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Question #43074

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit consists of a capacitor of capacitance $C = 6 \text{ mF}$ and a non-linear inductor. The capacitor is initially charged to a voltage $V_0 = 6 \text{ V}$ , and the initial current in the circuit is zero. Due to magnetic core saturation, the total magnetic flux linkage $\Phi$ of the inductor varies non-linearly with the current $I$ as $\Phi(I) = \alpha I + \beta I^3$ , where $\alpha = 30 \times 10^{-3} \text{ H}$ and $\beta = 4 \times 10^{-3} \text{ H/A}^2$ . Assuming no resistance in the circuit, what is the maximum current $I_{max}$ that will flow through the inductor?

Options

A.
$2 \text{ A}$
B.
$\frac{3}{\sqrt{2}} \text{ A}$
C.
$\frac{6}{\sqrt{5}} \text{ A}$
D.
$\sqrt{6} \text{ A}$

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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 Oscillations Energy Cycle|Self-Inductance of Solenoid|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of magnetic potential energy in non-linear magnetic circuits.

A common misconception among students is blindly applying the standard formula $U_B = \frac{1}{2}LI^2$ , either by defining an effective static inductance $L_{eff} = \frac{\Phi}{I} = \alpha + \beta I^2$ (leading to Option B) or by completely ignoring the non-linear term (leading to Option C).

Both approaches fail because they ignore the continuous accumulation of energy as the differential inductance changes.

The correct physical principle requires integrating the electrical work done against the induced emf, $U_B = \int I d\Phi$ .

By introducing a current-dependent flux linkage, the problem forces students to abandon rote formulas and derive the energy from Faraday's Law and power integration, exposing the mathematical elitism of non-linear differential equations in LC circuits.

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