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Question #42167

Difficulty: 8/10 | 🟢 Free

The Question

A stationary circular conducting loop of radius $a$ and resistance $R$ (with negligible self-inductance) is placed in a uniform magnetic field $\vec{B}$ perpendicular to its plane. The magnetic field is varied from $B_0$ to $-B_0$ in a fixed time interval $T$ through two different processes. In Process I, the magnetic field varies sinusoidally as $B(t) = B_0 \cos\left(\frac{\pi t}{T}\right)$ . In Process II, the magnetic field varies linearly as $B(t) = B_0 \left(1 - \frac{2t}{T}\right)$ . Let $W_I$ and $W_{II}$ be the total heat dissipated in the loop during Process I and Process II, respectively. Which of the following represents the ratio $\frac{W_I}{W_{II}}$ ?

Options

A.
$\frac{\pi^2}{8}$
B.
$\frac{\pi^2}{4}$
C.
$\frac{8}{\pi^2}$
D.
1

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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

State Function vs Path Function|Lenz’s Law Energy Conservation|Magnetic Field Line Behavior in Matter

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between state functions and path functions in electromagnetism.

A common misconception among students is treating the energy dissipated due to a changing magnetic field as a state function (analogous to $\Delta U$ in conservative fields), leading them to incorrectly assume that identical initial and final flux states result in the same energy dissipation (Option D).

However, the induced electric field is non-conservative.

The heat dissipated ( $W = \int i^2 R dt$ ) depends on the square of the rate of change of flux, making it highly path-dependent.

By comparing a sinusoidal and a linear variation, the problem mathematically proves that the kinematics of the flux change dictate the total energy lost, reinforcing the non-conservative nature of Faraday's Law.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	State Function vs Path Function
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free