A series $L C R$ circuit is connected across a source of alternating emf of changing frequency and resonates…
A series $L C R$ circuit is connected across a source of alternating emf of changing frequency and resonates at frequency $f_0$. Keeping capacitance constant, if the inductance $(L)$ is increased by $\sqrt{3}$ times and resistance is increased $(R)$ by 1.4 times, the resonant frequency now is
$3^{1 / 4} f_0$
$\sqrt{3} f_0$
$(\sqrt{3}-1)^{1 / 4} f_0$
$\left(\frac{1}{3}\right)^{1 / 4} f_0$
Solution
We knows in LCR circuits
and
$
\begin{aligned}
& f=\frac{1}{2 \pi \sqrt{L C}} \\
& f \propto \frac{1}{\sqrt{L}}
\end{aligned}
$
Here, in given condition
$\begin{aligned}
& \frac{f_1}{f_2}=\sqrt{\frac{L_1}{L_2}}=\sqrt{\frac{L}{\sqrt{3} L}} \\
& \Rightarrow \quad f_1=\left(\frac{1}{3}\right)^{1 / 4} f_2 \\
&
\end{aligned}
$