Before You Begin

Question #43879

Difficulty: 8/10 | 🟢 Free

The Question

A uniform circular conducting ring of radius $r$ , resistance $R$ , and self-inductance $L$ is placed in a uniform magnetic field $B$ directed along its central axis. Initially, the ring is at rest. At $t=0$ , the ring is subjected to a constant angular acceleration $\alpha$ about its central axis. The free charge carriers in the conductor are electrons of mass $m_e$ and charge $-e$ . Assume the kinetic inductance of the electrons is negligible compared to $L$ . Which of the following expressions correctly gives the magnitude of the induced current $I(t)$ in the ring at time $t$ ?

Options

A.
$\frac{\pi B r^2 \alpha}{R} \left[ t - \frac{L}{R}\left(1 - e^{-\frac{R}{L}t}\right) \right]$
B.
$\frac{2\pi m_e r^2 \alpha}{e R} \left( 1 - e^{-\frac{R}{L}t} \right)$
C.
$\frac{2\pi m_e r^2 \alpha}{e R} \left( 1 - e^{-\frac{R}{L}t} \right) + \frac{\pi B r^2 \alpha}{R} \left[ t - \frac{L}{R}\left(1 - e^{-\frac{R}{L}t}\right) \right]$
D.
$\frac{m_e r^2 \alpha}{e R} \left( 1 - e^{-\frac{R}{L}t} \right)$

Check Your Answer

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

Resistor Phase Invariance|Transient Current in RL Circuit|Frame of Reference Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between inertial EMF and motional EMF in a non-inertial frame.

The primary trap is the presence of the axial magnetic field $B$ .

Students commonly misapply Faraday's Law or the motional EMF formula ( $\vec{v} \times \vec{B}$ ), incorrectly assuming that rotation within a uniform magnetic field generates a tangential EMF (as it would in a Faraday disk).

However, for a closed ring, $\vec{v} \times \vec{B}$ is purely radial and yields zero net EMF around the loop.

The only driving force for the current is the fictitious inertial force acting on the charge carriers due to the angular acceleration, which mimics an effective tangential electric field $E = m_e r \alpha / e$ .

Students who fail to recognize this will erroneously include a time-varying magnetic EMF term in their RL circuit differential equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Resistor Phase Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free