\def\R{{\bf R}}
\def\curl{\mathop{\rm curl}}

Daniel Berwick Evans.

Maxwell's equations and classical electromagentism.

Fields: $E$, $B$: time dependent vector fields on~$\R^3$.
$\div E=\rho/\epsilon_0$, $\curl E=-\partial B/\partial t$,
$\div B=0$, $c^2\curl B=j/\epsilon_0+\partial E/\partial t$.
$\rho$ is ``charge density'' (coulombs per cubic meter),
$j$ is ``current'' (coulombs per second per square meter).

Gau\ss' law is an application of Stokes' theorem:
$q/\epsilon_0=\int_U\rho/\epsilon_0=\int_U\div E=\int_{\partial U}E\cdot n$.

Ampere's law:
$-\int_D\partial B/\partial t=-d\Phi_B/dt=\int_D\curl E=\int_{\partial D}E\cdot ds$.

No magenetic charges (monopoles): $\curl B=0$.
If $\rho=j=0$ then we get a symmetry between $E$~and~$B$.

Charge conservation:
$\div j/\epsilon_0=-\div(\partial E/\partial t)=-(\partial/\partial t)\div E
=-(\partial/\partial t)\rho/\epsilon_0$, hence $\div j=-\partial\rho/\partial t$.

Static examples: $\partial B/\partial t=\partial E/\partial t=0$.

Point charge: If $\rho=\delta(x)$ and $j=0$, then $E=\pm(4\pi\epsilon_0)^{-1}r/|r|^3$
and $B=0$.

Wave solutions: $\rho=j=0$.
$E=(0,E_y,E_z)$, $B=(0,B_y,B_z)$.
$E_y=f(x-ct)+g(x+ct)$, $E_z=F(x-ct)+G(x+ct)$,
$cB_z=f(x-ct)-g(x+ct)$, $cB_y=-F(x-ct)+G(x+ct)$.
Wave with speed of propogation~$c$.

Solenoid solution: $E=0$ outside some cylinder, $\rho=0$ everywhere,
$B$ is pointing upwards, $j$ goes around the cylinder.

$\int B\cdot ds=|B|h=j/(\epsilon_0c^2)=jnh/(\epsilon_0c^2)$.
$|B|=jn/(\epsilon_0c^2)$.
$n$ is the number of turns per length.

\bye
