A transmitting station transmits radio waves of wavelength 300 m. The inductance of the coil required with a capacitor of capacity \( 1.0\,\mu\mathrm{F} \) in a resonant circuit will be
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 2.53 \times 10^{-8}\,\mathrm{H} \)
The LC circuit is resonant at the frequency of the wave, \( f = 1/(2\pi\sqrt{LC}) \). First find the frequency of the radio wave from its wavelength.
\( \displaystyle f = \frac{c}{\lambda} = \frac{3\times10^{8}\,\mathrm{m\,s^{-1}}}{300\,\mathrm{m}} = 10^{6}\,\mathrm{Hz} \)
Squaring the resonance condition and solving for \( L \):
\( \displaystyle L = \frac{1}{4\pi^{2} f^{2} C} \)
\( C = 1.0\,\mu\mathrm{F} = 10^{-6}\,\mathrm{F} \)
\( 4\pi^{2} = 39.5 \)
\( \displaystyle L = \frac{1}{39.5 \times (10^{6})^{2} \times 10^{-6}} = \frac{1}{3.95\times10^{7}} = 2.53\times10^{-8}\,\mathrm{H} \)