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Question #43310

Difficulty: 8/10 | 🟢 Free

The Question

Two rigid, arbitrarily shaped conducting loops, $L_1$ and $L_2$ , are placed in a vacuum. $L_1$ is fixed and carries a constant current $I_1 = 5 \text{ A}$ . $L_2$ has a resistance $R = 2 \ \Omega$ and is initially at infinity. In Process I, $L_2$ is moved quasi-statically to a final position $P$ , requiring external work $W_I$ against magnetic forces. In Process II, $L_2$ is moved rapidly to the same position $P$ along the identical path, generating an entropy increase $\Delta S > 0$ due to Joule heating, and requiring external work $W_{II}$ against magnetic forces. Let $M_{12}$ be the magnetic flux through $L_1$ per unit current in $L_2$ , and $M_{21}$ be the magnetic flux through $L_2$ per unit current in $L_1$ , evaluated at position $P$ after Process II. Which of the following relationships is correct?

Options

A.
$M_{12} = M_{21}, \; W_{II} > W_{I}$
B.
$M_{12} \neq M_{21}, \; W_{II} > W_{I}$
C.
$M_{12} = M_{21}, \; W_{II} = W_{I}$
D.
$M_{12} \neq M_{21}, \; W_{II} = W_{I}$

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|Motional EMF for Arbitrary Shapes|Entropy and Reversibility|Thermodynamic Work

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between purely geometric electromagnetic properties and path-dependent thermodynamic quantities.

A common trap is assuming that the reciprocity theorem ( $M_{12} = M_{21}$ ) relies on thermodynamic reversibility, leading students to incorrectly conclude that time-asymmetric entropy generation (Joule heating) breaks this symmetry.

Furthermore, students often misapply the concept of conservative fields to magnetic forces, incorrectly assuming the work done moving a coil in a magnetic field is independent of the rate of motion ( $W_{II} = W_I$ ).

By recognizing that mutual inductance is governed by Neumann's geometric formula and that motional EMF induces velocity-dependent non-conservative drag forces, the correct relationship is established.

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