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In a simple DC circuit, I understand the concept of voltage (“V” - electrical potential energy difference per charge) and current (“I” – rate of flow of charge, = V/R). I also understand Ohm’s law (V=IR), and I realize that, where Ohm’s law applies, there are 3 ways of expressing power – V·I, I²R, and V²/R. What I’m trying to do is interpret what each one of those formulas mean on a “stand-alone” basis.

The first one, P=V·I, is easy because I’ve seen the definition in numerous places. It’s (energy dissipated per unit of time) = (energy dissipated per charge passing through a resistor) × (charge passing through a resistor per unit of time). This makes perfectly good sense to me, since the charge in the denominator of voltage cancels the charge in the numerator of current, leaving energy per unit of time, which is the definition of power.

P=I²R is a bit trickier. Why is P proportional to the square of I (and proportional to R)? I haven't found an interpretation of this; the only explanation I’ve seen is basically a restatement of P=V·I using Ohm’s law. This isn’t what I’m looking for, though. So I’ve come up with my own interpretation - it makes sense to me, but I don’t know if it’s actually correct. The way I think of it is that, in a current passing through a resistor, electrons are moving around and colliding with the other particles in the resistor. These collisions result in kinetic energy being transferred to the resistor, which is essentially the energy dissipated. Since KE is proportional to the square of velocity, it makes sense that power would be proportional to the square of current, since a higher current means higher electron velocity (assuming a constant R). And if the current is kept constant, doubling (for example) the resistance would in effect double the number of particles the electrons can collide with, thereby doubling the rate of collisions and doubling the total energy dissipated. So like I said, all this makes sense to me, and I’ve done it without explicitly talking about voltage, but I’m wondering if it is in fact a reasonably accurate description. And if it isn’t correct, what is the correct interpretation?

P= V²/R is worse. Why is power proportional to the square of voltage, and inversely proportional to R? I can’t for the life of me come up with an interpretation of this that doesn’t require me to consider current, which basically defeats the purpose of what I’m trying to do. Is there a way to look at this formula that doesn’t essentially require it to be restated as P= V·I?

Thanks in advance for any insight you can provide.