What is the formula for electrical power in a DC circuit?

[ximath]

I am studying DC circuits and trying to derive formula for power; that is in fact
$$P = I.Vab$$

where Vab is the potential difference between two terminals of a circuit element.

From my previous studies, I know that.

$$P = dW / dt$$

I assume that for a small interval, dt, a single charge q has a small displacement , dx.

Then;

$$dW = E q dx$$

where E is magnitude of the electric field.

Hence, from the formula above,

$$P = E q dx / dt$$

This is all I could come up with. I want to go on with this idea to prove

$$P = I Vab$$


I know I need to substitute dq somewhere (to get I ) , somehow.. Simply replacing q with dq does not seem to work; it leads to an incorrect formula. (I find P = I dV ; I guess, if I do that.)

 

[MATLABdude]

...I assume that for a small interval, dt, a single charge q has a small displacement , dx.

Then;

$$dW = E q dx$$

where E is magnitude of the electric field.

It would be more advantageous to assume that you have a differential charge dq moving a distance x in the electric field E. (And does dq/dt ring a bell?) Also I believe this approach is more correct, since you have numerous charges (instead of a single one) and you're attempting to find their aggregate behaviour when you use Ohm's law.

This derivation is also a very macroscopic one; the standard first-principles approach is given at Wikipedia:
http://en.wikipedia.org/wiki/Ohm's_law