Consider an E-field line of force, i.e., a continuous line everywhere parallel to the local direction of \(\mathbf{E}\), deflected at the boundary between two uniform media. Show that the exit line of force lies in the plane determined by the entrance line and the normal to the boundary surface and that the angles of incidence \(\theta_{1}\) and exit \(\theta_{2}\), measured with respect to the normal, are related by the Snell's law equation $$ \frac{1}{\kappa_{\theta 1}} \tan \theta_{1}=\frac{1}{K_{\theta 2}} \tan \theta_{2} \text {. } $$ What are the corresponding equations for \(\mathbf{B}, \mathbf{D}\), and \(\mathbf{H}\) ?

Answer: The Snell's law equation for the E-field line is given by: \frac{1}{\kappa_{\theta 1}} \tan \theta_{1} = \frac{1}{K_{\theta 2}} \tan \theta_{2} To find the corresponding Snell's law equations for B, D, and H fields, we can use similar boundary conditions and analysis as we did for the E-field, resulting in the following equations: B-field: \frac{1}{\mu_{1}\sin\theta_{1}} = \frac{1}{\mu_{2}\sin\theta_{2}} D-field: \frac{\epsilon_{1}\sin\theta_{1}}{\epsilon_{2}\sin\theta_{2}} = 1 H-field: \frac{1}{\mu_{1}\tan\theta_{1}} = \frac{1}{\mu_{2}\tan\theta_{2}} where θ1 and θ2 are the angles of incidence and exit, ε1 and ε2 are the permittivity of the two media, and μ1 and μ2 are the magnetic permeability of the two media.