How a Tesla coil works
Two coupled coils are tuned to the same resonant frequency, so energy pumped from primary to secondary over several cycles produces a voltage rise far greater than the turns ratio alone. The visible discharge is air breaking down at the terminal.
The physics behind the machines
Ohm’s law, the field of a solenoid, Faraday’s law of induction, sinusoidal alternating voltage, the transformer turns ratio, and the synchronous speed of a polyphase motor — six relationships that between them explain the whole electrical system.