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Question #42415

Difficulty: 8/10 | 🟢 Free

The Question

A small bead of mass $m$ and positive charge $q$ is constrained to move with a constant speed $v_0$ along a semicircular wire of radius $R$ in the $xy$ -plane. The wire connects the points $(R, 0, 0)$ and $(-R, 0, 0)$ passing through $(0, R, 0)$ . The region contains a non-uniform magnetic field $\vec{B} = B_0 y \hat{k}$ , where $B_0$ is a positive constant. What is the time-averaged magnetic force exerted on the bead as it travels from $(R, 0, 0)$ to $(-R, 0, 0)$ ?

Options

A.
$\frac{1}{2} q v_0 B_0 R \hat{j}$
B.
$\frac{2}{\pi} q v_0 B_0 R \hat{j}$
C.
$\frac{4}{\pi^2} q v_0 B_0 R \hat{j}$
D.
q v_0 B_0 R \hat{j}

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

Instantaneous vs Average|Moving Charge in Non-Uniform B|Vector Calculus: Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between the average of a product and the product of averages in a non-uniform field.

Students often incorrectly calculate the average magnetic force by multiplying the average velocity with the average magnetic field (yielding Option C) or by multiplying the constant speed with the spatial average of the B-field (yielding Option B).

The trap highlights that the Lorentz force $\vec{F} = q\vec{v} \times \vec{B}$ must be integrated instantaneously along the curved trajectory because the velocity vector and the spatially varying magnetic field are continuously coupled.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Instantaneous vs Average
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free