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Question #43389

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q$ is projected with an initial speed $v_0$ in a vertical plane (assume gravity is negligible). A uniform horizontal magnetic field of magnitude $B$ exists perpendicular to this plane. The region contains a viscous medium that exerts a drag force $\vec{F}_d = -k\vec{v}$ on the particle, where $k$ is a positive constant and $\vec{v}$ is the instantaneous velocity. Which of the following expressions correctly gives the total distance $S$ traveled by the particle before coming to rest, and the radius of curvature $R$ of its path when its velocity vector has rotated by an angle $\theta$ from its initial direction?

Options

A.
$S = \frac{mv_0}{k}$ , $R = \frac{mv_0}{qB} e^{-\frac{k\theta}{qB}}$
B.
$S = \frac{mv_0}{qB}$ , $R = \frac{mv_0}{k} e^{-\frac{qB\theta}{k}}$
C.
$S = \frac{mv_0}{k}$ , $R = \frac{mv_0}{qB} e^{-\frac{qB\theta}{k}}$
D.
$S = \frac{mv_0}{k} e^{-\frac{k\theta}{qB}}$ , $R = \frac{mv_0}{qB}$

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

Cyclotron Frequency Invariance|Vertical Circular Motion|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple orthogonal forces in a non-conservative system.

A common trap is assuming that because the speed decays, the angular velocity or orbital period must also change.

Students often misapply the relation $T = 2\pi R/v$ , thinking that a decreasing radius implies a changing period, failing to realize that $R$ and $v$ decay proportionally.

By separating the tangential equation (which yields $v(t) = v_0 e^{-(k/m)t}$ ) from the normal equation (which yields a constant $\omega = qB/m$ ), students can map the time domain to the angular domain to find the exact geometric properties of the logarithmic spiral trajectory.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Cyclotron Frequency Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free