Before You Begin

Question #43875

Difficulty: 8/10 | 🟢 Free

The Question

A uniform circular conducting wire of radius $R$ , cross-sectional area $A$ , volumetric mass density $\rho$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ is tightly fitted inside a rigid, non-conducting cylindrical cavity of inner radius $R$ at temperature $T_0$ . The wire carries a steady current $I$ and is placed in a uniform magnetic field $B$ parallel to the axis of the cylinder. The entire system is then rotated about the cylinder's axis with a constant angular velocity $\omega$ , and the temperature of the wire is increased by $\Delta T$ . If the normal force exerted by the cavity wall on the wire is exactly zero, which of the following expressions gives the required magnitude of the magnetic field $B$ ? (Assume the magnetic force is directed radially inward).

Options

A.
$\frac{A}{I R} \left( Y \alpha \Delta T + \rho R^2 \omega^2 \right)$
B.
$\frac{A}{I R} \left( \rho R^2 \omega^2 - Y \alpha \Delta T \right)$
C.
$\frac{A}{I R} \left( Y \alpha \Delta T - \rho R^2 \omega^2 \right)$
D.
$\frac{A}{2 I R} \left( Y \alpha \Delta T + \rho R^2 \omega^2 \right)$

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

Force on Current-Carrying Loop|Thermal Stress in Constrained Rods|Thermal Expansion of Area

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify the direction of thermal stress in a constrained geometry and couple it with electromagnetic and fictitious forces.

When the temperature of the constrained loop increases, it undergoes compressive strain, resulting in a compressive hoop stress.

A common trap is mistakenly treating this as a tensile stress (which would provide an inward restoring force) or failing to recognize that a compressed ring exerts an outward radial force.

Consequently, students incorrectly subtract the thermal term from the centrifugal term (Option B).

The problem requires a rigorous free-body diagram of a differential element balancing thermal, centrifugal, and Lorentz forces.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Force on Current-Carrying Loop
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free