In this monologue, i try to reason as why the power dissipated in a resistor with resistance $R$ carrying a current $I$ is $I^2R$.
One can of course study material properties and collisions etc and come up with something.
But the simple proof comes from law of conservation of energy..
A constant current $I$ indicates that every electron is travelling at an average constant velocity through the circuit.
But as it passes through the resistor, it moves through a potential difference of $R.I$.
But, why is the potential difference R.I. ? For that, one definitely needs to consider vibrating atoms and collisions, and the mean time between collisions etc :-)
Or, it can be simplified in this way.. The electrons need some motivation to keep passing through the resistor. That motivation is the potential difference between the two ends of the resistor. However experimental observation shows that charge passing through the resistor does not accelerate but instead a steady flow is observed. So the Voltage V instead of causing an acceleration only causes a velocity leading to constant current I. We are calling the proportionality factor as resistance R.
So each electron loses q.R.I of potential energy during its travel through the resistor where q is charge of electron in coulombs
Since the velocity is constant, the kinetic energy is constant and has not changed while the electron moved through the resistor. But, its potential energy went down by $q.R.I$ ,
The power dissipated by each electron is $(q.R.I)/t$ where '$t$' is the time taken by the electron to travel through the resistor because power is nothing but the 'rate' at which work is done.
If there are $N$ such electrons falling through the resistor every $t$ seconds, then $(N.q.R.I)/t$ is the total power dissipated. But, by definition $(N.q)/t$ is the charge transferred per second.
Why? Because, $N.q$ is the charge falling through the length of the resistor in $t$ seconds. So $(N.q)/t$ is the charging fall through the cross section of the resistor per second, which is nothing but the current $I$.
Thus, power dissipated turns out to be $I^2.R$
So, lets end this with a thanks to law of conservation of energy :-)
