Develop Poynting's theorem for the general material medium of relative permittivity \(\kappa_{\&}\) and permeability \(\kappa_{m}\) introduced in Prob. \(8.2 .2\); i.e., substitute the Maxwell curl equations \((8.2 .18)\) and \((8.2 .20)\) in the expansion of \(\nabla \cdot(\mathbf{E} \times \mathbf{H})\) to obtain $$ \oint_{S}(\mathbf{E} \times \mathbf{H}) \cdot d \mathbf{S}+\int_{V}\left(\mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t}+\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}\right) d v+\int_{V} \mathbf{E} \cdot \mathbf{J} d v=0 $$ from which it follows that the Poynting vector is $$ s=\mathbf{E} \times \mathbf{H} $$ and the energy density is $$ \begin{aligned} \left(W_{1}\right)_{\text {leld }} &=\frac{1}{2} \mathbf{E} \cdot \mathbf{D}+\frac{1}{2} \mathbf{H} \cdot \mathbf{B} \\ &=\frac{1}{2} \kappa_{\ell} \epsilon_{0} E^{2}+\frac{1}{2 \kappa_{\mathrm{m}} \mu_{0}} B^{2} \end{aligned} $$ What restrictions on \(\kappa_{\&}\) and \(\kappa_{m}\) are necessary to obtain \((8.4 .24)\) ?

We are given the Maxwell curl equations \((8.2 .18)\) and \((8.2 .20)\) and have to substitute these in the expansion of \(\nabla \cdot(\mathbf{E} \times \mathbf{H})\). The Maxwell curl equations are: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ We know that \(\mathbf{D} = \kappa_{\&} \epsilon_0 \mathbf{E}\) and \(\mathbf{B} = \kappa_{m} \mu_0 \mathbf{H}\). Substituting these relations in the above equations, we get: $$ \nabla \times \mathbf{E} = -\frac{\partial (\kappa_{m}\mu_0 \mathbf{H})}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial (\kappa_{\&} \epsilon_0 \mathbf{E})}{\partial t} $$ Now, we take the divergence of the Poynting vector: $$ \nabla \cdot (\mathbf{E} \times \mathbf{H}) = \nabla \cdot (\mathbf{E} \times (\frac{1}{\kappa_{m}\mu_0} \mathbf{B})) $$ Substituting the Maxwell curl equations, we obtain Poynting's theorem as: $$ \oint_{S}(\mathbf{E} \times \mathbf{H}) \cdot d \mathbf{S} + \int_{V} \left(\mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t} + \mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}\right) d v + \int_{V} \mathbf{E} \cdot \mathbf{J} d v = 0 $$

From the Poynting's theorem, we can find the Poynting vector as: $$ s = \mathbf{E} \times \mathbf{H} $$ The energy density can be found by considering the two components in the above integral expression: $$ \begin{aligned} \left(W_{1}\right)_{\text {leld }} &=\frac{1}{2} \mathbf{E} \cdot \mathbf{D}+\frac{1}{2} \mathbf{H} \cdot \mathbf{B} \\ &=\frac{1}{2} \kappa_{\ell} \epsilon_{0} E^{2}+\frac{1}{2 \kappa_{\mathrm{m}} \mu_{0}} B^{2} \end{aligned} $$

Now, we need to find the restrictions on \(\kappa_{\&}\) and \(\kappa_{m}\) necessary to obtain equation \((8.4 .24)\). Equation \((8.4 .24)\) is given by: $$ \frac{1}{2} \epsilon_{0} E^{2}+\frac{1}{2 \mu_{0}} B^{2} $$ Comparing this equation with the energy density equation derived above, we can see that the restrictions on \(\kappa_{\&}\) and \(\kappa_{m}\) are given by: $$ \kappa_{\&} = 1 $$ $$ \kappa_{m} = 1 $$ We need these restrictions for the energy density equation to match with equation \((8.4 .24)\).