Before You Begin

Question #41996

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit contains a capacitor of constant capacitance $C$ and an inductor whose inductance varies with time as $L(t) = L_0 \ln(1 + \alpha t)$ , where $L_0$ and $\alpha$ are positive constants. If $q(t)$ represents the charge on the capacitor at time $t$ , which of the following represents the dimensional formula of $\alpha$ and the correct differential equation governing the charge $q(t)$ ?

Options

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

Check Your Answer

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

Common Mistakes

This question tests the student's ability to apply Faraday's Law from first principles in a non-autonomous system.

A common trap is assuming the standard inductor voltage formula $V_L = L \frac{dI}{dt}$ holds universally, which ignores the $I \frac{dL}{dt}$ term arising from the time-varying inductance.

This missing term acts as a non-linear damping factor.

Additionally, it tests dimensional consistency by requiring students to recognize that the argument of a transcendental function like the natural logarithm must be dimensionless, forcing $\alpha$ to have units of inverse time.

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	7
Target Solve Time	160 sec
Access	Free