Question

The inductance of the oscillatory circuit of a radio station is $10mH$ and its capacitance is $0.25\mu F$.Taking the effect of resistance negligible, wavelength of the broadcasted waves will be (velocity of light $=3.0\times 10^{8}m/s,\pi =3.14$ )

• A. $9.42\times {10}^{4}m$
• B. $18\times 8.{10}^{4}m$
• C. $4.5\times {10}^{4}m$
• D. none of these

Answer 1

Answer: (A)

When inductive reactance is equal to capacitive reactance circuit is in resonance.
In an $L-C$ circuit the impedance of circuit is $Z=X_L-X_C$
When $X_L=X_C$, then $Z=0.$
In this situation the amplitude of current in the circuit would be infinite. It will be condition of electrical resonance and frequency is given by
$f=\frac{1}{2\pi \sqrt{LC}}$
$=\frac{1}{2\times 3.14\times \sqrt{10\times 10^{-3}\times 0.25\times 10^{-6}}}$
$=3184.7$ cycles/s.
Also frequency $=\frac{\text{ velocity }}{\text{ wavelength }}$
$\Rightarrow \lambda =\frac{c}{f}=\frac{3\times 10^8}{3184.7}$
$\Rightarrow \lambda =9.42\times 10^{4}m$