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Question #43368

Difficulty: 8/10 | 🟢 Free

The Question

A small rectangular coil of area $A$ and $N$ turns is mechanically rotated with a constant angular velocity $\omega$ in a uniform magnetic field $B$ . The terminals of the coil are connected to a stationary external series RLC circuit consisting of a resistor $R = 30\sqrt{2} \text{ }\Omega$ , an inductor $L = 50 \text{ mH}$ , and a capacitor $C = 20 \text{ }\mu\text{F}$ . The internal resistance and inductance of the rotating coil are negligible. The mechanical agent adjusting the coil's rotation can set $\omega$ to any desired constant value. At what angular velocity $\omega$ is the average electrical power dissipated in the resistor maximized?

Options

A.
$1000 \text{ rad/s}$
B.
$1250 \text{ rad/s}$
C.
$750 \text{ rad/s}$
D.
$1500 \text{ rad/s}$

Check Your Answer

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

Maximum Power Transfer|Damped Oscillation Equation|Quality Factor (Q-factor)

Why This Question Is Tricky

Common Mistakes

This question tests a critical misconception regarding Maximum Power Transfer and resonance in AC circuits.

Students instinctively equate maximum power dissipation with the standard resonance condition ( $X_L = X_C$ , or $\omega = 1/\sqrt{LC}$ ), which minimizes the circuit's impedance.

However, they fail to account for the fact that the source is an electromagnetic generator whose induced EMF amplitude ( $V_0 = NBA\omega$ ) is directly proportional to the driving frequency $\omega$ .

Consequently, the power function $P(\omega)$ has an $\omega^2$ term in the numerator.

Maximizing this function requires applying calculus to find the true peak, which shifts the optimal frequency away from the natural undamped resonance.

This traps students who rely on memorized formulas rather than deriving the physical behavior from first principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Maximum Power Transfer
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free