The voltage gain of a Tesla coil can be determined from the primary and secondary inductances \$L_1\$ and \$L_2\$ using the following formula (Wikipedia has a simple derivation): $$ \frac{V_2}{V_1} = \sqrt{\frac{L_2}{L_1}} $$
I've seen several websites show this formula and claim that it is the reason for Tesla coils' high gain. But the same formula also applies to an ordinary transformer built from coils of the same inductance - as @user287001 said, \$ \sqrt{\frac{L_2}{L_1}} \$ is the turns ratio \$ \frac{N_2}{N_1} \$.
My question is:
- Why would I want to build a Tesla coil rather than a simpler transformer of equal gain?
- In particular, do Tesla coils really have higher gain than an ordinary transformer with the same ratio of turns? I keep hearing this, but it contradicts what I said above about voltage gains.
The only explanation I can think of is that having a capacitor in the primary circuit allows us to first slowly charge it, and then transfer the energy without drawing extreme currents from voltage supply. Either that, or my reasoning about voltage gains is wrong.