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Question #43319

Difficulty: 8/10 | 🟢 Free

The Question

An ideal capacitor of capacitance $C$ , initially charged to a charge $Q_0$ , is connected at $t=0$ across an ideal solenoid to form a closed LC circuit. The solenoid has $N$ turns, length $l$ , and cross-sectional area $A$ . It is completely filled with a non-linear magnetic core material. The magnetic field $B$ and the magnetic intensity $H$ inside the core are related by the equation $H = \alpha B + \beta B^3$ , where $\alpha$ and $\beta$ are positive constants. Assuming no resistance in the circuit and no magnetic flux leakage, what is the maximum magnitude of the magnetic field $B_{\text{max}}$ inside the solenoid during the subsequent oscillations?

Options

A.
$\left( \frac{\sqrt{\alpha^2 + \frac{2 \beta Q_0^2}{C A l}} - \alpha}{\beta} \right)^{1/2}$
B.
$\left( \frac{\sqrt{\alpha^2 + \frac{4 \beta Q_0^2}{C A l}} - \alpha}{2\beta} \right)^{1/2}$
C.
$\left( \frac{\sqrt{\alpha^2 + \frac{6 \beta Q_0^2}{C A l}} - \alpha}{3\beta} \right)^{1/2}$
D.
$\left( \frac{\sqrt{\alpha^2 + \frac{8 \beta Q_0^2}{C A l}} - \alpha}{4\beta} \right)^{1/2}$

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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 Circuit Oscillations|Energy Density Locality|Magnetic Field Line Behavior in Matter

Why This Question Is Tricky

Common Mistakes

This question tests the student's fundamental understanding of magnetic energy density in non-linear media.

A common trap is to blindly apply the linear formula $u = \frac{1}{2} B H$ (or $U = \frac{1}{2} L i^2$ ), which assumes a constant permeability $\mu$ .

In a non-linear core where $H(B)$ is a polynomial, the true energy density must be derived from first principles using the work done by the source: $u = \int H dB$ .

Students who incorrectly use $u = \frac{1}{2} B H$ will obtain an incorrect coefficient for the non-linear term, leading them to Option B.

Those who confuse energy with co-energy ( $u = \int B dH$ ) will select Option C.

The problem forces a return to the fundamental electrodynamic definition of work rather than relying on memorized formulas for linear oscillators.

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