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Question #42992

Difficulty: 8/10 | 🟢 Free

The Question

An ideal capacitor of capacitance $C$ is connected in a closed series loop with a long ideal solenoid of self-inductance $L$ . A small conducting ring of resistance $R$ and negligible self-inductance is placed coaxially inside the solenoid. The mutual inductance between the solenoid and the ring is $M$ . The capacitor is initially charged, and at $t=0$ , the circuit is allowed to oscillate. Let $I$ be the current in the solenoid and $Q$ be the charge on the capacitor such that $I = \frac{dQ}{dt}$ . If $E_0 = \frac{Q^2}{2C} + \frac{1}{2} L I^2$ represents the sum of the energy stored in the capacitor and the self-magnetic energy of the solenoid, which of the following expressions correctly represents the rate of change of $E_0$ with respect to time?

Options

A.
$- \frac{M^2}{R} \left(\frac{dI}{dt}\right)^2$
B.
$\frac{M^2}{R} I \frac{d^2I}{dt^2}$
C.
$- \frac{M^2}{R} I \frac{d^2I}{dt^2}$
D.
$- \frac{M^2}{R} \left[ \left(\frac{dI}{dt}\right)^2 + I \frac{d^2I}{dt^2} \right]$

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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

Energy Stored in Inductor|Phase Space Trajectory|Eddy Currents Dissipation

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of energy conservation in coupled inductive circuits, specifically targeting the misconception that the rate of change of the standard LC energy ( $E_0$ ) equals the negative of the Joule heating power ( $-P_{dissipated}$ ).

Students often incorrectly choose Option A by blindly applying $dE_0/dt = -i_r^2 R$ .

However, $E_0$ is not the total energy; it neglects the mutual magnetic energy $U_M = M I i_r$ between the solenoid and the ring.

The true power balance is $d(E_0 + U_M)/dt = -P_{dissipated}$ .

By forcing students to derive the rate of change purely from KVL and Faraday's Law, the problem exposes the flaw in using incomplete energy expressions for non-conservative, coupled systems.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Energy Stored in Inductor
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free