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Why does a faster clock require more power?

If you overclock a microcontroller, it gets hot.

If you overclock a microcontroller, it needs more voltage.

In some abstract way it makes sense: it is doing more computation, so it needs more energy (and being less than perfect, some of that energy dissipates as heat).

However, from a just plain old Ohm's law level electricity and magnetism, what is going on?

Why does the clock frequency have anything to do with power dissipation or voltage?

As far as I know, the frequency of AC has nothing to do with its voltage or power, and a clock is just a super-position of a DC and a (square) AC. Frequency doesn't affect the DC.

Is there some equation relating clock frequency and voltage or clock frequency and power?

I mean does a high speed oscillator need more voltage or power than a low speed one?