$(a)$ Derive the formula for the calculation of work done when current flows through a resistor

$(b)$ One electric bulb is rated $40\, W$ and $240\, V$ and other $25\, W$ and $240\, V$. Which bulb has higher resistance and how many times ?

$(a)$ Suppose a current $I$ is sent through a conductor of resistance $R$ for time $t$ under a potential difference of $V$ volt as shown in figure.

Then charge flowing through the conductor is $Q=I t$

The work done in taking $Q$ coulomb charge from one end of the conductor to the other end at a potential difference $V$ is

$W=V Q$ $....(1)$

or $\quad W=V I t$ $....(2)$

$(b)$ $\quad P_{1}=40 W, V_{1}=240 V$

$P_{1}=\frac{V_{1}^{2}}{R_{1}}$ or $R_{1}=\frac{V_{1}^{2}}{P_{1}}$

$=\frac{240 \times 240}{40}$

$P_{2}=25 W, V_{2}=240 V$

$P_{2}=\frac{V_{2}^{2}}{R_{2}}$ or $R_{2}=\frac{V_{2}^{2}}{P_{2}}=\frac{240 \times 240}{25}$

So $\frac{R_{1}}{R_{2}}=\frac{\frac{240 \times 240}{40}}{\frac{240 \times 240}{25}}=\frac{25}{40}$

$\frac{R_{1}}{R_{2}}=\frac{5}{8}$

Twenty$-$five watt bulb has more resistance.