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### Electromagnetic wave equation in non conducting media (i.e. Perfect dielectric or Lossless media)

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.

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

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

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

4. $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \overrightarrow{J}$

Modified Form:

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \left(\overrightarrow{J}+ \epsilon \frac{ \partial \overrightarrow{E}}{\partial t} \right)$

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:

$\overrightarrow{\nabla}. \overrightarrow{E}=\frac{\rho}{\epsilon} \qquad(1)$
$\overrightarrow{\nabla}. \overrightarrow{B}=0 \qquad(2)$
$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} \qquad(3)$
$\overrightarrow{\nabla} \times \overrightarrow{B}= 0$

Modified form for non-conducting media:

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t} \qquad(4)$

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 ($\overrightarrow{B}$).

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

Now from Maxwell's equation $(3)$

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