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  4. A transistor-oscillator using a resonant circuit with an inductor L (of negligible resistance) and a capacitor C in series produce oscillations of frequency f . If L is doubled and C is changed to 4C, the frequency will be

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Q. A transistor-oscillator using a resonant circuit with an inductor $L$ (of negligible resistance) and a capacitor $C$ in series produce oscillations of frequency $ f $ . If $L$ is doubled and $C$ is changed to $4C$, the frequency will be

Haryana PMTHaryana PMT 2008Alternating Current

Solution:

In a series $LC$ circuit, frequency of $LC$ oscillations is given by
$f=\frac{1}{2 \pi \sqrt{L C}}$
or $f \propto \frac{1}{\sqrt{L C}} $
$\Rightarrow \frac{f_{1}}{f_{2}}=\sqrt{\frac{L_{2} C_{2}}{L_{1} C_{1}}}$
Given
$L_{1}=L, C_{1}=C, L_{2}=2 L, C_{2}=4 C, f_{1}=f$
$ \therefore \frac{f}{f_{2}}=\sqrt{\frac{2 L \times 4 C}{L C}}=\sqrt{8} $
$\Rightarrow f_{2}=\frac{f}{2 \sqrt{2}}$