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Question #42423

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q$ is projected with an initial speed $v_0$ in the $xy$ -plane. A time-varying uniform magnetic field $\vec{B} = B_0(1 + \alpha t)\hat{k}$ exists in the region, where $B_0$ and $\alpha$ are positive constants. The particle experiences a velocity-dependent drag force $\vec{F}_d = -k \vec{v}$ , where $k$ is a positive constant. Assuming the effects of the induced electric field are negligible, what is the magnitude of the total angle rotated by the velocity vector of the particle from $t = 0$ until the moment its speed reduces to $v_0/e$ ?

Options

A.
$\frac{q B_0}{k} \left( 1 + \frac{\alpha m}{2 k} \right)$
B.
$\frac{q B_0}{k} \left( 1 + \frac{\alpha m}{k} \right)$
C.
$\frac{q B_0}{2k} \left( 1 + \frac{\alpha m}{k} \right)$
D.
$\frac{q B_0}{k} \left( 1 - \frac{\alpha m}{2 k} \right)$

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

Free Body Diagram Isolation|Cyclotron Frequency Invariance|Change in Velocity (Δv)

Why This Question Is Tricky

Common Mistakes

This question tests the breakdown of the adiabatic invariant in a time-varying regime coupled with non-conservative forces.

Students often incorrectly assume that the cyclotron frequency $\omega = qB/m$ remains constant, or they attempt to use the standard circular motion formula $R = mv/qB$ without accounting for the simultaneous time-dependence of the magnetic field and the velocity decay due to drag.

By introducing a complex velocity vector, the non-homogeneous coupled differential equations elegantly decouple.

This reveals the trap: the angular rotation depends strictly on the integral of the time-varying magnetic field, while the speed decay is entirely and independently governed by the drag force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Free Body Diagram Isolation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free