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Question #43005

Difficulty: 8/10 | 🟢 Free

The Question

Two identical LC circuits, each consisting of a capacitor of capacitance $C$ and an inductor of inductance $L$ , are placed near each other. The total energy of the system is given by $E = \frac{q_1^2}{2C} + \frac{q_2^2}{2C} + \frac{1}{2} L i_1^2 + \frac{1}{2} L i_2^2 + M i_1 i_2$ , where $q_1, q_2$ are the instantaneous charges on the capacitors, $i_1 = \frac{dq_1}{dt}, i_2 = \frac{dq_2}{dt}$ are the currents, and $M = 0.8 L$ is the mutual inductance between the inductors. At $t=0$ , the currents in both circuits are zero, the charge on the first capacitor is $q_1(0) = Q_0$ , and the charge on the second capacitor is $q_2(0) = Q_0/2$ . What is the maximum charge that will appear on the second capacitor during the subsequent oscillations?

Options

A.
$\frac{Q_0}{\sqrt{2}}$
B.
$\frac{\sqrt{5}}{2} Q_0$
C.
Q_0
D.
$\frac{Q_0}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

LC Circuit Oscillations|Phase Constant (Epoch)|Superposition of SHMs

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of energy conservation and superposition in coupled oscillators.

A common trap is assuming that in a coupled LC system, 100% of the initial energy can eventually transfer to one capacitor, leading students to equate the total initial energy $\frac{5Q_0^2}{8C}$ to $\frac{q_{max}^2}{2C}$ , yielding the incorrect option $\frac{\sqrt{5}}{2}Q_0$ .

Another trap is directly adding the normal mode amplitudes ( $0.75 Q_0 + 0.25 Q_0 = Q_0$ ) without accounting for the phase constant (epoch) mismatch.

Because the initial conditions excite both normal modes with specific amplitudes, the modes interfere.

The maximum charge is dictated by the time-dependent superposition of these modes, which never perfectly align to localize all energy into the second capacitor.

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