A tram is powered by overhead wire, the wire has alternating voltage of 1000 V RMS, the frequency of the alternating voltage is 50 Hz. The rails are the other wire.
The tram is moving at speed 100 m/s away from the place where the overhead wire and the rails are connected to the electric grid.
Is the frequency of the electricity Doppler-shifted in the tram frame? Is the energy of the electricity reduced proportionally to the reduction of the frequency of the electricity?
Treating the wire and rails as a "transmission line," one obtains the wavelength of a 50 cycles per sec wave as $\lambda = c/f$, = $300×10^6 / 50 = 6×10^6m.$
A "peak" detector on the tram detects the start of the wave and at the end of 1/50 of a second, it would have detected the end of the wave if it had not moved. However, since it is moving at 100m/s, at the end of 1/50 sec, the detector will move 100m/s × 1/50 sec = 2m. So the wavelength as measured on the tram, would be $6.000002×10^6$m. This gives a frequency of 49.999983 cycles per second, so the frequency is Doppler-shifted by 0.000017 cps.
For the second question, assuming that the load is purely resistive (power factor = 1), the shift in frequency would have no effect on the energy being delivered.
