Electromagnetic Waves

In order to derive the electromagnetic wave equation in the general form, the properties of the medium can be considered [Hecht89]. If we consider a medium we get 3 additional constitutive equations. If the medium is a dielectric medium and is subjected to an applied electric field, then the internal charge distribution distorts under its influence. This leads to the generation of electric dipole moments which contribute an additional field moment. The resultant dipole moment per unit volume is called electric polarization P. Since the field in the material has changed, a new field quantity has to be defined, the displacement $\vec{D}$

$$\vec{D}(\vec{r},t)= \varepsilon \vec{E}(\vec{r},t) + \vec{P}(\vec{r},t)$$ (A.17)

The internal electric field $\vec{E}$ is therefore the difference between the field $\vec{D}/\varepsilon_0$, which would exist in absence of a polarization and the field $P/\varepsilon_0$ arising from the polarization. In the same way we can define a magnetic polarization or magnetization vector $\vec{M}$ as the magnetic dipole moment per volume. In order to cover the effect of the magnetically polarized medium a vector $\vec{H}$, known as the magnetic field intensity, is introduced.

$$\vec{H}(\vec{r},t)= \frac{1}{\mu} \vec{B}(\vec{r},t) + \vec{M}(\vec{r},t)$$ (A.18)

The third constitutive equation is given by

$$\vec{J}(\vec{r},t)= \sigma (\vec{r},t) \vec{E}(\vec{r},t) + \vec{J}_{e}(\vec{r},t)$$ (A.19)

$\varepsilon_0$ is the electric permittivity and $\mu_0$ the magnetic permeability in free space. For media other than free space the relative dielectricity $\bar{\varepsilon}(\vec{r})$, the relative permeability tensor $\bar{\mu}(\vec{r})$ and the electric conductivity tensor $\bar{\sigma}(\vec{r})$ are generally dependent on the spacial location $\vec{r}$ in the medium. For homogeneous media the $\bar{\varepsilon}(\vec{r}) = \varepsilon_r$, $\bar{\mu}(\vec{r})=\mu_r$ and $\bar{\sigma}(\vec{r}) =\sigma$ are constant for any location $\vec{r}$ in the medium. A medium is called isotropic if $\bar{\varepsilon}$, $\bar{\mu}$ and $\bar{\sigma}$ are scalar quantities. For linear (non-ferroelectric and non-ferromagnetic) media $\vec{P}$ and $\vec{E}$ as well as $\vec{B}$ and $\vec{M}$ are parallel and proportional.
Therefore, for homogeneous, isotropic, linear media the constitutive relations can be written as

$$\vec{D}(\vec{r},t) = \varepsilon_0 \varepsilon_r \vec{E}(\vec{r},t)$$

$$\vec{B}(\vec{r},t) = \mu_0 \mu_r \vec{H}(\vec{r},t)$$

$$\vec{J}(\vec{r},t) = \sigma \vec{E}(\vec{r},t) + \vec{J}_{e}(\vec{r},t)$$ (A.20)

where $\mu_r$ denotes the relative permeability and $\varepsilon_r$ is the relative permittivity of the medium.

Under this conditions the Maxwell equations for linear homogeneous media can be written as

$$\nabla \times \vec{E}(\vec{r},t) = - \frac {\partial }{\partial t} \vec{B}(\vec{r},t)$$ (A.21)

$$\nabla \times \vec{B}(\vec{r},t) = \mu_0 \mu_r \sigma \vec{E}(\ve... ..._r \varepsilon_0 \varepsilon_r \frac {\partial} {\partial t} \vec{E}(\vec{r},t)$$ (A.22)

$$\nabla \cdot \vec{E}(\vec{r},t) = \frac{1}{\varepsilon_0 \varepsilon_r} \rho$$ (A.23)

$$\nabla \cdot\vec{B}(\vec{r},t)= 0$$ (A.24)

One of the most interesting consequences arising from Maxwell 's equations is the prediction of the existence of electromagnetic waves, long before Heinrich Rudolf Hertz (1857-1894) experimentally proved their existence in 1887.
The general form of the wave motion equation is

$$\nabla^2 \Psi- \frac{1}{v^2}\frac{\partial}{\partial t^2}\Psi = g(\vec{r},t)$$ (A.25)

where $\Psi$ is one of the field quantities, v the propagation velocity and the disturbance term $g(\vec{r},t)$ indicates the source of wave generation.
Assuming a homogeneous, isotropic, linear media we can prove, that the Maxwell equations yield a wave equation. To achieve this, we form some second derivatives with respect to the space variables. Taking the curl of (A.23) we obtain.

$$\nabla \times (\nabla \times \vec{E}(\vec{r},t)) = - \nabla \time... ...\vec{r},t) = - \frac {\partial }{\partial t} (\nabla \times \vec{B}(\vec{r},t))$$ (A.26)

If we now eliminate $\nabla \times \vec{B}(\vec{r},t)$ by using (A.22) the equation above becomes

$$\nabla \times (\nabla \times \vec{E}(\vec{r},t)) = - \mu_0 \mu_r ... ...\varepsilon_0 \varepsilon_r \frac {\partial^2}{\partial t^2} \vec{E}(\vec{r},t)$$ (A.27)

Utilizing the operator identity ( $\nabla \times (\nabla \times ) = \nabla \cdot(\nabla \cdot ) - \nabla^2$) and, as a consequence of (A.25), that $\nabla \cdot(\nabla \cdot \vec{E}(\vec{r},t))=\nabla(\rho / \varepsilon_0 \varepsilon_r$) we find for an medium with $\nabla\rho =0$

$$\nabla^2 \vec{E}(\vec{r},t) - \mu_0 \mu_r \sigma \frac{\partial }... ...repsilon_0 \varepsilon_r \frac {\partial^2}{\partial t^2} \vec{E}(\vec{r},t) =0$$ (A.28)

Similarly, the corresponding result for the $\vec{B}$-field can be shown to be

$$\nabla^2 \vec{B}(\vec{r},t) - \mu_0 \mu_r \sigma \frac{\partial }... ...repsilon_0 \varepsilon_r \frac {\partial^2}{\partial t^2} \vec{B}(\vec{r},t) =0$$ (A.29)

If we assume the absence of an externally applied current we can use (A.20) to rewrite (A.28) as

$$\nabla^2 \vec{E}(\vec{r},t) - \mu_0 \mu_r \varepsilon_0 \varepsil... ...vec{E}(\vec{r},t) = - \mu_0 \mu_r\frac{\partial }{\partial t}\vec{J}(\vec{r},t)$$ (A.30)

which shows the existence of electromagnetic waves, caused by a time varying electric current. In nonconducting media ($\sigma =0$) these equations become

$$\nabla^2 \vec{E}(\vec{r},t) - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \frac {\partial^2}{\partial t^2} \vec{E}(\vec{r},t) =0$$ (A.31)

$$\nabla^2 \vec{B}(\vec{r},t) - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \frac {\partial^2}{\partial t^2} \vec{B}(\vec{r},t) =0$$ (A.32)

By comparison with (A.25) it turns out that the wave propagation velocity is given by

$$v= \frac {1}{\sqrt{\mu_0 \mu_r \varepsilon_0 \varepsilon_r}}=\frac {c}{\sqrt{\mu_r \varepsilon_r }}= \frac {c}{n}$$ (A.33)

Where $c = 1/\sqrt{\mu_0\varepsilon_0}$ is the propagation velocity in free space and $n = \sqrt{\mu_r\varepsilon_r}$ the refraction index of the medium.