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It is not the electrons that are flowing through an electric wire, it is a charge (energy) transferred from electron to electron within the wires. But even this motion is much slower than you think, and depends on different factors, such as transfer medium. Even lightning traveling through ionized air can only travel, at most, half the speed of light. (see the related link)

There are several different ways to define the speed of electricity. In an AC circuit the electrons do not move forward at all, they just vibrate back and forth in place.

At the same time, the amount of time that it takes between when you flip a switch and a light turns on is actually much faster than the speed of light, so that it actually takes longer for the light of the bulb to reach your eyes than it takes for the light to turn on after flipping the switch.

The easiest way to imagine this is to picture a mile-long pipe that is just larger than a marble and is filled from beginning to end with marbles. When you put another marble in one side, a marble comes out the other side nearly (but not quite) instantly. This has nothing to do with the "speed of marbles," just like electrical current has nothing to do with the speed of light.

Other things such as radio waves, magnetic fields, and so on are a completely different story.

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