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Question #42984

Difficulty: 8/10 | 🟢 Free

The Question

A uniform conducting rod of length $L = 1.0 \text{ m}$ and resistance $R = 5.0 \ \Omega$ is pivoted at one end and rotates with a constant angular velocity $\omega = 10 \text{ rad/s}$ in a horizontal plane. A uniform vertical magnetic field $B = 2.0 \text{ T}$ exists throughout the region. The free end of the rod slides on a fixed, perfectly conducting circular ring. The pivot and the ring are connected by a stationary wire containing an ideal battery of EMF $V_0 = 5.0 \text{ V}$ . The battery is oriented such that it opposes the motional EMF generated by the rod. Let $P_B$ be the net rate of work done by the magnetic force on all charge carriers in the rod, $P_{emf}$ be the rate at which energy is converted from mechanical to electrical form by the motional EMF, and $P_{ext}$ be the mechanical power supplied by the external agent to maintain the constant angular velocity. Which of the following sets of values is correct?

Options

A.
$P_B = 0 \text{ W}, \ P_{emf} = 10 \text{ W}, \ P_{ext} = 10 \text{ W}$
B.
$P_B = 10 \text{ W}, \ P_{emf} = 10 \text{ W}, \ P_{ext} = 10 \text{ W}$
C.
$P_B = 0 \text{ W}, \ P_{emf} = 5.0 \text{ W}, \ P_{ext} = 5.0 \text{ W}$
D.
$P_B = 10 \text{ W}, \ P_{emf} = 5.0 \text{ W}, \ P_{ext} = 10 \text{ W}$

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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

Motional EMF in Rotating Rods|Conservative vs Non-conservative|Integration

Why This Question Is Tricky

Common Mistakes

This question targets a profound misconception regarding the work-energy theorem in electrodynamics.

Students frequently assume that the magnetic force does the work to drive the current, erroneously calculating $P_B = \mathcal{E}I = 10 \text{ W}$ .

In reality, the magnetic force strictly does zero net work as it is always perpendicular to the charge's velocity; it merely acts as a mediator, redirecting the mechanical work done by the external agent into electrical energy.

Furthermore, students may incorrectly calculate $P_{emf}$ or $P_{ext}$ using the net EMF ( $5.0 \text{ V}$ ) rather than the motional EMF ( $10 \text{ V}$ ), failing to recognize that the external agent must supply enough power to overcome both the Joule heating ( $I^2R = 5.0 \text{ W}$ ) and the power absorbed by the opposing battery ( $V_0 I = 5.0 \text{ W}$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Motional EMF in Rotating Rods
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free