Circuits don't operate at speed of light, though. People like to say this over and over again, but electron mobility through a circuit is only around 2/3s the speed of light, IIRC. They may have very tiny mass, but they still have mass.
No, it's the electromagnetic field which propagates at around 2/3c in the conductor. The electron drift velocity is on the order of a fraction of a mm/s.
