Huygens construction
The Huygens construction explains Cerenkov radiation as a phenomena very similar to the shockwave produced by supersonic jets. Suppose a particle emits a wave pulse every time t, and also that these pulses move out spherically at speed c'. Now allow the particle to move with speed v. If v < c', then the wavecrests merely bunch up in the direction of motion, but do not cross. When v > c' , however, the wavecrests do cross, adding up constructively at some angle θ, defined as
This wavefront moves out at the speed of light in the dielectric.
Dielectric Polarization
This explanation is closer to the actual matter at hand, as the wavecrest explanation is far less helpful when one considers the case where the particle is moving outside the dielectric (as is the case in the experiment at hand). It also does not address why “superluminous” motion would create a shockwave. Thinking about the polarization of the dielectric can help to explain this aspect of Cerenkov radiation a little better [1]. Consider that a charged particle, in passing by the atoms in the dielectric, momentarily polarizes them (pushing like charges in the atom away, and inducing a dipole state). Once the particle has passed, this polarized state collapses, causing each atom to emit Cerenkov radiation (if the velocity constraint is met, of course). For slow moving particles, the polarization is perfectly symmetrical, resulting in no electric field at long distances (and thus no radiation). When the particle is moving very quickly (beyond the constraint 
, hereafter written with the abbreviation 
as 
), however, the polarization is no longer perfectly symmetrical.
Electron passing through a dielectric at 
.
Electron passing through a dielectric at 
.
The state is still symmetric in the azimuthal plane, but no longer along the axis of motion (a “cone” of dipoles develops behind the electron). There would now be distinct dipole field established in the dielectric, one that can only be collapsed with the emission an electromagnetic pulse (Cerenkov radiation). Radiation would be emitted perpendicular to the surface of this cone, with the angle of the cone found using the Huygens construction.
Theoretical Analysis
The following treatment of the problem makes several simplifying assumptions (from the Frank and Tamm theory detailed in [1]):
(i) The dielectric is considered as a continuum, in as much as the material is defined simply by its dielectric constant with all micro-structure ignored.
(ii) Dispersion is ignored.
(iii) Radiation reaction is ignored.
(iv) The material is considered to be a perfect isotropic dielectric, with no absorption of radiation (conductivity is zero, permeability is unity).
(v) The beam is considered to move at a constant velocity (no slowing due to ionization, of Coulomb scattering).
(vi) The material is unbounded and the beam path is infinite.
Electron sheet passing over a dielectric.
Even given these assumptions, there has been relatively little theoretical work done on the experimental geometry considered here. The closest work would be that of Danos [2, 3], who considered the motion of an infinitesimally thin sheet of charge moving at some distance from the vacuum-dielectric boundary. The following derivations follow the work presented in [2].
One begins by defining a charge density ρ
where
and, as before,
Also, 
is the Fourier coefficient of the charge density corresponding to the frequency 
, and 
is the average DC charge density. Of course, this is for a bunched sheet, moving in the z-axis, with zero extent in the x-axis (perpendicular to the surface of the dielectric).
The potentials in free space (above the vacuum-dielectric boundary) are
where 
is constant and 
is the z-axis unit vector.
Considering only the AC part of the charge density, one arrives at
where
To solve the homogeneous equation 
one makes the trial solution
By the Lorentz condition, 
, it follows that 
. Thus
The vacuum fields that go with the electron beam are
From above
Here signx is defined as
and
Above, the quantities 
and 
are parameters that can be adjusted to match the boundary conditions.
The vector potential inside the dielectric is just
where
From this the fields can be calculated
Now one can impose the boundary condition (the tangential field must match) at 
From here one can find the equations of the outgoing wave
where
and
The electron beam radiates an average energy defined by the Poynting vector
where
Poynting vector decomposition.
The normal component of this vector corresponds to the energy radiated into the dielectric:
where
with N being the number of electrons in one coulomb (
). Thus this component of the Poynting vector is in watts per centimeter squared if I is in amps per centimeter.
One should also note that the angle of the radiation (
) is given by
To simplify things, lets put the above derivation in terms of known quantities:
If the beam sheet in this example has width, this can be applied in the Poynting vector as such (in simplified notation)
where 
is measured from the surface of the dielectric.
A similar analysis was conducted by Linhart [4] for a single electron passing over an infinite plane interface between a vacuum and a dielectric (with similar constraints as above with the addition of 
and 
). The total radiation output was found to be
where e is the charge of the electron, and the energy radiated per unit angular frequency is
Sketch of frequency dependence of power.
Consequently, there exists a frequency, and thus a wavelength, at which most of the radiation is emitted:
Summary
It has been shown that the general design used in this experiment, namely the passing of an electron beam over a dielectric, will produce Cerenkov radiation given the correct dielectric. This was done by Danos [3] when he generated microwave Cerenkov radiation by passing a 10 KeV electron beam over a plate of titanium dioxide . Since we are using a low-energy beam of ~3KeV, we need a dielectric constant of at least 85. This is found by
Of course, one can move easily from a potential difference like 3KV to a value for 
using
Solving one gets
