A.37 Maxwell’s wave equations
This note derives the wave equations satisfied by electromagnetic
fields. The derivation will use standard formulae of vector analysis,
as found in, for example, [40, 20.35-45].
The starting point is Maxwell’s equations for the electromagnetic
field in vacuum:
Here is the electric field, the magnetic field,
the charge density, the current density, the
constant speed of light, and is a constant called the
permittivity of space. The charge and current densities are related
by the continuity equation
To get a wave equation for the electric field, take the curl,
, of (3) and apply the standard vector identity
(D.1), (1) and (4) to get
Similarly, for the magnetic field take the curl of (4) and use (2) and
(3) to get
These are uncoupled inhomogeneous wave equations for the components of
and , for given charge and current densities.
According to the theory of partial differential equations, these
equations imply that effects propagate no faster than the speed of
light. You can also see the same thing pretty clearly from the fact
that the homogeneous wave equation has solutions like
which are waves that travel with speed in the -direction.
The wave equations for the potentials and are next.
First note from (2) that the divergence of is zero. Then vector
calculus says that it can be written as the curl of some vector. Call
that vector .
Plug this into (3) to show that the curl of is zero.
Then vector calculus says that it can be written as minus the gradient
of a scalar. Call this scalar . Plug that into the
expression above to get
Next, note that if you define modified versions
and of and by setting
where is any arbitrary function of , ,
, and , then still
since the curl of a gradient is always zero, and
because the two terms drop out against each other.
The fact that and produce the
same physical fields is the famous
gauge property of
the electromagnetic field.
Now you can select so that
That is known as the “Lorenz condition.” A corresponding gauge function is a
To find the gauge function that produces this condition, plug
the definitions for and in terms of
and into the left hand side of the Lorentz condition.
That produces, after a change of sign,
That is a second order inhomogeneous wave equation for .
Now plug the expressions (6) and (7) for and in
terms of and into the Maxwell’s equations.
Equations (2) and (3) are satisfied automatically. From (2),
after using (8),
From (4), after using (8),
You can still select the two initial conditions for . The
smart thing to do is select them so that and its time
derivative are zero at time zero. In that case, if there is no charge
density, will stay zero for all time. That is because its
wave equation is then homogeneous. The Lorenz condition will then
ensure that is zero too.
Instead of the Lorenz condition, you could select to make
zero. That is called the “Coulomb gauge” or “transverse gauge” or “transverse gauge.” It requires that satisfies the
Then the governing equations become
Note that now satisfies a purely spatial Poisson equation.