Question

The inductance of the oscillatory circuit of a radio station is $10 \,mH$ 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
$f=\frac{1}{2 \pi \sqrt{L C}}$
$=\frac{1}{2 \times 3.14 \times \sqrt{10 \times 10^{-3} \times 0.25 \times 10^{-6}}}$
$=3184.7$ cycles/s.
Also frequency $=\frac{\text { velocity }}{\text { wavelength }}$
$\Rightarrow \lambda=\frac{c}{f}=\frac{3 \times 10^{8}}{3184.7}$
$\Rightarrow \lambda=9.42 \times 10^{4}\, m$