When a potential difference is applied across the ends of a conductor, the free electrons are accelerated and acquire kinetic energy. As the electrons move through, they collide with the positive ions and atoms of the conductor and transfer their kinetic energy to them. Between two collisions, the electrons again pick up kinetic energy from the electric field. As a result, the kinetic energy of vibration of these lattice ions or atoms increases. This increases the thermal energy of the lattice, which means that the temperature of the conductor increases. Since the source of emf (e.g., a battery) is maintaining current in the conductor, the electric energy supplied by the battery is converted into heat in the conductor.
To find out how much heat is produced, consider the circuit below.
Let a battery maintain a steady current I through the circuit, and a potential difference V between the two ends 'a' and 'b' of the resistor. Let Dt be the time taken by the charge to flow from a to b, and Dq the amount of charge crossing point a in time Dt. Then
The loss in electrical potential energy in time Dt is given by
dw = Dq (Va - Vb) = Dq. V
= VI Dt = I2 R Dt
This gets converted to heat in the resistor. So, for a steady current I, the amount of heat produced in time t is
H = I2 Rt
These equations comprise Joule's law of heating. It implies that the heat produced in a conductor is directly proportional to
(i) square of current for given R
(ii) resistance for a given I
(iii) time for which current flows; and inversely proportional to
(iv) resistance for a given V.
