$(a)$ 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 another is $25 \ W$ and $240 \ V$. Which bulb has higher resistance and how many times?

(N/A) Suppose a current $I$ flows through a conductor of resistance $R$ for time $t$ under a potential difference of $V$ volts.
The charge flowing through the conductor is $Q = I \times t$.
The work done $(W)$ in moving a charge $Q$ across a potential difference $V$ is given by:
$W = V \times Q$
Substituting $Q = I \times t$ into the equation:
$W = V \times I \times t$
$(b)$ Given:
For bulb $1$: $P_1 = 40 \ W$,$V_1 = 240 \ V$
For bulb $2$: $P_2 = 25 \ W$,$V_2 = 240 \ V$
We know that $P = \frac{V^2}{R}$,so $R = \frac{V^2}{P}$.
$R_1 = \frac{240^2}{40} = \frac{57600}{40} = 1440 \ \Omega$
$R_2 = \frac{240^2}{25} = \frac{57600}{25} = 2304 \ \Omega$
Comparing $R_1$ and $R_2$,$R_2 > R_1$.
Ratio: $\frac{R_2}{R_1} = \frac{2304}{1440} = 1.6$
Thus,the $25 \ W$ bulb has higher resistance,and it is $1.6$ times the resistance of the $40 \ W$ bulb.