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Question #43318

Difficulty: 8/10 | 🟢 Free

The Question

An ideal capacitor of capacitance $C$ is initially charged to a voltage $V_0$ . At time $t=0$ , it is connected in a closed series loop with an inductor of inductance $L$ and a time-varying resistor. The resistance of the resistor increases linearly with time as $R(t) = \alpha t$ , where $\alpha$ is a positive constant in $\Omega \cdot \text{s}^{-1}$ . Assume that the circuit remains highly underdamped during the time interval of interest (i.e., the fractional energy loss per cycle is very small). Which of the following expressions gives the time $t_1$ at which the total electromagnetic energy of the circuit decreases to $1/e$ of its initial value?

Options

A.
$\sqrt{\frac{2L}{\alpha}}$
B.
$\sqrt{\frac{L}{\alpha}}$
C.
$\sqrt{\frac{L}{2\alpha}}$
D.
$\sqrt{\frac{4L}{\alpha}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

LC Circuit Oscillations|Internal Resistance Variation|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze non-autonomous dissipative systems, specifically an LC circuit with a time-varying internal resistance.

The primary trap is the premature assumption of an invariant decay constant.

Students often memorize the energy decay formula for a constant resistance, $E(t) = E_0 e^{-Rt/L}$ , and incorrectly substitute $R(t) = \alpha t$ directly into the exponent to get $E(t) = E_0 e^{-\alpha t^2/L}$ .

This leads to the incorrect answer $\sqrt{L/\alpha}$ .

By doing so, they bypass the fundamental principle of energy conservation ( $\frac{dE}{dt} = -P$ ) and fail to integrate the time-varying damping factor, missing the crucial factor of $1/2$ that arises from the integration of $t$ with respect to time.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	LC Circuit Oscillations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free