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A transistor-oscillator using a resonant circuit with an inductor $L$ (of negligible resistance) and a capacitor $C$ in series produces oscillations of frequency $f$. If $L$ is doubled and $C$ is changed to $4C$, the frequency will be :-
A
$\dfrac{f}{4}$
B
$8f$
C
$\dfrac{f}{2\sqrt{2}}$
D
$\dfrac{f}{2}$
Detailed Solution
Frequency of oscillation of an LC circuit: $f = \dfrac{1}{2\pi\sqrt{LC}}$
When $L$ is changed to $2L$ and $C$ to $4C$: $f' = \dfrac{1}{2\pi\sqrt{(2L)(4C)}}$
$f' = \dfrac{1}{2\pi\sqrt{8LC}} = \dfrac{1}{2\sqrt{2}} \times \dfrac{1}{2\pi\sqrt{LC}}$
Therefore $f' = \dfrac{f}{2\sqrt{2}}$
When $L$ is changed to $2L$ and $C$ to $4C$: $f' = \dfrac{1}{2\pi\sqrt{(2L)(4C)}}$
$f' = \dfrac{1}{2\pi\sqrt{8LC}} = \dfrac{1}{2\sqrt{2}} \times \dfrac{1}{2\pi\sqrt{LC}}$
Therefore $f' = \dfrac{f}{2\sqrt{2}}$