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Question #42342

Difficulty: 8/10 | 🟢 Free

The Question

A fixed solid conducting sphere of radius $R$ carries a total charge $+Q$ . A smooth, rigid, insulating circular wire of radius $a$ ( $a > R$ ) is fixed in the vertical $x-y$ plane, concentric with the sphere. The $+y$ -axis points vertically upwards. A small bead of mass $m$ and charge $+q$ is threaded on the wire. A uniform magnetic field $\vec{B} = -B_0 \hat{k}$ exists throughout the space. The bead is given a negligible push in the $+x$ direction from the highest point of the wire. What is the magnitude of the normal force exerted by the wire on the bead when it reaches the lowest point? (Assume $k = \frac{1}{4\pi\epsilon_0}$ and acceleration due to gravity is $g$ in the $-y$ direction)

Options

A.
$5mg + \frac{k qQ}{a^2} + 2qB_0\sqrt{ga}$
B.
$5mg + \frac{k qQ}{a^2} - 2qB_0\sqrt{ga}$
C.
$5mg + 2qB_0\sqrt{ga}$
D.
$mg + \frac{k qQ}{a^2} + 2qB_0\sqrt{ga}$

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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

Vertical Circular Motion|Charged Conducting Sphere|Mechanical Energy Conservation|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

The question tests the synthesis of the Work-Energy theorem and radial force dynamics in a multi-field environment.

The primary psychological trap is the 'zero-work implies zero-effect' fallacy: students correctly deduce that the non-uniform electric field does no work along the equipotential circular path, but erroneously omit the electrostatic force from the radial force balance at the lowest point.

Additionally, it tests spatial reasoning with the Lorentz force right-hand rule; a sign error in the cross product due to misidentifying the velocity direction at the bottom of the loop leads to an incorrect subtraction of the magnetic force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Vertical Circular Motion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free