Discussion
In a critically damped series LCR circuit :
Answer: A
For critical damping \(R^{2} = 4L/C\).
The charge in a series LCR circuit obeys \(L\ddot{q} + R\dot{q} + q/C = 0\), with characteristic equation \(Ls^{2} + Rs + 1/C = 0\). Critical damping is the boundary between oscillation and slow decay, where the discriminant is zero:
\[ \begin{aligned} R^{2} - \frac{4L}{C} &= 0 \\ R &= 2\sqrt{\frac{L}{C}} \end{aligned} \]
If \(R^{2}\) is smaller, the circuit oscillates (underdamped), and if larger, it is overdamped. \(R = \sqrt{L/C}\) is half the critical value, and the other forms do not even have the dimensions of resistance squared.
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