**Future developments.**

An array of TEM waves which are mutually trapped (to form a crystal) appears
to have ample degrees of freedom to enable it to construct a classic crystal
with flat exterior surfaces composed of rows and columns of 'atoms'. It is
regrettable that the intrusion of the particle, or photon, into an otherwise
straightforward system with rich development potential should obstruct forward
progress. The political compromise nearly a century ago which caused 'modern
physics' to exploit the pedigrees of both wave theory and particle theory^{[1]} has inevitably led to a sterile
century with no development, and it blocks development today.

Keeping within the wave theoretical system, it is possible to explain why so-called 'particles' should appear to have equal size, although a totally wave theory appears to be scalable and therefore incompatible with the apparently recurrent electron and hydrogen particles with consistent size. One method would be to discuss the collision of two such particles, and the resulting energy/matter exchange. There are three possibilities. Either the larger steals from the smaller, or there is no transfer, or the smaller steals from the larger. The fact that there is more than one 'particle' in today's galaxy indicates that if a galaxy is very old, the first possibility must be wrong. The second possibility is unlikely. The third would fully explain the gradual equalizing out of 'particles' in a galaxy over time. (This approach only explains why all hydrogen particles are equal, and needs extension to explain the existence of more than one type of particle.)

**The analogy between L, C and R.**

In this chapter we develop a useful analogy which leads to simplified calculation in all cases and to a simple technique for measurement in those cases which do not yield to calculation.

We shall consider the special case of a parallel-plate transmission line (Figure 24). a<<b.

We shall discuss the resistance, capacitance and inductance per unit length of the line.

**Resistance.**

If the medium between the plates has resistivity ρ, then the resistance between the plates per unit length is

where we have defined a/b as a geometrical factor which is a dimensionless
function of the dimensions of the line^{[2]}.

**Capacitance.**

For a parallel-plate capacitor, the capacitance per unit length is

.

**Inductance.**

For a parallel-plate transmission line, the self-inductance per unit length is

Note that the same geometrical factor f occurs in each case. This useful result holds not only in the case of parallel-plate geometries, but is true in general.

We now calculate the characteristic impedance Zo and velocity of propagation
*C*.

*
*

Let us look first at the result for . In the parallel-plate case we can substitute

to obtain

In general we can obtain a value for by noting the analogy between the equations for , where we note that the formula for is the same as that for except that has been replaced by . This means that we can obtain the geometrical factor by calculating the resistance between the conductors and multiplying by the factor . In cases where a calculation cannot be made, measurements using resistive paper can be used (Ref.15). Here the conductors are painted onto the resistive paper using conducting paint and the resistance between them measured with an ohmmeter. The equivalent to the resistivity is the resistance between two sides of a square of paper.

Note also that the velocity of a wavefront *C* is independent of the geometry,
and is a property only of the medium in which the conductors are placed.

We shall use the results just derived to obtain the impedance of a co-axial line.

**Impedance of a co-axial line.**

The outwards resistance of a thin co-axial shell at radius r is

The resistance between inner and outer conductors for a unit length of cable is

_{
MPSetEqnAttrs('eq2217','',3,[[194,29,11,0,0],[260,37,15,0,0],[325,47,18,-1,-1],[],[],[],[812,118,46,-1,-1]])
MPEquation()
}

Thus in this case, the geometrical factor is

and therefore the impedance is

which is the standard result for the impedance of a co-axial line.

**The L-C Model for the transmission line.**

It is common for textbooks to represent a transmission line as shown in Figure 26. It is possible, on the basis of this model and making use of the Laplace transform to derive the equations of step propagation. However, this method has little to recommend it, especially since it appears to lead to a high frequency cutoff which is quite spurious. There is of course no high frequency cutoff inherent in any transmission line geometry. The only factor which can lead to high frequency cutoff is frequency-dependent behaviour in the dielectric. If the dielectric is a vacuum there is no high frequency cutoff.

Malcolm Davidson has pointed out that since a capacitor is a transmission line
(Ref.16), the model models a transmission line
in terms of itself, which is absurd^{[3]}, see Figure 27.

**The Transmission Line Reconsidered.**

The traditional view is that when a TEM step travels down a two wire transmission line, it is bounded by electric current on each side and displacement current at the front. However, as well as advancing down the dielectric, the concept of skin depth tells us that it penetrates sideways into the conductors. We will however investigate the idea that the penetration into the conductors is of the same nature as the forward penetration down the dielectric, and no electric current is involved.

^{[1]}

ignoring Einstein's opposition to the quantum. "... what the basic
axioms in physics will turn out to be. The quantum or the particle will
surely not be amongst them; ...." (Ref.14).

^{[2]}

Before moving on to capacitance and inductance, we replace the resistive
medium by a dielectric with infinite resistivity.

^{[3]
}"Big fleas have little fleas

Upon their backs to bite 'em,

And these fleas have lesser fleas,

And so ad infinitum."