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Question #42911

Difficulty: 8/10 | 🟢 Free

The Question

A non-conducting disc of radius $R$ has a non-uniform areal mass density $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ and a non-uniform surface charge density $\sigma(r) = \sigma_0 \left(\frac{r}{R}\right)$ , where $r$ is the radial distance from the center. The total mass of the disc is $M$ and its total charge is $Q$ . The disc is initially at rest and is free to rotate about its fixed central axis. A uniform magnetic field, perpendicular to the plane of the disc, is switched on and increases from $0$ to $B_0$ . Neglecting any friction or radiation losses, what is the final angular velocity $\omega$ of the disc?

Options

A.
$\frac{Q B_0}{M}$
B.
$\frac{Q B_0}{2 M}$
C.
$\frac{3 Q B_0}{5 M}$
D.
$\frac{5 Q B_0}{6 M}$

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

Continuous Charge Distribution|Angular Momentum conservation|Magnetic Dipole Moment of Revolving Electron|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Faraday's Law and rotational mechanics against the common heuristic of the gyromagnetic ratio.

Students frequently fall into the trap of assuming the gyromagnetic ratio $\gamma = \frac{Q}{2M}$ applies universally, leading them to directly apply the uniform disc formula $\omega = \frac{Q B_0}{2M}$ .

By introducing decoupled, non-uniform mass and charge distributions, the problem invalidates this shortcut.

The 'lethal' trap lies in failing to independently integrate the induced torque (which depends heavily on the outer radii due to the $r^4$ dependence) and the moment of inertia.

If a student assumes uniform distributions or blindly applies Larmor precession concepts, they will confidently select the incorrect option.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Continuous Charge Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free