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Question #43341

Difficulty: 8/10 | 🟢 Free

The Question

A series LCR circuit with a quality factor $Q = 50$ and resonant frequency $\omega_0 = 1000 \text{ rad s}^{-1}$ is connected to an alternating voltage source $V(t) = V_0 \cos(\omega_0 t)$ at time $t=0$ . The initial charge on the capacitor and the initial current in the inductor are both zero. Assuming the circuit is highly underdamped ( $Q \gg 1$ ), the total current in the circuit can be approximated as an oscillating function with a time-varying amplitude $I(t)$ . Find the time $t_0$ at which the amplitude $I(t)$ reaches $(1 - e^{-1})$ of its final steady-state value.

Options

A.
$0.10 \text{ s}$
B.
$0.05 \text{ s}$
C.
$0.02 \text{ s}$
D.
$0.20 \text{ s}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Bandwidth in RLC|Series LCR Impedance Vector|Transient Current in RL Circuit|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of transient dynamics in an AC-driven LCR circuit, specifically targeting a common misconception regarding time constants.

Students frequently confuse the time constant of a simple RL circuit ( $\tau = L/R$ ) with the amplitude decay time constant of an underdamped second-order LCR circuit.

Because the characteristic equation of the LCR circuit yields a real part of $-R/2L$ , the transient amplitude envelope decays as $e^{-Rt/2L}$ , making the true time constant $\tau = 2L/R$ .

A student who blindly applies the steady-state bandwidth formula ( $\Delta \omega = R/L$ ) and assumes the transient envelope decays as $e^{-(\Delta \omega) t}$ will incorrectly choose $0.05 \text{ s}$ .

This highlights the critical distinction between energy decay rate (proportional to $e^{-Rt/L}$ ) and amplitude decay rate (proportional to $e^{-Rt/2L}$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Bandwidth in RLC
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free