Question

Using the mathematical expressions for the strength of current, potential difference and ohm's law, derive an expression for the heat produced if the current is purely resistive

Answer 1

We know that,
$$\begin{gathered} Current\left(I\right)=\frac{Charge\left(Q\right)}{Time\left(t\right)} \\ or,Q=It................\left(i\right) \end{gathered}$$

If W is the amount of work done in carrying a charge from one end to the other end of a conductor, then by definition of potential difference,
$$\begin{gathered} V=\frac{W}{Q} \\ or,W=VQ..............\left(ii\right) \end{gathered}$$

From equation (i) and (ii),
$W=V(It)=VIt.............\left(iii\right)$

According to Ohm's law,
$V=IR...............\left(iv\right)$

From equation (iii) and (iv),
$W=(IR)It=I^{2}Rt............\left(v\right)$

This work done (W) by the current measures the electric energy which is supplied by the battery or any other source of electrical energy. If the conductor is a heating appliance and current is purely resistive, then this energy (W) is converted into heat (H). Thus,
$W=H=I^{2}Rt$

This is the expression for heat produced.