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Question #42419

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly conducting rod of length $L$ rotates with a constant angular velocity $\omega$ in a horizontal plane about a fixed vertical axis passing through one of its ends. A uniform vertical magnetic field $B$ exists in the region, directed downwards. The rotation of the rod is counterclockwise when viewed from above. If $m_e$ and $-e$ are the mass and charge of an electron respectively, what is the exact potential difference $V_{\text{center}} - V_{\text{end}}$ between the pivoted end (center) and the free end of the rod in the steady state?

Options

A.
$\frac{1}{2} B \omega L^2 + \frac{m_e \omega^2 L^2}{2e}$
B.
$\frac{1}{2} B \omega L^2 - \frac{m_e \omega^2 L^2}{2e}$
C.
$\frac{1}{2} B \omega L^2$
D.
$\frac{1}{2} B \omega L^2 + \frac{m_e \omega^2 L^2}{e}$

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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

Frame of Reference Inertial vs Non-Inertial|Motional EMF in Rotating Rods|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously apply Newton's laws and electromagnetic induction in a non-inertial reference frame.

The standard JEE trap is to blindly apply the motional EMF formula $\mathcal{E} = \frac{1}{2}B\omega L^2$ while ignoring the inertial stresses on the charge carriers.

By placing the observer in the rotating frame, a centrifugal pseudo-force $m_e \omega^2 r$ acts on the electrons in addition to the magnetic Lorentz force.

Students must recognize that the macroscopic potential difference is the integral of the induced electric field, which must balance both the magnetic and centrifugal forces at equilibrium.

Failing to account for the electron's mass leads to the standard, yet incomplete, approximation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Frame of Reference Inertial vs Non-Inertial
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free