Consider a wave incident on a good conductor from a dielectric. If the permittivity and permeability
of the conductor are similar to those in the dielectric, then, since σ ≫ ωε (by definition, for a good
conductor), the impedance of the conductor will be much less than the impedance of the dielectric.
Fresnel’s equations become (for both N and P polarization)
E
0R
E
0I
≈ −1,
E
0T
E
0I
≈ 0. (164)
There is (almost) perfect reflection of the wave (with a change of phase); and very little of the wave
penetrates into the conductor.
At optical frequencies and below, most metals are good conductors. In practice, as we expect from
the above discussion, most metals have highly reflective surfaces. This is of considerable importance for
RF systems in particle accelerators, as we shall see when we consider cavities and waveguides, in the
following sections.
8.3 Fields on the boundary of an ideal conductor
We have seen that a good conductor will reflect most of the energy in a wave incident on its surface.
We shall define an ideal conductor as a material that reflects all the energy in an electromagnetic wave
incident on its surface
1
. In that case, the fields at any point inside the ideal conductor will be zero at all
times. From the boundary conditions (136) and (140), this implies that, at the surface of the conductor,
B
⊥
= 0, E
k
= 0. (165)
That is, the normal component of the magnetic field, and the tangential component of the electric field
must vanish at the boundary. These conditions impose strict constraints on the patterns of electromag-
netic field that can persist in RF cavities, or that can propagate along waveguides.
The remaining boundary conditions, (137) and (139), allow for discontinuities in the normal com-
ponent of the electric field, and the tangential component of the magnetic field, depending on the presence
of surface charge and surface current. In an ideal conductor, both surface charge and surface current can
be present: this allows the field to take non-zero values at the boundary of (and within a cavity enclosed
by) an ideal conductor.
9 Fields in cavities
In the previous section, we saw that most of the energy in an electromagnetic wave is reflected from
the surface of a good conductor. This provides the possibility of storing electromagnetic energy in the
form of standing waves in a cavity; the situation will be analogous to a standing mechanical wave on,
say, a violin string. We also saw in the previous section that there are constraints on the fields on the
surface of a good conductor: in particular, at the surface of an ideal conductor, the normal component
of the magnetic field and the tangential component of the electric field must both vanish. As a result,
the possible field patterns (and frequencies) of the standing waves that can persist within the cavity are
determined by the shape of the cavity. This is one of the most important practical aspects for RF cavities
in particle accelerators. Usually, the energy stored in a cavity is needed to manipulate a charged particle
beam in a particular way (for example, to accelerate or deflect the beam). The effect on the beam is
determined by the field pattern. Therefore, it is important to design the shape of the cavity so that the
fields in the cavity interact with the beam in the desired way; and that undesirable interactions (which
always occur to some extent) are minimized. The relationship between the shape of the cavity and the
different field patterns (or modes) that can persist within the cavity will be the main topic of the present
section. Other practical issues (for example, how the electromagnetic waves enter the cavity) are beyond
our scope.
1
It is tempting
to identify superconductors with ideal conductors; however, superconductors are rather complicated materials
that show sometimes surprising behaviour not always consistent with our definition of an ideal conductor.