Calc3.8

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.