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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
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The correct answer is:
remains unchanged.
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.
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.
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