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An initially charged undriven $L C R$ circuit having inductance $L$, capacitance $C$ and resistance $R$ will be
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The correct answer is:
oscillate with damping, if $R^2 < \frac{4 L}{C}$
In an $L-C-R$ circuit (without any AC input), when $\frac{R^2}{4 L^2}=\frac{1}{L C}$, charge decays exponentially and no oscillations occurs.
When $\frac{R^2}{4 L^2}>\frac{1}{L C}$, then also charge decays exponentially and no oscillations occurs.
But when $\frac{R^2}{4 L^2} < \frac{1}{L C}$ or when $R^2 < \frac{4 L}{C}$, then oscillations occurs with frequency,
$$
f=\frac{1}{2 \pi} \sqrt{\frac{1}{L C}-\frac{R^2}{4 L^2}}
$$
When $\frac{R^2}{4 L^2}>\frac{1}{L C}$, then also charge decays exponentially and no oscillations occurs.
But when $\frac{R^2}{4 L^2} < \frac{1}{L C}$ or when $R^2 < \frac{4 L}{C}$, then oscillations occurs with frequency,
$$
f=\frac{1}{2 \pi} \sqrt{\frac{1}{L C}-\frac{R^2}{4 L^2}}
$$
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