Tapered Wire

A tapered wire transmission line

For easy visualization of the forces and their relationships, a microcosm of space is simulated with a tapered wire model. The tapered wires can be used to create a gradient in the speed of energy. This is because the Z0 of a tapered transmission line is inversely proportional to the cross-sectional area of the line. Therefore, the speed of energy will be higher in regions where the cross-sectional area of the line is smaller.

Here are equations showing aspects of how tapered conductors can be used to create a model space:

The distribution of electrons in a tapered transmission line

ρ = ε0 * E / V

where:

ρ is the charge density of the electrons

ε0 is the permittivity of the dielectric material

E is the electric field

V is the voltage

The observation that the tapered transmission line charge distribution is related to ε0 is a very important one. The permittivity of free space, ε0, is the parameter that determines the strength of the electric field in a vacuum.

The electric field in a tapered transmission line

E = V / d

where:

E is the electric field

V is the voltage

d is the distance between the conductors

By substituting the equations for the electric field and the capacitance into the equation for the charge density, we get the following equation:

ρ = ε0 * V2 / (d2)

This equation shows that the charge density of the electrons is inversely proportional to the square of the distance between the conductors. Therefore, the charge density of the electrons is higher in regions with lower impedance.

The generation of magnetic flux in the impedance along the line would indeed control the speed at which the charge is moving. This is because the magnetic flux would create a force on the charge that would oppose its motion. The strength of this force would be proportional to the strength of the magnetic flux and the velocity of the charge.

The relationship between the magnetic flux, the impedance, and the speed of the charge

F = B * I * L

where:

F is the force on the charge

B is the magnetic flux density

I is the current flowing through the conductor

L is the length of the conductor