A transmitting station transmits radio waves of wavelength 300 m. The inductance of the coil required with a capacitor of capacity 1.0 μF in a resonant circuit will be

$2.53 × 10^{-8} H$

The correct answer is Option (2) → $2.53 × 10^{-8} H$

Given:

Wavelength, $\lambda = 300\ \text{m}$

Capacitance, $C = 1.0\ \mu\text{F} = 1.0 \times 10^{-6}\ \text{F}$

Speed of light, $c = 3 \times 10^8\ \text{m/s}$

Frequency of wave: $f = \frac{c}{\lambda} = \frac{3 \times 10^8}{300} = 1 \times 10^6\ \text{Hz}$

For resonance in an LC circuit:

$f = \frac{1}{2\pi \sqrt{LC}}$

Rearrange for $L$:

$L = \frac{1}{(2\pi f)^2 C}$

Substitute values:

$L = \frac{1}{(2\pi \cdot 10^6)^2 \cdot 1 \times 10^{-6}}$

$L = \frac{1}{4\pi^2 \cdot 10^{12} \cdot 10^{-6}}$

$L = \frac{1}{4\pi^2 \cdot 10^6}$

$L \approx \frac{1}{39.5 \cdot 10^6} \approx 2.53 \times 10^{-8}\ \text{H}$