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Question #43419

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q > 0$ is injected at time $t=0$ from the origin $(0,0,0)$ with an initial velocity $\vec{v}_0 = v_0 \hat{i}$ into a region of space. The region contains a non-uniform magnetic field $\vec{B} = B_0(1 + \alpha x)\hat{k}$ and the particle experiences a velocity-dependent drag force $\vec{F}_{drag} = -b\vec{v}$ , where $B_0, \alpha,$ and $b$ are positive constants. What is the initial rate of change of the normal component of the particle's acceleration, $\left. \frac{da_n}{dt} \right|_{t=0}$ ?

Options

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

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

Moving Charge in Non-Uniform B|Current Density Vector|Average vs. Instantaneous Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple tangential and normal components of acceleration in a non-conservative, spatially varying vector field.

The primary trap is the assumption of an adiabatic invariant (constant speed or constant magnetic moment) often misapplied in magnetic field problems.

Students frequently ignore the drag force's effect on the instantaneous speed (leading to Option B) or fail to apply the convective derivative (chain rule) for the spatially varying magnetic field (leading to Option C).

By requiring the time derivative of the normal acceleration, the problem forces the synthesis of the power equation (to find speed decay) and spatial gradients (to find field variation) at the exact moment of injection.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Moving Charge in Non-Uniform B
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free