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. These equations, called the Maxwell equations after James Clerk Maxwell, 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.