Q. A transistor-oscillator using a resonant circuit with an inductor $L$ (of negligible resistance) and a capacitor $C$ in series produce oscillation of frequency $f$. If $L$ is doubled and $C$ is changed to $4C$, the frequency will be
A) $f/2\sqrt{2}$
B) $f/2$
C) $f/4$
D) $8f$
Explanation
(a) Frequency of $LC$ oscillation $= \frac{1}{2\pi\sqrt{LC}}$
$\Rightarrow \frac{f_{1}}{f_{2}} = \frac{1}{\sqrt{L_{1}C_{1}}}\sqrt{L_{2}C_{2}} = \left( \frac{L_{2}C_{2}}{L_{1}C_{1}} \right)^{1/2}$
$= \left( \frac{2L \times 4C}{L \times C} \right)^{1/2} = (8)^{1/2}$
$\therefore\frac{f_{1}}{f_{2}} = 2\sqrt{2} \Rightarrow f_{2} = \frac{f_{1}}{2\sqrt{2}}$ or, $f_{2} = \frac{f}{2\sqrt{2}}\ \ \lbrack\because f_{1} = f\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.