Derive that $P = I^2R$ As our homework, we were asked to derive $P = I^2R$. 
Now, I started off with the basic relation $P = \frac{W}{T}$. I was not able to think of anything from here, so I started plugging in random formulas from a purely algebraic point of view. 
I tried the one for voltage $V = \frac{W}{q} \implies Vq = W$ and got $P = \frac{Vq}{T}$. And since $I = \frac{q}{T}$, $IT = q$, plugging this in we get $P = IV$. A simple application of Ohm's Law gives us the required equation. But, when I tried to get an intuitive grasp of what I'd just done, I found a few discrepancies:


*

*In the definition of voltage, $W$ refers to the energy an electron loses when it goes from the negative to the positive terminal. In the definition of power, $W$, refers to the total energy consumed by the appliances. How can you relate the two?

*In the definition of current $T$, refers to the time taken by the electron to go from one terminal to the other, whereas, in power $T$ refers to the time for which the appliances run or the energy is supplied. How can these two be the same?
Is my derivation incorrect? If not, then how would you counter the arguments I mentioned above?
 A: 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 $P=I^2 R$ 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 $P=\frac{dE}{dt}$ 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 $e V_0$ of energy, where $V_0$ 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 $W=NqV_0$, 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 $P=\frac{Nq}{T}V_0$.
I'll leave the rest to your imagination.
A: This is very Simple..
P = VI   _____ (1)
V = IR  ______(By Ohm's Law)
VI = I^2 R  _(Multiplying both sides by I)
P = I^2 R  _(From eq.1)
Hence Proved !
;)
