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Question #42746

Difficulty: 8/10 | 🟢 Free

The Question

A stationary circular loop of radius $a$ is formed by joining two semicircular conducting wires of resistances $R_1$ and $R_2$ . The loop is placed in a uniform magnetic field $\vec{B}$ perpendicular to its plane. The magnitude of the magnetic field varies with time as $B(t) = \alpha t$ , where $\alpha$ is a positive constant. An ideal voltmeter is connected between the two junction points $P$ and $Q$ of the semicircles. If the connecting wires of the voltmeter lie exactly along the straight diameter joining $P$ and $Q$ , what is the magnitude of the reading of the voltmeter? (Assume all given parameters are in SI units, and the options represent the reading in Volts).

Options

A.
$\frac{\pi a^2 \alpha |R_1 - R_2|}{2(R_1 + R_2)}$
B.
$\frac{\pi a^2 \alpha |R_1 - R_2|}{R_1 + R_2}$
C.
$\frac{\pi a^2 \alpha R_1}{R_1 + R_2}$
D.
0

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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

Kirchhoff’s Loop Rule|Work against Conservative Force|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests a critical misconception regarding Kirchhoff's Voltage Law in the presence of non-conservative electric fields.

Students often incorrectly assume that the potential difference between two points is path-independent, treating it as a static potential $V$ , and thus erroneously conclude the voltmeter reads either $I R_1$ , $I R_2$ , or $0$ (due to symmetry).

In reality, a time-varying magnetic flux creates a non-conservative induced electric field.

The voltmeter measures the line integral of the total electric field $\int \vec{E} \cdot d\vec{l}$ strictly along its own physical leads.

To solve it, students must extend Kirchhoff's Loop Rule by applying Faraday's Law to the newly formed geometric sub-loop containing the voltmeter's path, reconciling the lumped-element approximation with Maxwell's equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Kirchhoff’s Loop Rule
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free