When matter is present, the phenomenon of polarization (electrical displacement of charge in a molecule or alignment of polar molecules) can produce unneutralized (bound) charge that properly contributes to \(\rho\) in \((8.2 .1)\). Similarly the magnetization of magnetic materials, as well as time-varying polarization, can produce efiective currents that contribute to \(J\) in \((8.24)\). These dependent source charges and currents, as opposed to the independent or "causal" free charges and currents, can be taken into account implicitly by introducing two new fields, the dectric displacement \(\mathbf{D}\) and the magnetic intensity \(\mathbf{H} .+\) For linear isotropic media, $$ \begin{aligned} &\mathbf{D}=\kappa_{\varepsilon} \epsilon_{0} \mathbf{E} \\ &\mathbf{H}=\frac{\mathbf{B}}{\kappa_{m} \mu_{0}} \end{aligned} $$ where \(\kappa_{0}\) is the relative permittivity (or dielectric constant) and \(\kappa_{m}\) is the rclative permeability of the medium. In this more general situation, Maxwell's equations are $$ \begin{aligned} &\nabla \cdot \mathbf{D}=\rho_{\text {ree }} \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{H}=\mathbf{J}_{\text {frea }}+\frac{\partial \mathbf{D}}{\partial l} \end{aligned} $$ Show that in a homogeneous material medium without free charges or currents, the fields obey the simple wave equation with a velocity of propagation $$ c^{\prime}=\frac{1}{\left(x_{q} \operatorname{tos}_{m} \mu_{0}\right)^{1 / 2}}=\frac{c}{\left(\alpha_{q} K_{m}\right)^{1 / 2}} $$ and that consequently the refractive index of the medium is given by $$ n=\left(x_{q} K_{m}\right)^{1 / 2} $$

Step by step solution

01

General form of the wave equation

The general form of the wave equation is given by: $$ \nabla^2 \mathbf{F} - \frac{1}{v^2}\frac{\partial^2 \mathbf{F}}{\partial t^2} = 0 $$ where \(\mathbf{F}\) represents a generic field, and \(v\) is the propagation velocity.

02

Substitute the expressions of \(\mathbf{D}\) and \(\mathbf{H}\) in Maxwell's equations

Let's substitute the expression for \(\mathbf{D}\) and \(\mathbf{H}\) in the Maxwell's equations: $$ \begin{aligned} &\nabla \cdot (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})=\rho_{\text {fee }} \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right)=\mathbf{J}_{\text {free }}+\frac{\partial (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t} \end{aligned} $$

03

Consider no free charges and currents

As there are no free charges or currents in the material medium, we set \(ρ_{\text{free}} = 0\) and \(J_{\text {free}} = 0\): $$ \begin{aligned} &\nabla \cdot (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})=0 \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right)=\frac{\partial (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t} \end{aligned} $$

04

Use the curl of the curl identity

We will use the following identity for the curl of the curl: $$ \nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2 \mathbf{F} $$ Applying this identity to the second and fourth equation: $$ \begin{aligned} &-\nabla \times (\nabla \times \mathbf{E}) = \frac{\partial^2 \mathbf{B}}{\partial t^2} \\ &\nabla(\nabla \cdot \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right)) - \nabla^2 \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right) = \frac{\partial^2 (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t^2} \end{aligned} $$

05

Show that the resulting equations satisfy the wave equation

Now, we have the following equations: $$ \begin{aligned} &-\nabla \times (\nabla \times \mathbf{E}) = \frac{\partial^2 \mathbf{B}}{\partial t^2} \\ &\nabla^2 \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right) = \frac{\partial^2 (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t^2} \end{aligned} $$ These equations have the form of the wave equation, as shown in Step 1.

06

Find the propagation velocity and refractive index

From the wave equation, we can find the propagation velocity \(c'\) and the refractive index \(n\) of the medium: $$ c^{\prime}=\frac{1}{\sqrt{\kappa_{\varepsilon} \kappa_{m} \epsilon_{0} \mu_{0}}}=\frac{c}{\sqrt{\kappa_{\varepsilon} \kappa_{m}}} $$ and $$ n=\sqrt{\kappa_{\varepsilon} \kappa_{m}} $$

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