6.2 Electromagnetism at an interface

Em at an interface

We want to understand with Electromagnetism what happens at a surface. From Maxwell's equations we can understand what happens to the components of the $\vec{E}$ and $\vec{B}$ fields: First lets look at the $\vec{E}$ field using Gauss' law. Recall $$\oint \epsilon \vec{E}\cdot d\vec{s}=\int \rho dV$$

Similarly Gauss' law of Magnetism $$\oint \vec{B}\cdot d\vec{s}=0$$ gives $$B_{i\perp }+B_{r\perp }=B_{t\perp }$$ or $${\widehat{u}}_n\cdot {\vec{B}}_i+{\widehat{u}}_n\cdot {\vec{B}}_r={\widehat{u}}_n\cdot {\vec{B}}_t$$

Amperes law can also be applied to an interface.Then

Now we will not consider cases with surface currents. Also we can shrink the vertical ends of the loop so that the area of the box is 0 so that $\int \epsilon \vec{E}\cdot d\vec{s}=0$ . Thus we get at asurface $$\frac{B_{i\parallel }+B_{r\parallel }}{{\mu }_i}=\frac{B_{t\parallel }}{{\mu }_t}$$ or $$\frac{{\widehat{u}}_n\times {\vec{B}}_i}{{\mu }_i}+\frac{{\widehat{u}}_n\times {\vec{B}}_r}{{\mu }_r}=\frac{{\widehat{u}}_n\times {\vec{B}}_t}{{\mu }_t}$$

Similarly we can use Faraday's law $$\int \vec{E}\cdot d\vec{l}=-\frac{d}{dt}\int \vec{B}\cdot d\vec{s}$$ and play the same game with the edges to get $$E_{i\parallel }+E_{r\parallel }=E_{t\parallel }$$ or $${\widehat{u}}_n\times {\vec{E}}_i+{\widehat{u}}_n\times {\vec{E}}_r={\widehat{u}}_n\times {\vec{E}}_t$$ (notice $\epsilon$ does not appear)

In summary we have derived what happens to the $\vec{E}$ and $\vec{B}$ fields at the interface between two media: $${\epsilon }_i{\widehat{u}}_n\cdot {\vec{E}}_i+{\epsilon }_i{\widehat{u}}_n\cdot {\vec{E}}_r={\epsilon }_t{\widehat{u}}_n\cdot {\vec{E}}_t$$

$${\widehat{u}}_n\cdot {\vec{B}}_i+{\widehat{u}}_n\cdot {\vec{B}}_r={\widehat{u}}_n\cdot {\vec{B}}_t$$

$$\frac{{\widehat{u}}_n\times {\vec{B}}_i}{{\mu }_i}+\frac{{\widehat{u}}_n\times {\vec{B}}_r}{{\mu }_i}=\frac{{\widehat{u}}_n\times {\vec{B}}_t}{{\mu }_t}$$

$${\widehat{u}}_n\times {\vec{E}}_i+{\widehat{u}}_n\times {\vec{E}}_r={\widehat{u}}_n\times {\vec{E}}_t$$