# The Maxwell Equations

One the the most important applications of the concepts you have learned are the equations which describe classical electricity and magnetism. When we set $\mathbf{E} \$ as the electric field,  $\mathbf{B} \$ as the magnetic field, $\mathbf{D} \$ as the electric displacement field, and $\mathbf{H} \$ as the magnetizing field, then the equations describing these quantities are

$\nabla \cdot \mathbf{D} = \rho$

$\nabla \cdot \mathbf{B} = 0$

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

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

We can integrate all these equations to get the integral forms:

$\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V$

$\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$

$\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}$

$\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}$

The four laws are called, respectively, Coulombs/Gauss' Law, absence of magnetic monopoles, Faraday's/Induction Law, and Ampère's Law. Together, they are called the Maxwell equations, after James Clerk Maxwell.