Maxwell's Equations:
Maxwell's equation of the electromagnetic wave is a collection of four equations i.e. Gauss's law of electrostatic, Gauss's law of magnetism, Faraday's law of electromotive force, and Ampere's Circuital law. Maxwell converted the integral form of these equations into the differential form of the equations. The differential form of these equations is known as Maxwell's equations for free space.

- $\overrightarrow{\nabla}. \overrightarrow{E}= \frac{\rho}{\epsilon_{0}}$

- $\overrightarrow{\nabla}. \overrightarrow{B}=0$

- $\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$

- $\overrightarrow{\nabla} \times \overrightarrow{H}= \overrightarrow{J}$

Modified Form:

$\overrightarrow{\nabla} \times \overrightarrow{H}= \overrightarrow{J}+ \frac{\partial \overrightarrow{D}}{\partial t}$

For non-conducting media:

Current density $(\overrightarrow{J})=0$

Volume charge distribution $(\rho) \neq 0 $

Permittivity of non-conducting media= $\epsilon$

Permeability of non-conducting media=$\mu$

Now, Maxwell's equation for non-conducting media:

$\overrightarrow{\nabla}. \overrightarrow{E}=\frac{\rho}{\epsilon} \qquad(1)$

$\overrightarrow{\nabla}. \overrightarrow{B}=0 \qquad(2)$

$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} \qquad(3)$

$\overrightarrow{\nabla} \times \overrightarrow{H}= 0$

Modified form for non-conducting media:

$\overrightarrow{\nabla} \times \overrightarrow{H}= \frac{\partial \overrightarrow{D}}{\partial t} \qquad(4)$

Now, On solving Maxwell's equation for a non-conducting media i.e perfect dielectric and Lossless media, we get the electromagnetic wave equation for a non-conducting media. The electromagnetic wave equation has both an electric field vector and a magnetic field vector. So Maxwell's equations for non-conducting media give two equations for electromagnetic waves i.e. one is for electric field vector($\overrightarrow{E}$) and the second is for magnetic field vector ($\overrightarrow{H}$).

Electromagnetic wave equation for non-conducting media in terms of $\overrightarrow{E}$:

Now from equation $(3)$

$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} $

Now take the curl on both sides of the above equation$

$\overrightarrow{\nabla} \times (\overrightarrow{\nabla} \times \overrightarrow{E})=-\overrightarrow{\nabla} \times \frac{\partial \overrightarrow{B}}{\partial t} $

$(\overrightarrow{\nabla}. \overrightarrow{E}).\overrightarrow{\nabla} - (\overrightarrow{\nabla}. \overrightarrow{\nabla}).\overrightarrow{E}=-\frac{\partial}{\partial t} (\overrightarrow{\nabla} \times \overrightarrow{B}) \qquad(5)$

But for non-conducting media:

$\overrightarrow{\nabla}. \overrightarrow{E}=\frac{\rho}{\epsilon} \qquad \left( From \: equation (1) \right)$

$\overrightarrow{\nabla}.\overrightarrow{\nabla}=\nabla^{2}$

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \frac{\partial \overrightarrow{D}}{\partial t} \qquad \left( From \: equation (4) \right)$

Now substitute these values in equation $(5)$. So

$ \nabla \left(\frac{\rho}{\epsilon}\right) -\nabla^{2}.\overrightarrow{E}=-\frac{\partial}{\partial t} (\mu \frac{\partial \overrightarrow{D}}{\partial t})$

For non conducting media, If the volume charge distribution is uniform then the gradient of volume charge density is very small (or almost zero) so neglect the term gradient of volume charge density $(\nabla \rho)$ in above eqaution.

Or

The wave propagation does not contain the charges in most of the cases. Therefore we get

$ -\nabla^{2}.\overrightarrow{E}=-\mu \frac{\partial^{2} \overrightarrow{D}}{\partial t^{2}} $

$ \nabla^{2}.\overrightarrow{E}=\mu \frac{\partial^{2}}{\partial t^{2}} (\epsilon \overrightarrow{E}) \qquad \left(\because \overrightarrow{D}=\epsilon \overrightarrow{E} \right)$

$\nabla^{2}.\overrightarrow{E}=\mu \epsilon \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}}$

The value of $\frac{1}{\sqrt{\mu \epsilon}}= v$. Where $v$ is the speed of the wave in the non-conducting media. So the above equation is often written as

$\nabla^{2}.\overrightarrow{E}=\frac{1}{v^{2}} \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}}$

*
This is an electromagnetic wave equation for non-conducting media in terms of electric field vector ($\overrightarrow{E}$).
*

Electromagnetic wave equation for non-conducting media in terms of $\overrightarrow{H}$:

Now from equation $(4)$

$\overrightarrow{\nabla} \times \overrightarrow{H}= \frac{\partial \overrightarrow{D}}{\partial t}$

Now take the curl on both sides of the above equation

$\overrightarrow{\nabla} \times (\overrightarrow{\nabla} \times \overrightarrow{H})=\overrightarrow{\nabla} \times \frac{\partial \overrightarrow{D}}{\partial t} $

$(\overrightarrow{\nabla}. \overrightarrow{H}).\overrightarrow{\nabla} - (\overrightarrow{\nabla}. \overrightarrow{\nabla}).\overrightarrow{H}=\frac{\partial}{\partial t} (\overrightarrow{\nabla} \times \overrightarrow{D}) \qquad(6)$

But for non-conducting media:

$\overrightarrow{\nabla}. \overrightarrow{B}=0 \qquad \left( from \: equation (2) \right)$

$\overrightarrow{\nabla}.\overrightarrow{\nabla}=\nabla^{2}$

$\overrightarrow{\nabla} \times \overrightarrow{D}= -\epsilon \frac{\partial \overrightarrow{B}}{\partial t} \qquad \left( from \: equation (3) \right)$

Now substitute these values in equation $(6)$. So

$ -\nabla^{2}.\overrightarrow{H}=-\frac{\partial}{\partial t} (\epsilon \frac{\partial \overrightarrow{B}}{\partial t})$

$ -\nabla^{2}.\overrightarrow{H}=-\epsilon \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}} $

$ \nabla^{2}.\overrightarrow{H}=\epsilon \frac{\partial^{2}}{\partial t^{2}} (\mu \overrightarrow{H}) \qquad \left(\because \overrightarrow{B}=\mu \overrightarrow{H} \right)$

$\nabla^{2}.\overrightarrow{H}=\mu \epsilon \frac{\partial^{2} \overrightarrow{H}}{\partial t^{2}}$

The value of $\frac{1}{\sqrt{\mu \epsilon}}= v$. Where $v$ is the speed of the wave in the non-conducting media. So the above equation is often written as

$\nabla^{2}.\overrightarrow{H}=\frac{1}{v^{2}} \frac{\partial^{2} \overrightarrow{H}}{\partial t^{2}}$

*
This is an electromagnetic wave equation for non-conducting media in terms of electric field vector ($\overrightarrow{H}$).
*