Derive the equation for p=i^2r
Answers
Explanation:
What you did is a good proof, though as you are finding for it to be convincing you have to think carefully about what everything means. The really hard thing is P=VI, which I'm actually surprised isn't a given formula. Once you have that, as you showed, getting to P=I2R is easy, so let's focus on getting P=VI.
It's much cleaner to use calculus if you know how to use that: for example you can be able to use the definition of power P=dEdt as well as a formula for the energy (and also the definition of current).
However if you don't know calculus, then you have to think through what the time interval T and the energy W are, as you are rightly worried about. (However you are going above and beyond by thinking carefully through what these things mean, and that is great!)
I would start with T. It doesn't really matter what you take for T, but you do have to be consistent about it. Let's say you want T to be the time it takes the electron to make one complete circuit.
OK, now we go through the other formulas. Can we find an interpretation for all of them with this definition of T?
First compute the energy gained/lost by one electron. That would be E=qV, where V is the voltage gained/lost by an electron in that period of time. Actually... to be very precise we only want the energy lost by the electrons as they moved through the appliance, we don't want the energy gained by the electron as it went through the battery. With that caveat, a single electron loses eV0 of energy, where V0 is the voltage of the battery.
Then, how much energy does the device use in that time? Call it W.
Then use conservation of energy. All the energy the device used came from electrons. So W=NqV0, where N is the number of electrons that are in the wire.
What was the power? It was the energy that was used in the time T. So it's P=NqTV0.
I'll leave the rest to your imaginatio