(19) | |||

(20) | |||

(21) |

where and are the Bessel functions of the first kind. I can discard the term because it diverges at , which is not physical for this particular problem (it would be if there were a current passing along the -axis).

I can further eliminate some other variables. First, I know that the
field is symmetric about the -plane, so must be zero. If
were not zero, the function would be asymmetric about .
Similarly, I assert that at time , the magnetic field is maximum,
so must also be zero (this also follows because there is no charge
or current in the system). Thus, a solution of the original partial
differential equation, with the known symmetry conditions, is

Equation (22) can be considered a *standing wave*, or
eigenmode, solution to Maxwell's equations for the toroidal flux loop
initial condition. In other words, this solution provides waves are
fixed in space and that oscillate for infinite time. However, it is
not the same kind of standing wave that Schallger refers to.
Specifically, this solution has waves that extend infinitely
throughout space, so they do not reflect (as yet) a concentrated flux
loop bundle. Schallger conjectures that the magnetic flux can be
contained in a tight loop on the plane and essentially nowhere
else. Thus, at this step we cannot verify that a flux loop is a valid
standing wave solution.

However, we can take the analysis further. Equation (22)
is *only one* solution of an infinite set of solutions. Any
combination of and will provide valid solutions.
(Recall that
, so is not an
independently choosable variable). Also, because the partial
differential equation is linear, any *sum* of solutions will also
be a solution. Thus, the most general solution is of the form:

(24) |

Thus, the problem is essentially solved, with the only really difficult thing remaining to accomplish is to determine the constants to match the initial conditions of a flux loop.