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Question #43004

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The Question

An ideal inductor of inductance $L$ is connected in a closed loop with a non-linear capacitor. The voltage $V$ across the capacitor depends on the charge $q$ stored in it according to the relation $V(q) = \frac{q}{C_0} + \beta q^3$ , where $C_0$ and $\beta$ are positive constants. The circuit is set into oscillation such that the maximum charge on the capacitor is $Q_0$ . Assuming the non-linear term is small ( $\beta Q_0^2 \ll \frac{1}{C_0}$ ), what is the approximate fundamental angular frequency $\omega$ of the oscillation?

Options

A.
$\frac{1}{\sqrt{L C_0}} \left( 1 + \frac{3}{8} \beta C_0 Q_0^2 \right)$
B.
$\frac{1}{\sqrt{L C_0}} \left( 1 + \frac{1}{4} \beta C_0 Q_0^2 \right)$
C.
$\frac{1}{\sqrt{L C_0}} \left( 1 + \frac{1}{2} \beta C_0 Q_0^2 \right)$
D.
$\frac{1}{\sqrt{L C_0}}$

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

LC Circuit Oscillations|Frequency Sensitivity of Z|Small Angle/Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze non-linear oscillators where standard Linear Time-Invariant (LTI) impedance logic fails.

The primary trap lies in misapplying energy conservation.

A student might equate maximum inductive energy to maximum capacitive energy: $\frac{1}{2} L I_{max}^2 = \int_0^{Q_0} V(q) dq = \frac{Q_0^2}{2 C_0} + \frac{\beta Q_0^4}{4}$ .

If they incorrectly assume the current remains perfectly harmonic ( $I_{max} = \omega Q_0$ ), they will obtain the erroneous coefficient of $1/4$ (Option B).

This naive energy method fails because the non-linear restoring force generates higher-order harmonics, meaning the kinetic energy is not purely characterized by the fundamental frequency.

The method of harmonic balance (or perturbation theory) correctly isolates the fundamental frequency shift, yielding the $3/8$ coefficient.

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