In a potentiometer, the emf of a cell is directly proportional to its balancing length:

$$E \propto l \implies \frac{E_1}{E_2} = \frac{l_1}{l_2}$$

Given:

• $E_1 = 1.5\,\text{V}$, $l_1 = 45.0\,\text{cm}$
• $l_2 = 75.0\,\text{cm}$

Substituting the values:

$$E_2 = E_1 \times \frac{l_2}{l_1} = 1.5 \times \frac{75.0}{45.0} = 1.5 \times \frac{5}{3} = 2.5\,\text{V}$$

First, calculate the individual resistance of the heater and the bulb:

$$R_H = \frac{V^2}{P_{H,\text{rated}}} = \frac{220^2}{750} = \frac{48400}{750} \approx 64.53\,\Omega$$ $$R_B = \frac{V^2}{P_{B,\text{rated}}} = \frac{220^2}{200} = \frac{48400}{200} = 242.0\,\Omega$$

Total resistance in series:

$$R_{\text{total}} = R_H + R_B = 64.53 + 242.0 = 306.53\,\Omega$$

Circuit current under $220\,\text{V}$ supply:

$$I = \frac{V}{R_{\text{total}}} = \frac{220}{306.53} \approx 0.7177\,\text{A}$$

Power consumed by the bulb:

$$P_B = I^2 R_B = (0.7177)^2 \times 242 \approx 0.5151 \times 242 \approx 124.8\,\text{W}$$

Power consumed by the heater:

$$P_H = I^2 R_H = (0.7177)^2 \times 64.53 \approx 0.5151 \times 64.53 \approx 33.25\,\text{W}$$

Let the emf of the cell be $E$ and its internal resistance be $r$.
The power delivered to external resistance $R$ is:

$$P = I^2 R = \left(\frac{E}{R + r}\right)^2 R$$

Given $P_1 = P_2$ for resistances $R_1$ and $R_2$:

$$\frac{E^2 R_1}{(R_1 + r)^2} = \frac{E^2 R_2}{(R_2 + r)^2}$$

Taking the square root on both sides:

$$\frac{\sqrt{R_1}}{R_1 + r} = \frac{\sqrt{R_2}}{R_2 + r}$$ $$\sqrt{R_1}(R_2 + r) = \sqrt{R_2}(R_1 + r)$$ $$R_2\sqrt{R_1} + r\sqrt{R_1} = R_1\sqrt{R_2} + r\sqrt{R_2}$$ $$r(\sqrt{R_2} - \sqrt{R_1}) = R_2\sqrt{R_1} - R_1\sqrt{R_2} = \sqrt{R_1 R_2}(\sqrt{R_2} - \sqrt{R_1})$$

Since $R_1 \neq R_2$:

$$r = \sqrt{R_1 R_2}$$

Mobility of electrons $\mu$ is defined as the magnitude of drift velocity per unit electric field:

$$\mu = \frac{v_d}{E} = \frac{e\tau}{m}$$

• Dependence on Electric Field / Potential Difference: Since drift velocity $v_d = \frac{e E \tau}{m}$ is directly proportional to electric field $E = \frac{V}{L}$, the ratio $\frac{v_d}{E}$ is independent of the applied electric field and potential difference. Thus, statement C is correct, while A and B are incorrect.
• Dependence on Temperature: As temperature increases, lattice ions vibrate more vigorously, reducing the average relaxation time $\tau$, which decreases mobility ($\mu \propto \tau$). Conversely, as temperature decreases, $\tau$ increases, so mobility increases. Thus, statement D and E are incorrect.