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Question #42899

Difficulty: 8/10 | 🟢 Free

The Question

A stationary circular ring of radius $a$ (in meters) consists of two ideal semicircular conducting wires connected at diametrically opposite nodes P and Q. A lumped resistor of resistance $R$ is inserted in one semicircle, and a lumped ideal inductor of self-inductance $L$ is inserted in the other semicircle. The ring is placed in a uniform magnetic field perpendicular to its plane. The magnetic field is present everywhere inside the ring and its magnitude varies with time as $B(t) = \alpha t$ , where $\alpha > 0$ is a constant in $\text{T s}^{-1}$ . At $t=0$ , the current in the circuit is zero. What is the magnitude of the scalar potential difference $|V_P - V_Q|$ between the nodes P and Q at time $t = \frac{L}{R} \ln 3$ ?

Options

A.
$\frac{\pi a^2 \alpha}{6}$
B.
$\frac{2\pi a^2 \alpha}{3}$
C.
$\frac{\pi a^2 \alpha}{3}$
D.
$\frac{\pi a^2 \alpha}{2}$

Check Your Answer

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

Superposition Principle|Induced Electric Fields|Transient Current in RL Circuit

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental distinction between conservative (Coulombic) and non-conservative (induced) electric fields in a circuit.

Students commonly misapply Kirchhoff's Voltage Law (KVL) by assuming the potential difference across the nodes is simply $iR$ or $L(di/dt)$ , treating the circuit as if it were driven by a localized battery.

However, when the induced EMF is distributed spatially due to a changing magnetic flux through the loop, the net electric field is the superposition of the Coulombic field and the induced field.

The scalar potential is only associated with the Coulombic field.

By forcing the calculation of the true scalar potential difference, the trap exposes the flaw in using standard lumped-circuit KVL in the presence of non-conservative fields.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Superposition Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free