1. Condition for maximum power in an $LC$ combination

For the motor coil of inductance $L$ connected with a capacitor $C$, maximum power is imparted at resonance.

At resonance, $X_L = X_C$ where $X_L = \omega L, \qquad X_C = \frac{1}{\omega C}$

So, $\omega L = \frac{1}{\omega C}$ $C = \frac{1}{\omega^2 L}$

2. Given data

• Self inductance: $L = 10\,\text{H}$
• Frequency: $f = 50\,\text{Hz}$

Angular frequency: $\omega = 2\pi f = 2\pi(50) = 100\pi\,\text{rad/s}$

3. Calculate capacitance

$C = \frac{1}{(100\pi)^2 \cdot 10}$

$C = \frac{1}{10000\pi^2 \cdot 10}$ $C = \frac{1}{100000\pi^2}$

Using $\pi^2 \approx 9.87$, $C \approx \frac{1}{9.87 \times 10^5}$ $C \approx 1.01 \times 10^{-6}\,\text{F}$

$C \approx 1\,\mu\text{F}$

4. Match with options

• A: $8\mu F$
• B: $4\mu F$
• C: $2\mu F$
• D: $1\mu F$

Hence, the correct option is: $\boxed{D\; (1\mu F)}$

5. Comparison with stored correct answer

Stored correct answer: D

Our derived answer: D

So, they agree.