Joule Heating Law
Introduction1
The current density in a conductor $\mathbf{J}$ is expressed according to Ohm’s law as follows.
$$ \mathbf{J} = \sigma \mathbf{E} $$
Then the 🔒(26/05/23)charge $q$ in the conductor experiences the force $q\mathbf{E}$, so by Newton’s second law it should be accelerated, and the current would increase with time. In reality this does not occur, because charges collide while moving in the conductor. As charges collide they lose energy, and that energy is converted into heat. In other words, the work done by the electric force on the charges is converted into heat. The power converted into heat can be written as follows.
$$ P = \dfrac{\text{work}}{\text{time}} = \dfrac{\text{work}}{\text{charge}} \dfrac{\text{charge}}{\text{time}} $$
The work per unit charge is the potential difference, and the charge per unit time is the current, so we obtain $P = VI$. And by Ohm’s law $V = IR$ we obtain the following expression.
Law
In a resistor $R$ the heat dissipated by a current $I$ is proportional to $I^{2}R$.
$$ P = I^{2}R $$
Explanation
This is called Joule heating. It was reported in 1840 by the British physicist James Prescott Joule.2