0$ - $\hat{n}\times \bigl[ \lvert \vec{B} \rvert \bigr]=\mu_{0}\vec{K}$ - $\vec{B}$ terminates parallel to boundary Note: $\bigl[ \lvert \mathcal{X} \rvert \bigr]=\mathcal{X}^{a}-\mathcal{X}^{b}$, where $\hat{n}$ points from region $b$ into region $a$ ### General Interface (time-varying fields, any media) With $\hat n$ pointing from $b$ into $a$, and surface current $\vec K$ and surface charge $\sigma_s$: $ \hat n\times[\![\vec E]\!]=0,\qquad \hat n\times[\![\vec H]\!]=\vec K,\qquad \hat n\cdot[\![\vec D]\!]=\sigma_s,\qquad \hat n\cdot[\![\vec B]\!]=0 $ **Two dielectrics (no free surface sources):** - Tangential $\vec E$ and tangential $\vec H$ are continuous. These two are the independent conditions. - Normal $\vec D$ and normal $\vec B$ are continuous. These follow from the tangential ones for time-harmonic fields.