# Maxwell’s Equations in Vacuum and Matter

Access list of astrophysics formulas download page:

### 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.