Reflection Coefficient with Electric Field and Voltage

Question

In different situations the input reflection coefficient (of a waveguide, of a transmission line etc) is defined differently.

Wikipedia says:

The reflection coefficient may also be established using other field or circuit pairs of quantities whose product defines power resolvable into a forward and reverse wave. For instance, with electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the forward wave (or magnetic fields, again with a minus sign).

So, let's consider for instance a coaxial cable. We may define its input reflection coefficient as the ratio between reverse and the direct travelling voltage waves (V-(z)/V+(z)), but also as the ratio between the reverse and the direct travelling electric fields (E-(z)/E-(z)).

Are these two reflection coefficients the same?

Answer 1

A couple weeks ago, I answered one of your earlier questions like this:

Remember that when we defined the electrostatic potential difference (aka "voltage"),

$$V=-\int \vec{E}\cdot d\vec\ell,$$

we called it the electrostatic potential difference because it is only strictly valid in electrostatics. When we use this concept in AC circuits, we're using it as an approximation only (usually described as the lumped circuit approximation). In particular, in the presence of time-varying magnetic fields, we can't count on this \$V\$ to be independent of the path over which we take the integral.

In transmission lines, we are definitely dealing with time-varying magnetic fields, so we can't expect the electrostatic potential difference to be well defined.

We define an approximate potential at a point along the transmission line as the negative integral of the electric field from one conductor to the other at that point.

[emphasis added]

The voltage at a point on a transmission line is defined by an integral of the electric field between the conductors at that point. So if you double the electric field, you double the voltage, or if you halve the electric field you halve the voltage.

So your two definitions of the reflection coefficient are equivalent, provided the transmission line is well-behaved enough to let you actually define a voltage on it.