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Question #43037

Difficulty: 8/10 | 🟢 Free

The Question

An AC generator consists of a flat coil of $N$ turns and area $A$ , rotating at a constant angular velocity $\omega$ in a uniform magnetic field of induction $B$ . The axis of rotation is perpendicular to the magnetic field. The internal resistance and self-inductance of the generator coil are negligible. The output of the generator is connected across a galvanometer having a resistance $R$ and self-inductance $L$ . To protect the galvanometer, a pure resistor of resistance $R_S$ is connected in parallel with it as a shunt. The value of $R_S$ is chosen such that the average electrical power dissipated in the shunt is exactly equal to the average electrical power dissipated in the galvanometer. Determine the average mechanical torque $\tau$ that an external agent must apply to the generator coil to maintain its constant angular velocity $\omega$ . (Assume no mechanical friction).

Options

A.
$\frac{N^2 A^2 B^2 \omega R}{R^2 + \omega^2 L^2}$
B.
$\frac{N^2 A^2 B^2 \omega (2R^2 + \omega^2 L^2)}{2R(R^2 + \omega^2 L^2)}$
C.
$\frac{N^2 A^2 B^2 \omega}{R}$
D.
$\frac{N^2 A^2 B^2 \omega R}{2(R^2 + \omega^2 L^2)}$

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

Shunt Resistance Conversion|AC Generator Back-EMF|Power Dissipation Ratio

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from DC circuit logic to AC impedance matching while applying the principle of energy conservation (Lenz's Law).

The primary trap lies in the calculation of the shunt resistance $R_S$ .

In a DC circuit, equal power dissipation in parallel branches implies equal resistances ( $R_S = R$ ).

However, in an AC circuit, the galvanometer's self-inductance creates a reactive back-EMF, increasing its impedance to $\sqrt{R^2 + \omega^2 L^2}$ .

Students who incorrectly apply DC logic will assume $R_S = R$ , leading to an incorrect total equivalent admittance and a wrong mechanical torque (Option B).

Furthermore, the problem elegantly couples AC circuit theory with rotational mechanics, requiring students to equate the average mechanical power ( $\tau \omega$ ) to the total active electrical power dissipated, reinforcing that reactive power does not contribute to the average mechanical work required to rotate the generator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Shunt Resistance Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free