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:

(10.1) |

(10.2) |

In addition to these two inhomogeneous equations that normal and
parallel fields at the surface to sources, we have the usual two
homogeneous equations:

(10.3) | |||

(10.4) |

Note that these are pretty much precisely the boundary conditions for a static field and should come as no surprise. For perfect conductors, we expect the fields inside to vanish, which in turn implies that outside must be normal to the conducting surface and outside must lie only parallel to the conducting surface, as usual.

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:

(10.5) | |||

(10.6) |

where . However, this creates a problem! If this field varies rapidly in some direction (and it does) it will generate an electric field according to Faraday's law! If the direction of greatest variation is ``into the conductor'' (as the field is being screened by induced surface currents) then it will generate a small electric field parallel to the surface, one which is neglected (or rather, cannot occur) in the limit that the conductivity is infinite. This electric field, in turn, generates a current, which causes the gradual cancellation of as less and less the total bulk current is enclosed by a decending loop boundary.

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:

- First, we assume that outside the conductor we have only and from the statement of the boundary conditions assuming that the fields are instantly cancelled at the surface.
- Assume
*along*the surface - the skin depth is much less than a wavelength and the fields (whatever they may be)*vanish*across roughly this length scale, so we can neglect variation (derivatives) with respect to coordinates that lie along the surface compared to the coordinate perpendicular to the surface. - Use this approximation in Maxwell's Equations, along with the assumed boundary conditions for a perfect conductor, to derive relations between the fields in the transition layer.
- These relations determine the
*small*corrections to the presumed boundary fields both just outside and just inside the surface.

Thus (from 1):

(10.7) |

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:

(10.8) | |||

(10.9) |

become

(10.10) | |||

(10.11) |

As we might expect, high frequencies create relatively large induced electric fields as the magnetic fields change, but high conductivity limits the size of the supported electric field for any given magnetic field strength in a

Now we need to implement assumption 2 on the
operator.
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 :

(10.12) |

(10.13) |

(Note well the deliberate use of to emphasize that there may well be components of the fields in the normal direction or other couplings between the components in the surface, but those components do not

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:

(10.14) |

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.

(where and ) and find that it does! The fact that the electric field varies most rapidly in the () direction picks out its component in the plane

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:

or

(10.15) |

This is a well-known differential equation that can be written any of
several ways. Let
. It is equivalent
to all of:

(10.16) | |||

(10.17) | |||

(10.18) | |||

(10.19) |

Where:

(10.20) |

(10.21) |

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,

(10.22) |

(10.23) |

(consider ).

Now we need to find an expression for
, which we do by
backsubstituting into Ampere's Law:

(10.24) |

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):

- not in phase, but out of phase by .
- Rapid decay as wave penetrates surface.
- ( ``large'', ``small'') so energy is primarily magnetic.
- - fields are predominantly parallel to the surface and mutually transverse, they propagate ``straight into'' surface, attenuating rapidly as they go.
- Recall:

(10.25)

(10.26) *outside*the surface - the field is approximately continuous! At this level of approximation, , is parallel to the surface, and there is a small to the surface of the same general order of magnitude as .

Since both
and
at the surface
() there must be a power flow *into* the conductor!

(10.27) |

(10.28) |

(10.29) | |||

(10.30) |

which just happens to correspond to the flux of the pointing vector through a surface !

Finally, we need to define the ``surface current'':

(10.31) |

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!