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Question #79297

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The Question

An ideal capacitor of capacitance $C$ is connected in a closed series loop with a time-varying inductor. The inductance varies with time $t$ as $L(t) = L_0 \ln(1 + \alpha t)$ , where $L_0$ is a positive constant and $\alpha$ is a dimensional parameter. Let $q(t)$ represent the instantaneous charge on the capacitor, and assume the current in the loop is given by $I = \frac{dq}{dt}$ . Which of the following represents the correct differential equation governing the charge $q(t)$ , and what is the dimensional formula for $\alpha$ ?

Options

A.
$L_0 \ln(1 + \alpha t) \frac{d^2q}{dt^2} + \frac{L_0 \alpha}{1 + \alpha t} \frac{dq}{dt} + \frac{q}{C} = 0$ , $[\alpha] = [\text{T}^{-1}]$
B.
$L_0 \ln(1 + \alpha t) \frac{d^2q}{dt^2} + \frac{q}{C} = 0$ , $[\alpha] = [\text{T}^{-1}]$
C.
$L_0 \ln(1 + \alpha t) \frac{d^2q}{dt^2} + \frac{L_0 \alpha}{1 + \alpha t} \frac{dq}{dt} + \frac{q}{C} = 0$ , $[\alpha] = [\text{T}]$
D.
$L_0 \ln(1 + \alpha t) \frac{d^2q}{dt^2} - \frac{L_0 \alpha}{1 + \alpha t} \frac{dq}{dt} + \frac{q}{C} = 0$ , $[\alpha] = [\text{T}^{-1}]$

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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

Dimensional Consistency and Transcendental Functions|LC Circuit Oscillations|L-C Circuit Analogy

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Common Mistakes

This question tests the student's fundamental understanding of Faraday's Law and dimensional analysis in non-autonomous systems.

A common trap for students is to blindly apply the standard LC oscillation equation $L \frac{d^2q}{dt^2} + \frac{q}{C} = 0$ by treating inductance as a quasi-static parameter.

However, when inductance is time-varying, the magnetic flux linkage is $\Phi = L(t)I$ , and the induced EMF requires the product rule: $\frac{d\Phi}{dt} = L\frac{dI}{dt} + I\frac{dL}{dt}$ .

Ignoring the $I\frac{dL}{dt}$ term leads to an incorrect differential equation that misses the non-linear damping-like term.

Additionally, the question filters out dimensional errors by testing the principle that arguments of transcendental functions must be strictly dimensionless.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Dimensional Consistency and Transcendental Functions
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free