Joule’s Laws of Heating

When current passes through a conductor, the electrons flow in the conductor from the negative terminal to the positive terminal. These flowing electrons collide with atoms (ions) of the conductor which causes the resistance of the conductor. Due to collision, some amount of energy of electrons is converted into heat. In this way, heat is produced in a conductor when current passes through it. This is called heating effect of current. The amount of heat produced was first calculated by Dr. Joule of Manchester on his experimental results in 1841. From his experiments, he gave some laws regarding production of heat in a conductor carrying current which are known as Joule’s laws of heating.

Joule’s laws of heating are as follows;

1. Joule’s first law (Law of current): The amount of heat produced in a conductor in a given time is directly proportional to the square of current i.e. $H ∝ I^2 \text{ when R and t are constant}$ $\text{or, }\frac{H_1}{H_2}=\frac{I_1^2}{I_2^2}=\text{Constant}$
2. Joule’s second law (Law of resistance): The amount of heat produced in a conductor in a given time by a given current is directly proportional to the resistance of the conductor i.e. $H ∝ R\text{ when I and t are constant}$ $\text{or, }\frac{H_1}{H_2}=\frac{R_1}{R_2}=\text{Constant}$
3. Joule’s third law (Law of time): The amount of heat produced in a conductor by a given current is directly proportional to the time i.e. $H ∝ t \text{ when I and R are constant}$ $\text{or, }\frac{H_1}{H_2}=\frac{t_1}{t_2}=\text{Constant}$

Expression for the heat developed in a conductor carrying electric current using Joule’s Laws of Heating

Consider a conductor having resistance $R$ is connected to a cell of emf $E$.

Let $V$ be the potential difference across the conductor and $I$ be the current passing through the conductor. Then, according to Ohm’s law, $V=IR \text{ ___(1)}$

If $q$ is the total charge that flows through the conductor in time $t$, then, $I=\frac{q}{t}$ $q=It \text{ ___(2)}$

The potential difference is work done per unit charge i.e. $V=\frac{W}{q}$ $W=Vq \text{ ___(3)}$

From equation $(1)$, $(2)$ and $(3)$ $W=IR.It$ $W=I^2Rt$

This work done is converted into heat energy. Thus, if $H$ is the heat produced, then according to the law of conservation of energy, $H=I^2Rt$

This equation gives the heat developed in a conductor on passing current.

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