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Question #42995

Difficulty: 8/10 | 🟢 Free

The Question

Two LC circuits are coupled via mutual inductance $M$ . Circuit 1 consists of an inductor of inductance $L_1 = L$ and a capacitor of capacitance $C_1 = C$ . Circuit 2 consists of an inductor of inductance $L_2 = L$ and a capacitor of capacitance $C_2 = C/4$ . The mutual inductance between the inductors is $M = \frac{\sqrt{7}}{4} L$ . At time $t=0$ , the capacitor in Circuit 1 has an initial charge $Q_0$ , while the capacitor in Circuit 2 is uncharged. The initial currents in both circuits are zero. Let $E_0$ be the initial total energy of the system. What is the maximum energy stored in the capacitor of Circuit 2 during the subsequent oscillations? (Assume ideal components and no resistance)

Options

A.
$\frac{7}{27} E_0$
B.
$\frac{7}{16} E_0$
C.
$E_0$
D.
$\frac{4}{7} E_0$

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

Mutual Inductance Reciprocity|LC Oscillations|Energy Conservation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze non-resonant coupled LC circuits without relying on the standard 'beats' approximation.

A common trap is assuming that energy completely transfers between the circuits ( $E_{max} = E_0$ ) as it does in identical weakly coupled oscillators, or incorrectly applying the coupling coefficient $k$ to guess the energy transfer fraction (e.g., $k^2 E_0$ ).

By setting up non-identical circuits, the normal modes have different amplitudes and frequencies.

The problem forces students to derive the exact characteristic equation, find the eigenfrequencies, and recognize the commensurate frequency ratio ( $\omega_1 = 3\omega_2$ ) to analytically maximize the superposition of the modes, punishing those who memorize formulaic LC oscillation behaviors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Mutual Inductance Reciprocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free