• It is a step-up transformer with an air core; used to produce high voltages at high frequencies. Google :D Edit: It is a giant metal thing that generates a huge amount of electricity that has no where else to go but out the metal ball thingy. So it's like those things you see at Spencer's gifts but on a much much larger scale.
• Yes, it's a step-up transformer with an air core... but a transformer can't PRODUCE new frequencies. That's what the air gap is for. So, it's not just a step-up transformer. It has a special sort of oscillator built in to drive it at the primary coil's resonant frequency, which is typically at or very near AM radio frequencies. The power coming in is at 60 hertz. The secondary coil then must be tuned to the resonant frequency of the primary. What happens is this. The incoming voltage is applied across a high voltage capacitor. In parallel with that, there's an air gap in series with the primary coil. An electrical arc occurs when enough voltage is applied through the air the current has to cross. This voltage is determined by how close the metal contacts are together. Air needs about 600 or 800 volts to break down for any distance, and then about 15 kV per inch of gap it has to travel through. But once it breaks down - once the arc starts, it only has to be held hot enough to keep it going. It doesn't need 15 kV per inch, it needs maybe 5% of that. So the capacitor discharges through the coil and actually reverses most of the way because of the coil's inductance, and only then does the arc die. It takes a while for the capacitor to be recharged by the incoming power, and once it reaches the breakdown voltage of the air gap again, it goes through another cycle. This happens many thousands of times per second, so even with a slow 60 Hz wave coming in, if everything is adjusted properly and going at the right frequnecies, it will end up being somewhat efficient. With a solid state oscillator - using modern electronics instead of an air gap, it's generally more efficient, though. This is what people mean when they talk about a 'solid state' tesla coil.
• It doesn't, it exists in the same limbo of pseudoscience as the Loch Ness Monster, perpetual motion and the Atkins diet. Shame, it would be great if it worked.
• in layman's terms: A tesla coil produces vibrations of electricity in the air. These vibrations are the basis upon nearly all wireless communication today. The reason they are popular is because a tesla coil can be (and usually is) adjusted to make what looks like small lightning bolts. The effect is beautiful to see, especially in person.

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