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Question #42835

Difficulty: 8/10 | 🟢 Free

The Question

A non-conducting thin disc of radius $R$ has a surface charge density $\sigma(r) = \sigma_0 \left(1 - \frac{r}{R}\right)$ , where $r$ is the radial distance from the center and $\sigma_0$ is a positive constant. The disc rotates with a constant angular velocity $\omega$ about its central geometric axis. The disc is placed entirely inside a long ideal solenoid such that the axis of rotation of the disc is perpendicular to the central axis of the solenoid. The solenoid has $n$ turns per unit length and is wound using a wire of cross-sectional area $S$ . The wire material has a free electron number density $\eta$ . Due to Joule heating, the temperature $T$ of the solenoid wire increases, causing the drift velocity of the free electrons to vary as $v_d(T) = v_0 e^{-\alpha T}$ , where $v_0$ and $\alpha$ are positive constants. Assuming the external magnetic field is uniform over the disc, what is the magnitude of the magnetic torque $\tau$ acting on the disc at temperature $T$ ?

Options

A.
$\frac{\pi \mu_0 \sigma_0 \omega R^4 n \eta e S v_0 e^{-\alpha T}}{20}$
B.
$\frac{\pi \mu_0 \sigma_0 \omega R^4 n \eta e S v_0 e^{-\alpha T}}{12}$
C.
$\frac{\pi \mu_0 \sigma_0 \omega R^4 n \eta e S v_0 e^{-\alpha T}}{6}$
D.
$\frac{\pi \mu_0 \sigma_0 \omega R^4 n \eta e S v_0 e^{-\alpha T}}{10}$

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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

Continuous Charge Distribution|Torque on Current Loop|Drift Velocity Mechanism

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize electromagnetism concepts across different scales and distributions.

The primary trap lies in the calculation of the magnetic dipole moment for a non-uniform continuous charge distribution.

Students frequently misapply the formula $\mu = \frac{1}{2} Q \omega R^2$ (valid only for a ring) or $\mu = \frac{1}{4} Q \omega R^2$ (valid only for a uniform disc).

By forcing an integration over $\sigma(r) = \sigma_0(1 - r/R)$ , the problem exposes reliance on memorized discrete formulas (leading to incorrect denominators like 12 or 6).

Furthermore, the question couples this with the microscopic drift velocity model ( $I = n_e e A v_d$ ) to determine the external magnetic field, testing whether students can correctly link temperature-dependent microscopic electron motion to macroscopic magnetic torque.

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