Let us consider for a moment what time dependent EM fields look like at the surface of a ``perfect'' conductor. A perfect conductor can move as much charge instantly as is required to cancel all fields inside. The skin depth as diverges - effectively all frequencies are ``static'' to a perfect conductor. This is how type I superconductors expel all field flux.
If we examine the fields in the vicinity of a boundary between a perfect
conductor and a normal dielectric/diamagnetic material, we get:
In addition to these two inhomogeneous equations that normal and
parallel fields at the surface to sources, we have the usual two
However, for materials that are not perfect conductors, the fields don't vanish instantly ``at'' the mathematical surface. Instead they die off exponentially within a few multiples of the skin depth . On scales large with respect to this, they will ``look'' like the static field conditions above, but of course within this cutoff things are very different.
For one thing, Ohm's law tells us that we cannot have an actual
``surface layer of charge'' because for any finite conductivity, the
resistance scales like the cross-sectional area through which charge
flows. Consequently the real boundary condition on
precisely at the surface is:
If the conductivity is large but not infinite, one way to figure out what happens is to employ a series of successive approximations starting with the assumption of perfect conductivity and using it to generate a first order correction based on the actual conductivity and wavelength. The way it works is:
Thus (from 1):
We both Ampere's law (assuming no displacement in the conductor
to leading order) and Faraday's law to obtain relations for the harmonic
fields in terms of curls of each other:
Now we need to implement assumption 2 on the
If we pick a coordinate to be perpendicular to the surface
pointing into the conductor (in the
direction) and insist
that only variations in this direction will be significant only on
length scales of :
These two equations are very interesting. They show that while the magnitude of the fields in the vicinity of the conducting surface may be large or small (depending on the charge and currents near the surface) the curls themselves are dominated by the particular components of and that are in the plane perpendicular to (and each other) because the field strengths (whatever they are) are most rapidly varying across the surface.
What this pair of equations ultimately does is show that if there is a magnetic field just inside the conductor parallel to its surface (and hence perpendicular to ) that rapidly varies as one descends, then there must be an electric field that is its partner. Our zeroth approximation boundary condition on above shows that it is actually continuous across the mathematical surface of the boundary and does not have to be zero either just outside or just inside of it. However, in a good conductor the field it produces is small.
This gives us a bit of an intuitive foundation for the manipulations of Maxwell's equations below. They should lead us to expressions for the coupled EM fields parallel to the surface that self-consistently result from these two equations.
We start by determining the component of
(the total vector
magnetic field just inside the conductor) in the direction
perpendicular to the surface:
Next we form a vector that lies perpendicular to both the normal
and the magnetic field. We expect
to lie along this
direction one way or the other.
However, this does not show that the two conditions can lead to a self-sustaining solution in the absence of driving external currents (for example). To show that we have to substitute Ampere's law back into this:
This is a well-known differential equation that can be written any of
several ways. Let
. It is equivalent
to all of:
As always, we have two linearly independent solutions. Either of them will work, and (given the already determined sign/branch associated with the time dependence ) will ultimately have the physical interpretation of waves moving in the direction of ( ) or in the direction of ( ). Let us pause for a moment to refresh our memory of taking the square root of complex numbers (use the subsection that treats this in the last chapter of these notes or visit Wikipedia of there is any problem understanding).
For this particular problem,
Now we need to find an expression for
, which we do by
backsubstituting into Ampere's Law:
Note well the direction! Obviously , (in this approximation) so must lie in the plane of the conductor surface, just like !
As before (when we discussed fields in a good conductor):
at the surface
() there must be a power flow into the conductor!
Finally, we need to define the ``surface current'':
Hopefully this exposition is complete enough (and correct enough) that any bobbles from lecture are smoothed out. You can see that although Jackson blithely pops all sorts of punch lines down in the text, the actual algebra of getting them, while straightforward, is not trivial!