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Question #43001

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit contains a capacitor of capacitance $C$ and an inductor whose inductance varies with time as $L(t) = L_0 + \alpha t$ , where $L_0$ and $\alpha$ are positive constants. The variation in inductance is achieved by an external agent mechanically moving a magnetic core. Let $Q(t)$ be the charge on the capacitor and $I(t) = \frac{dQ}{dt}$ be the current in the circuit. The differential equation governing the charge can be written in the form $\frac{d^2Q}{dt^2} + \gamma(t) \frac{dQ}{dt} + \omega^2(t) Q = 0$ . Which of the following correctly gives the effective damping coefficient $\gamma(t)$ and the rate of change of the total electromagnetic energy $\frac{dE}{dt}$ of the circuit at an instant when the current is $I$ ?

Options

A.
$\gamma(t) = \frac{\alpha}{L_0 + \alpha t}, \quad \frac{dE}{dt} = -\frac{1}{2} \alpha I^2$
B.
$\gamma(t) = \frac{\alpha}{L_0 + \alpha t}, \quad \frac{dE}{dt} = \frac{1}{2} \alpha I^2$
C.
$\gamma(t) = 0, \quad \frac{dE}{dt} = \frac{1}{2} \alpha I^2$
D.
$\gamma(t) = 0, \quad \frac{dE}{dt} = 0$

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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

LC Circuit Oscillations|SHM Definition via Differential Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply fundamental electromagnetic principles to a non-adiabatic, time-varying system.

The primary trap lies in the misapplication of Faraday's Law; students habitually use $V_L = L \frac{dI}{dt}$ instead of the generalized $V_L = \frac{d(LI)}{dt}$ .

By ignoring the $I \frac{dL}{dt}$ term, they incorrectly conclude that the system undergoes undamped simple harmonic motion ( $\gamma = 0$ ) and that the energy increases at a rate of $\frac{1}{2}\alpha I^2$ due to the changing inductance.

Furthermore, students often assume that an ideal LC circuit must strictly conserve energy ( $\frac{dE}{dt} = 0$ ), failing to recognize that the external agent mechanically changing the inductance does negative work on the circuit, leading to a continuous extraction of electromagnetic energy.

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