A $10 \Omega$ resistance, $5 \mathrm{mH}$ coil and $10 \mu \mathrm{F}$ capacitor are joined in series. When…
A $10 \Omega$ resistance, $5 \mathrm{mH}$ coil and $10 \mu \mathrm{F}$ capacitor are joined in series. When alternating current source of suitable frequency is joined to this combination, the circuit resonates. If the resistance is halved, the resonant frequency is
quadrupled.
halved.
doubled.
remains unchanged.
Solution
Concept based.
At resonance, in LCR circuit inductive and capacitive reactance are equal:
$\begin{aligned}
& X_L=X_C \\
& \Rightarrow 2 \pi \mathrm{fL}=\frac{1}{2 \pi \mathrm{fC}} \\
& \Rightarrow \mathrm{f}^2=\left(\frac{1}{4 \pi^2 \mathrm{LC}}\right)
\end{aligned}$
The resonant frequency is independent of the resistance.