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Question #42997

Difficulty: 8/10 | 🟢 Free

The Question

A long straight wire and a coplanar square loop of side $a$ lie on a smooth horizontal table. The wire carries a time-varying current $I(t) = I_0(1 + \alpha t)$ , where $I_0$ and $\alpha$ are positive constants. At time $t = 0$ , the near edge of the loop is parallel to the wire and at a distance $a$ from it. If the loop is being pulled away from the wire with a constant velocity $v$ perpendicular to the wire, which of the following expressions represents the magnitude of the total induced electromotive force (EMF) in the loop at $t = 0$ ?

Options

A.
$\frac{\mu_0 I_0}{2\pi} \left| a \alpha \ln 2 - \frac{v}{2} \right|$
B.
$\frac{\mu_0 I_0}{2\pi} \left( a \alpha \ln 2 + \frac{v}{2} \right)$
C.
$\frac{\mu_0 I_0 a \alpha}{2\pi} \ln 2$
D.
$\frac{\mu_0 I_0 v}{4\pi}$

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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

Mutual Inductance Reciprocity|Motional EMF|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply Faraday's Law in its complete form for a system where both the magnetic field source (current) and the geometry (position) are time-dependent.

A common trap is to blindly apply the static mutual inductance formula $\varepsilon = -M \frac{di}{dt}$ , completely ignoring the motional EMF generated by the loop translating through a spatially varying magnetic field.

Conversely, students might only calculate the motional EMF $\int (\vec{v} \times \vec{B}) \cdot d\vec{l}$ and forget the transformer EMF.

By requiring the full expansion of the flux derivative $\frac{d\Phi}{dt} = M(t)\frac{dI}{dt} + I(t)\frac{dM}{dt}$ , the problem forces the synthesis of both transformer and motional EMFs, while also testing the correct relative sign (Lenz's Law) between the two competing effects.

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