Relativistic Effects on Electromagnetic Wave Propagation

Question

So for a recent lab I had to calculate the length of a conductor by measuring the time it took a signal to reflect off of the open end. I used the very simple principle of $v=st$ and, knowing that the propagation of waves inside this specific conductor is $\frac{2}{3} c$, and keeping in mind the fact that the signal would have to travel through the length of the wire twice before I measured it again, I came to the following expression:

$$l = \frac{2}{3} c \frac{(250 \pm 2) \times 10^{-9} s}{2}$$

with $(250 \pm 2) \times 10^{-9} s$ being the time it took the wave to travel back to me. This gave me a very accurate result of $24.9m \pm 0.2m$, with the actual conductor being manufactured to be 25 meters long. My question is, since the wave is propagating at $\frac{2}{3}$ the speed of light, shouldn't length contraction play a role in some way? Why was I able to get such an accurate result despite not accounting for relativistic effects?

Answer 1

My question is, since the wave is propagating at 23 the speed of light, shouldn't length contraction play a role in some way? Why was I able to get such an accurate result despite not accounting for relativistic effects?

You were able to get accurate results despite not accounting for relativistic effects for two reasons:

1. Electromagnetism is a fully relativistic theory already. So, in fact, you did already account for relativistic effects. They are already built into the theory by which the $2/3 \ c$ velocity was found.

2. You were working in a single reference frame. Length contraction is a disagreement between two different reference frames. Since you are only working in a single frame there is no length contraction involved. If you were to also work in a frame moving at $2/3 \ c$ relative to the first, then you would have needed to transform all relevant quantities to that frame. The length would have been contracted.

This last point may be a little surprising considering the common explanation of magnetism as a relativistic consequence of length contraction. But note that that explanation also involves using different reference frames. The magnetic force in one frame is turned into an electric force in another frame. But in the first frame the force is already fully accounted for by the magnetic field. So here you already have the electric and magnetic fields in your rest frame, thus fully accounting for the behavior.