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Question #43384

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ (in $\text{kg}$ ) and positive charge $q$ (in $\text{C}$ ) is projected from the origin $(0,0,0)$ with an initial velocity $\vec{v} = v_0 \hat{i}$ (in $\text{m s}^{-1}$ ) into a region containing a uniform magnetic field $\vec{B} = B_0 \hat{k}$ (in $\text{T}$ ). The region is filled with a resistive medium that exerts a drag force $\vec{F}_d = -b\vec{v}$ (in $\text{N}$ ) on the particle, where $b$ is a positive constant (in $\text{kg s}^{-1}$ ) and $\vec{v}$ is the instantaneous velocity. The particle eventually comes to rest at a point $P(x_f, y_f, 0)$ . What are the coordinates $(x_f, y_f)$ of point $P$ ?

Options

A.
$\left( \frac{m v_0 b}{q^2 B_0^2 + b^2} \text{ m}, -\frac{m v_0 q B_0}{q^2 B_0^2 + b^2} \text{ m} \right)$
B.
$\left( \frac{m v_0 q B_0}{q^2 B_0^2 + b^2} \text{ m}, \frac{m v_0 b}{q^2 B_0^2 + b^2} \text{ m} \right)$
C.
$\left( \frac{m v_0}{b} \text{ m}, 0 \text{ m} \right)$
D.
$\left( \frac{m v_0 b}{q^2 B_0^2 - b^2} \text{ m}, -\frac{m v_0 q B_0}{q^2 B_0^2 - b^2} \text{ m} \right)$

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

Trajectory of Charged Particle|The Velocity-Position Identity|Phase Constant (Epoch)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-conservative drag forces coupled with velocity-dependent magnetic forces.

A common trap is assuming that because the magnetic force does no work, the stopping distance is simply $x_f = mv_0/b$ (Option C), ignoring that the magnetic field continuously redirects the velocity vector, causing the drag force to act along both the x and y axes over time.

Another trap is attempting to find the exact trajectory $y(x)$ by solving the coupled second-order differential equations, which is mathematically tedious and prone to errors under time pressure.

The elegant and intended JEE Advanced method requires recognizing that integrating the equations of motion directly with respect to time transforms the velocity derivatives into finite boundary conditions and the velocities into final coordinates, bypassing the explicit time-domain solution of the dying spiral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Trajectory of Charged Particle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free