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Question #43268

Difficulty: 8/10 | 🟢 Free

The Question

Three identical perfectly conducting circular loops, each having a self-inductance $L$ , are fixed in space with their planes parallel and their centers forming an equilateral triangle. The mutual inductance between any pair of loops is $M$ . Loop 1 contains a capacitor of capacitance $C$ initially charged to a voltage $V_0$ , and an open switch. Loops 2 and 3 are continuous, short-circuited loops with no active components. At $t=0$ , the switch in Loop 1 is closed. Assuming the ohmic resistance of all loops is strictly zero, what is the maximum magnitude of the induced current in Loop 2 during the subsequent LC oscillations?

Options

A.
$V_0 M \sqrt{\frac{C}{(L+M)(L-M)(L+2M)}}$
B.
$V_0 M \sqrt{\frac{C}{L(L^2 - 2M^2)}}$
C.
$V_0 M \sqrt{\frac{C}{L(L+M)^2}}$
D.
$V_0 M \sqrt{\frac{C}{L^3}}$

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

Superposition of Magnetic Fields|Kirchhoff’s Loop Rule|Mutual Inductance Reciprocity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the superposition principle of magnetic flux and Kirchhoff's loop rule (via Faraday's law for zero-resistance loops) to a multi-loop coupled transient system.

The primary trap lies in treating the mutual inductance interactions as isolated pairs.

Students often correctly identify the coupling between Loop 1 and Loop 2 but fail to account for the mutual coupling between the two induced currents in Loops 2 and 3 (ignoring the $M i_3$ term in Loop 2's flux equation).

This omission leads to the incorrect relation $i_2 = -M/L i_1$ and the trap answer B.

Another common mistake is ignoring the back-EMF exerted by the induced currents on the primary loop, treating Loop 1 as an uncoupled LC circuit ( $L_{eq} = L$ ), which leads to trap answers C or D.

The elegant solution requires recognizing symmetry to prevent mathematical bloat when solving the coupled differential equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Superposition of Magnetic Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free