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Question #42678

Difficulty: 8/10 | 🟢 Free

The Question

A uniform conducting rod of mass $m$ and length $L$ is pivoted at one of its ends and rotates in a horizontal plane with an initial angular velocity $\omega_0$ . A uniform vertical magnetic field $B$ exists throughout the region. The free end of the rod slides without friction on a fixed, horizontal, circular conducting ring of radius $L$ centered at the pivot. An uncharged capacitor of capacitance $C$ is connected between the pivot and the circular ring. The total resistance of the circuit (rod, ring, and wires) is $R$ . What is the steady-state charge $q_f$ on the capacitor after a long time?

Options

A.
$\frac{2 m C B L^2 \omega_0}{4 m + 3 C B^2 L^2}$
B.
$\frac{1}{2} C B L^2 \omega_0$
C.
$\frac{m C B L^2 \omega_0}{\sqrt{4 m^2 + 3 m C B^2 L^2}}$
D.
$\frac{m C B L^2 \omega_0}{m + 3 C B^2 L^2}$

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

Capacitance Independence|Motional EMF in Rotating Rods|Non-Uniform Acceleration Constraint

Why This Question Is Tricky

Common Mistakes

This question tests the 'Variable Acceleration Constraint' in electromagnetic induction.

The primary trap is assuming the rod's angular velocity remains constant, leading students to incorrectly calculate the charge using the initial static EMF (Option B).

A secondary, more subtle trap is applying the conservation of mechanical and electrical energy (Option C), where students equate initial kinetic energy to final kinetic energy plus the energy stored in the capacitor.

This is fundamentally flawed because charging a capacitor through any resistance (even internal) inherently dissipates energy as Joule heating.

The correct physical framework requires recognizing that the magnetic damping torque is directly proportional to the charging current, allowing the use of the Angular Impulse-Momentum theorem to bypass the complex time-dependent differential equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Capacitance Independence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free