2.1: Maxwell's Equations

Maxwell’s equations are given by

$$\vec{\nabla} \times \vec{H} = \vec{j} + \dfrac{\partial \vec{D}}{\partial t}, \nonumber$$

$$\vec{\nabla} \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t}, \label{eq2.1.2}$$

$$\vec{\nabla} \cdot \vec{D} = \rho, \nonumber$$

$$\vec{\nabla} \cdot \vec{B} = 0 \nonumber$$ .

The material equations accompanying Maxwell’s equations are:

$$\vec{D} = \epsilon_0 \vec{E} + \vec{P}, \nonumber$$

$$\vec{B} = \mu_0 \vec{H} + \vec{M}. \nonumber$$

Here, $\vec{E}$ and $\vec{H}$ are the electric and magnetic field, $\vec{D}$ the dielectric flux, $\vec{B}$ the magnetic flux, $\vec{j}$ the current density of free carries, $\rho$ is the free charge density, $\vec{P}$ is the polarization, and $\vec{M}$ the magnetization. By taking the curl of Equation $\ref{eq2.1.2}$ and considering $\vec{\nabla} \times (\vec{\nabla} \times \vec{E}) = \vec{\nabla} (\vec{\nabla} \vec{E}) - \Delta \vec{E}$, we obtain

$$\Delta \vec{E} - \mu_0 \dfrac{\partial}{\partial t} \left (\vec{j} + \epsilon_0 \dfrac{\partial \vec{E}}{\partial t} + \dfrac{\partial \vec{P}}{\partial t} \right ) = \dfrac{\partial}{\partial t} \vec{\nabla} \times \vec{M} + \vec{\nabla} (\vec{\nabla} \cdot \vec{E}) \nonumber$$

and hence

$$\left (\Delta - \dfrac{1}{c_0^2} \dfrac{\partial^2}{\partial t^2} \right ) \vec{E} = \mu_0 \left ( \dfrac{\partial vec{j}}{\partial t} + \dfrac{\partial^2}{\partial t^2} \vec{P} \right ) + \dfrac{\partial}{\partial t} \vec{\nabla} \times \vec{M} + \vec{\nabla} ( \vec{\nabla} \cdot \vec{E}).\label{eq2.1.8}$$

The vacuum velocity of light is

$$c_0 = \sqrt{\dfrac{1}{\mu_0 \epsilon_0}}. \nonumber$$