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Question #43311

Difficulty: 8/10 | 🟢 Free

The Question

A rectangular coil of $N = 100$ turns and area $A = 0.1 \text{ m}^2$ is rotated at a constant angular velocity $\omega = 10 \text{ rad s}^{-1}$ in a uniform magnetic field $B = 2 \text{ T}$ . The axis of rotation is perpendicular to the magnetic field. The terminals of the coil are connected via slip rings to a stationary external circuit consisting of a resistor $R = 3 \text{ } \Omega$ and an inductor $L = 0.4 \text{ H}$ in series. The resistance and self-inductance of the coil itself are negligible. An external agent applies a mechanical torque $\tau_{ext}$ to maintain the constant angular velocity of the coil. What is the maximum instantaneous mechanical torque $\tau_{max}$ that the external agent must apply during the steady-state rotation?

Options

A.
$640 \text{ N m}$
B.
$800 \text{ N m}$
C.
$240 \text{ N m}$
D.
$400 \text{ N m}$

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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

AC Generator Back-EMF|Work-Energy Theorem for Rotation|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that the maximum mechanical torque occurs simultaneously with the maximum induced EMF, calculating $\tau_{max} = N I_0 A B = 800 \text{ N m}$ .

This ignores the phase shift $\phi$ introduced by the inductor, which delays the current relative to the EMF.

Another common trap is equating the average mechanical power to the average electrical power to find an 'average' torque ( $240 \text{ N m}$ ), failing to recognize that the instantaneous torque fluctuates at $2\omega$ and requires a peak value significantly higher than the average to maintain constant angular velocity.

The correct approach requires expressing the instantaneous torque as a function of time and maximizing the resulting trigonometric expression.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	AC Generator Back-EMF
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free