Question

How do you get from Power = work/time to Power = v^2/r ?

Answer 1

Consider a conductor with end points A and B, in which a current I is flowing from A to B. The electric potential at A and B are denoted by V(A) and V(B) respectively. Since current is flowing from A to B, V(A) > V(B) and the potential difference across AB is V = V(A) – V(B) > 0.

In a time interval ?t, an amount of charge ?Q = I ?t travels from A to B. The potential energy of the charge at A, by definition, was Q V(A) and similarly at B, it is Q V(B) (basics rules from electrostatics). Thus, change in its potential energy ?Upot is ?Upot = Final potential energy – Initial potential energy = ?Q[(V (B) – V (A)] = –?Q V = –I V?t < 0 If charges moved without collisions through the conductor, their kinetic energy would also change so that the total energy is unchanged. Conservation of total energy would then imply that, ?K = –?Upot

While in an actual conductor, an amount of energy dissipated as heat in the conductor during the time interval ?t is, ?W = I V?t . The energy dissipated per unit time is the power dissipated P = ?W/?t and we have, P = I V Using Ohm’s law V = IR, we get P = I ² R = V ² /R as the power loss (“ohmic loss”) in a conductor of resistance R carrying a current I