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Question #43780

Difficulty: 8/10 | 🟢 Free

The Question

An alpha particle of mass $m$ and charge $+2e$ is projected from infinity with an initial speed $v_0$ and an impact parameter $b$ towards a fixed heavy nucleus of charge $+Ze$ located at the origin. The entire region contains a static, non-uniform magnetic field given by $\vec{B} = B_0 \left( \frac{x}{a} \hat{i} - \frac{y}{a} \hat{j} \right)$ , where $B_0$ and $a$ are positive constants. Due to the combined electric and magnetic forces, the alpha particle follows a complex 3D trajectory. If $r_{min}$ is the distance of closest approach to the nucleus, what is the speed $v$ of the alpha particle at this point of closest approach? (Assume the nucleus remains stationary, $\epsilon_0$ is the vacuum permittivity, and ignore electromagnetic radiation).

Options

A.
$\sqrt{v_0^2 - \frac{Z e^2}{\pi \epsilon_0 m r_{min}}}$
B.
$\frac{v_0 b}{r_{min}}$
C.
$\sqrt{v_0^2 - \frac{Z e^2}{\pi \epsilon_0 m r_{min}} - \frac{e^2 B_0^2 r_{min}^2}{m^2}}$
D.
$0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Distance of Closest Approach|Moving Charge in Non-Uniform B

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between conserved and non-conserved quantities in a complex field.

The presence of a non-uniform magnetic field breaks the central force symmetry, meaning the magnetic force exerts a torque about the origin ( $\vec{\tau} = \vec{r} \times (q\vec{v} \times \vec{B}) \neq 0$ ).

Students falling into the trap will erroneously apply conservation of angular momentum to conclude $v = v_0 b / r_{min}$ , or they will attempt to set up coupled Lorentz-Newton differential equations, leading to massive mathematical bloat.

The conceptual key is recognizing that a static magnetic field, regardless of its complexity or non-uniformity, does zero work.

Thus, the Work-Energy Theorem bypasses the trajectory's complexity, reducing the problem to pure mechanical energy conservation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Distance of Closest Approach
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free