# Maxwell’s Equations in Vacuum and Matter

### Maxwell’s Equations

The following forms of Maxwell’s equations in vacuum and matter are in SI units. (For vacuum, put $\mathbf{P}=0$, $\mathbf{M}=0$, $\epsilon_{r}=1$, $\mu_{r}=1$, see below for definitions.)

Integral Form

$\begin{array}{ccc} \oint_{S}{\mathbf{D. dS}} & = & q \\ \\ \oint_{S}{\mathbf{B. dS}} & = & 0 \\ \\ \\ \oint_{C}{\mathbf{E. dl}} & = & -\frac{d}{dt} \int_{S}{\mathbf{B. dS}} \\ \oint_{C}{\mathbf{H. dl}} & = & I \ + \ \frac{d}{dt} \int_{S}{\mathbf{D. dS}} \end{array}$

Differential Form
$\begin{array}{ccc} \mathbf{\nabla . D} & = & \rho \\ \\ \mathbf{\nabla . B} & = & 0 \\ \\ \mathbf{\nabla \times E} & = & – {\partial{\mathbf{B}}\over\partial{t}} \\ \\ \mathbf{\nabla \times H} & = & \mathbf{J} + {\partial{\mathbf{D}}\over\partial{t}} \\ \end{array}$

where

$\begin{array}{ccc} \mathbf{D} & = & \epsilon_{r}\epsilon_{0} \mathbf{E} \ + \mathbf{P} \\ \\ \mathbf{H} & = & \frac{\mathbf{B}}{\mu_{r}\mu_{0}} \ – \ \mathbf{M} \\ \end{array}$
Lorentz Force

$\mathbf{F} \ = \ q(\mathbf{E} \ + \ \mathbf{v} \mathbf{\times} \mathbf{B})$

Definitions of terms:

$\mathbf{S}$ and $\mathbf{dS}$ are surface area and element of area respectively.

$\int_{S}$ and $\oint_{S}$ denote integration over a surface and a closed surface respectively.

$\int_{C}$ and $\oint_{C}$ denote integration over a curve and a closed curve respectively.

$\mathbf{D} \ =$ displacement current.
$\mathbf{P} \ =$ polarization field due to bound charges.
$\mathbf{E} \ = \ \nabla V$ electric field, where $V$ is the electric potential.
$\mathbf{H} \ = \$ auxillary magnetic field.
$\mathbf{M} \ = \$ magnetization of region/medium.
$\mathbf{B} \ = \$ magnetic field.
$\mathbf{v} \ =$ velocity of the charge $q$.
$q \ = \$ enclosed free charge.
$I \ = \$ current.
$\mathbf{J} \ = \$ surface current density.
$\rho \ = \$ free charge density.
$\epsilon_{0} \ = \$ permittivity of vacuum.
$\epsilon_{r} \ = \$ permittivity of medium relative to that of free space.
$\mu_{0} \ = \ 1/(\epsilon_{0} c^{2}) \ = \$ permeability of vacuum.
$\mu_{r} \ = \$ permeability of medium relative to vacuum.