(a)\( Generalize the boundary conditions \)(8.6 .5)\( to \)(8.6 .8)\( to include the case where a surface charge density \)\sigma=\Delta q_{\text {tree }} / \Delta S\( and a surface current of magnitude \)K=\Delta I_{\text {fres }} / \Delta l\( exist on the boundary surface, establishing the conditions $$ \text { fi } \begin{aligned} & \cdot\left(x_{22} \mathbf{E}_{2}-x_{41} \mathbf{E}_{1}\right)=\frac{\sigma}{\epsilon_{0}} \\ \text { fi } \times\left(\frac{\mathbf{B}_{2}}{k_{m 2}}-\frac{\mathbf{B}_{1}}{k_{m 1}}\right) &=\mu_{0} \mathbf{K} . \end{aligned} $$ (b) Show that the boundary conditions remain valid when the boundary is not plane and when the respective media are not homogeneous. (c) What are the boundary conditions on the \)\mathbf{D}\( and \)\mathbf{H}$ fields?

The given boundary conditions are: $$ \text { f1 } \cdot\left(x_{22} \mathbf{E}_{2}-x_{41} \mathbf{E}_{1}\right)=\frac{\sigma}{\epsilon_{0}} \\ \text { f2 } \times\left(\frac{\mathbf{B}_{2}}{k_{m 2}}-\frac{\mathbf{B}_{1}}{k_{m 1}}\right) =\mu_{0} \mathbf{K}. $$ #Step 2: Generalize the boundary conditions for surface charge density and surface current#

In order to include the surface charge density \(\sigma\) and surface current \(K\), we can modify the boundary conditions as follows: For electric field: $$ \mathbf{f1} \cdot (\mathbf{E}_2 - \mathbf{E}_1) = \frac{\sigma}{\epsilon_{0}} $$ For magnetic field: $$ \mathbf{f2} \times \left(\frac{\mathbf{B}_2}{\mu_2} - \frac{\mathbf{B}_1}{\mu_1}\right) = \mu_{0} \mathbf{K} $$ #Step 3: Show that the boundary conditions are valid for non-plane boundaries and non-homogeneous media#

The above generalized boundary conditions do not depend on the assumption of a plane boundary or homogeneous media. They were derived based on Maxwell's equations and the assumption that the fields are continuous across the boundary. Therefore, it can be concluded that the boundary conditions remain valid for non-plane boundaries and when the media are not homogeneous. #Step 4: Derive boundary conditions for D and H fields#

Using the constitutive relations, we can derive the boundary conditions for D and H fields: For D field, we use \(\mathbf{D} = \epsilon \mathbf{E}\), where \(\epsilon_1 = \epsilon_0 x_{41}\) and \(\epsilon_2 = \epsilon_0 x_{22}\). $$ \mathbf{f1} \cdot (\epsilon_2 \mathbf{E}_2 - \epsilon_1 \mathbf{E}_1) = \sigma $$ Now, using the constitutive relation for the D field, this becomes: $$ \mathbf{f1} \cdot (\mathbf{D}_2 - \mathbf{D}_1) = \sigma $$ For the H field, we use \(\mathbf{H} = \frac{1}{\mu} \mathbf{B}\), where \(\mu_1 = \mu_0 k_{m1}\) and \(\mu_2 = \mu_0 k_{m2}\). $$ \mathbf{f2} \times \left(\frac{\mathbf{B}_2}{\mu_2} - \frac{\mathbf{B}_1}{\mu_1}\right) = \mu_{0} \mathbf{K} $$ Now, using the constitutive relation for the H field, this becomes: $$ \mathbf{f2} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K} $$ The final boundary conditions on the \(\mathbf{D}\) and \(\mathbf{H}\) fields are: $$ \mathbf{f1} \cdot (\mathbf{D}_2 - \mathbf{D}_1) = \sigma $$ $$ \mathbf{f2} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K} $$