\def\div{\mathop{\rm div}}
\def\curl{\mathop{\rm curl}}
\def\R{{\bf R}}
Daniel Berwick Evans.
Maxwell's equations and electromagnetism II.
Example: Current through a wire. $E=0$.
From this data we can derive Ampere's law: $\int B\cdot ds=j_e/(\epsilon_0c^2)=|B|\cdot2\pi$.
Hence $|B|=|j|/(2\pi\epsilon_0c^2)$.
Example: Solenoid. We assume that $B_o=0$.
$\int B\cdot ds=j_e/(\epsilon_0c^2)=N|j|/(\epsilon_0c^2)$.
Hence $|B|=n|j|/(\epsilon_0c^2)$.
Potentials are a nice way to cook up solutions to Maxwell's equations.
Since $\div B=0$ we can write $B=\curl A$ for some non-unique vector potential~$A$
if there are no topological obstructions.
Also $\curl(E+\partial_tA)=0$, hence $E+\partial_tA=-\div\phi$ for some
non-unique scalar potential~$\phi$.
Gauge transformations: $A\mapsto A+\div\psi$ and $\phi\mapsto\phi-\partial_t\psi$.
By choosing a particular $A$~and~$\phi$ we are ``fixing our gauge''.
For example, $\div A=-c^{-2}\partial_t\phi$ is called the Lorentz gauge.
Relativistic formulation.
Motivation.
Consider two wires with current. Lorentz' law: $F=q(E+v\times B)$
(force on a test particle with velocity~$v$ and charge~$q$).
(We cannot derive it from Maxwell's equations.)
Hence the wires are attracted to each other if the have the same direction of current
and repelled otherwise.
Now we move into a frame of reference in which electrons in wires are at rest.
We observe that now we have electric field instead of magnetic.
We introduce some forms: $B=B_x dy\wedge dz+B_y dz\wedge dx+B_z dx\wedge dy$,
$E=E_xdx+E_ydy+E_zdz$, and $F=B+E\wedge dt$ (electromagnetic field).
$J=\rho dt+j_x dx+j_y dy+j_z dz$.
Maxwell's equations: $dF=0$ and $*d*F=J$, where $*$ is the Hodge star.
In the vacuum, given a solution~$F$, $*F$ is another solution.
In particular we have self-dual and anti-self-dual solutions: $F=\pm*F$.
Self-dual solutions are left circularly polarized wave solutions
and anti-self-dual are right circularly polarized (for given orientation
on~$\R^{3+1}$ determined by~$*$.
Lagrangian formulation of Maxwell's equations:
fields are closed 2-forms. Lagrangian (vacuum case): $\int F\wedge*F$.
We can also take $A\in\Omega^1(\R^{3+1})$
and $L=\int dA\wedge*dA+*J\wedge A$.
Aharonov-Bohm effect: $S_q=\int\exp(i\hbar^{-1}q\int_\gamma A)$.
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