Before You Begin

Question #43073

Difficulty: 8/10 | 🟢 Free

The Question

An ideal inductor of inductance $L$ is connected in series with a non-linear capacitor. The voltage $V$ across the capacitor depends on the charge $Q$ on its plates as $V = kQ^3$ , where $k$ is a positive constant. The circuit has no resistance. At $t=0$ , the current in the circuit is zero and the charge on the capacitor is $Q_0$ . The current $I$ and charge $Q$ form a closed trajectory in the phase space. Given the definite integral $\int_{0}^{1} \frac{dx}{\sqrt{1-x^4}} = c$ , which of the following expressions correctly represents the time period $T$ of the oscillations in the circuit? (Assume all quantities are in SI units)

Options

A.
$\frac{4\sqrt{2}c}{Q_0} \sqrt{\frac{L}{k}}$
B.
$\frac{4c}{Q_0} \sqrt{\frac{L}{k}}$
C.
$\frac{2\pi}{Q_0} \sqrt{\frac{L}{k}}$
D.
$\frac{2\pi}{Q_0} \sqrt{\frac{L}{3k}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental origin of oscillatory time periods, trapping students who blindly apply the standard simple harmonic oscillator formula $T = 2\pi\sqrt{LC}$ .

For a non-linear capacitor, the restoring 'force' (voltage) is not proportional to the displacement (charge), meaning the Hamiltonian is non-quadratic and the phase space trajectory is not an ellipse.

Students fall into two major traps: either they attempt to define an effective capacitance ( $C = Q/V$ or $C = dQ/dV$ ) and plug it into the SHM formula, or they incorrectly assume the energy stored in the capacitor is still $U = \frac{1}{2}QV = \frac{1}{2}kQ^4$ instead of integrating $\int V dQ$ .

The correct approach requires constructing the exact energy conservation equation and integrating the phase space trajectory over a quarter cycle.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	LC Oscillations Energy Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free