Derivation: Electric Power

When charge $ Q $, due to an applied voltage $ U $ is transported in the conductor, potential energy is converted into kinetic energy. The kinetic energy $ W $ that a charge gains ($W$ positive)) or loses ($W$ negative)) by passing through the voltage $U$ is given by:

$$ \begin{align} W ~=~ Q \, U \end{align} $$

The power $P$ is defined as the converted energy $W$ per time period $t$:

$$ \begin{align} P ~=~ \frac{W}{t} \end{align} $$

The electric power is obtained by substituting Eq. 1 into 2:

$$ \begin{align} P ~=~ \frac{Q \, U }{t} \end{align} $$

The electric current $I$, is the charge $Q$ transported per time interval $t$: $I = Q/t $. The factor $Q/t$ is in Eq. 3, so we replace it with $I$ to eliminate the unknown and experimentally not easily accessible time $t$. Thus, the electric power becomes:

Electric power (with U, I)

$$ \begin{align} P ~=~ U \, I \end{align} $$

For an Ohmic conductor (these are those conductors for which Ohm's law applies), Equation 4 can be rewritten using $ U = R \, I $. The power $P$ can therefore be expressed by the resistance $R$ of the conductor (or a load) and the applied voltage $U$:

Electric power (with U, R)

$$ \begin{align} P ~=~ \frac{U^2}{R} \end{align} $$

A simple circuit.

Example: Constant voltage

If the voltage $U$ is kept constant, the regions of the conductor that have the smallest resistance $R$ will be the warmest, because that is where the converted power is the largest.

Of course you can also rearrange Ohm's law $ U = R\,I $ with respect to the current $ I = U/R $, and use it to rewrite the electric power:

Electric power (with I, R)

$$ \begin{align} P ~=~ R \, I^2 \end{align} $$

Example: Constant current

When the current $I$ is kept constant, the regions of the conductor that have the largest resistance $R$ will be the warmest, because that is where the power converted is the largest.