Question

Prove that the heat produced in a conductor of resistance $R$ on passing current $I$ through it for time $t$ is $H=I^{2}Rt$.

Answer 1

Proof-

Assume a conductor of resistance $R$, the current flowing through it and potential difference are $I$ and $V$ respectively. If the given amount of current flows for $t$ seconds, then the amount of electric charge, $Q$, transferred from one end to the other end of the conductor is given by

$Q=It$…………(1)

When an electric charge $Q$ moves against a potential difference$V$, the amount of work done, $W$, is given by,

$W=QV$……..(2)

According to Ohm's law,

$V=IR$……….(3)

Now substitute the value of $Q$ and $V$ from equations (1) and (3) into equation (2) of $W$.

$$\begin{gathered} W=It\times IR \\ W=I^{2}Rt \end{gathered}$$

This amount of work is known as electrical work. Now, this amount of work done is dissipated as heat, $H$ in the conductor. Therefore, $W=H$

So, it can be said that,

$H=I^{2}Rt$

The above expression gives the amount of heat production on the passing of current $I$ through the conductor. It is also known as Joule's law of heating.