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Question #86215

Difficulty: 5/10 | 🟢 Free

The Question

A rectangular wire loop of initial length $a$ (along the $x$ -axis) and initial width $b$ (along the $y$ -axis) is placed in the $xy$ -plane with one corner fixed at the origin $(0,0)$ . The loop is made of a material with a coefficient of linear thermal expansion $\alpha$ (in $\text{K}^{-1}$ ). Starting at $t=0$ , the loop is uniformly heated such that its temperature increases at a constant rate $R = \frac{dT}{dt}$ (in $\text{K s}^{-1}$ ). A time-varying, non-uniform magnetic field is applied perpendicular to the loop, given by $\vec{B}(x,t) = B_0 \left(\frac{x}{a}\right) e^{-\lambda t} \hat{k}$ , where $B_0$ (in $\text{T}$ ) and $\lambda$ (in $\text{s}^{-1}$ ) are positive constants. Assuming the loop remains planar and rectangular, what is the magnitude of the induced electromotive force in the loop at the exact instant $t=0$ ?

Options

A.
$\frac{1}{2} B_0 a b |3\alpha R - \lambda|$
B.
$\frac{1}{2} B_0 a b |2\alpha R - \lambda|$
C.
$B_0 a b |3\alpha R - \lambda|$
D.
$\frac{1}{2} B_0 a b \lambda$

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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

Magnetic Flux Scalar Sign|Thermal Expansion of Area and Volume|Calculus Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

Students often misapply Faraday's law when both the magnetic field and the loop's geometry are time-dependent.

A common trap is assuming the flux changes only due to the standard area expansion ( $\frac{dA}{dt} \approx 2\alpha R A_0$ ) and the field's explicit time dependence, leading to the incorrect factor of $2\alpha R$ .

However, because the magnetic field has a spatial gradient ( $B \propto x$ ), the expanding boundary $x = L(t)$ sweeps into regions of progressively stronger magnetic field.

This requires integrating the flux with time-dependent limits (Leibniz Integral Rule), revealing that the flux scales with $(1+\alpha R t)^3$ , yielding a $3\alpha R$ expansion term.

Alternatively, calculating motional EMF requires evaluating $\oint (\vec{v} \times \vec{B}) \cdot d\vec{l}$ on all moving sides, where the non-uniform field heavily weights the contribution of the rightmost wire.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Magnetic Flux Scalar Sign
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free