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Question #43773

Difficulty: 8/10 | 🟢 Free

The Question

An alpha particle of mass $m$ and charge $q$ is projected from infinity with an initial kinetic energy $K_0 = \frac{qQ}{2\pi\epsilon_0 R}$ towards a fixed, uniformly charged non-conducting solid sphere of radius $R$ and total charge $Q$ ( $q, Q > 0$ ). The impact parameter of the alpha particle is $b = \frac{R}{2}$ . Assuming the sphere remains fixed and unperturbed, what is the distance of closest approach, $r_{\text{min}}$ , of the alpha particle to the center of the sphere?

Options

A.
$R \sqrt{\frac{\sqrt{5}-1}{2}}$
B.
$R \left(\frac{\sqrt{5}+1}{4}\right)$
C.
$\frac{R}{2}$
D.
$R \sqrt{1 - \frac{1}{\sqrt{2}}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Impact Parameter Geometry|Angular Momentum conservation|Coulomb's Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from standard Rutherford scattering (point charge potential) to scattering by an extended charge distribution.

The primary trap lies in blindly applying the $V(r) \propto 1/r$ Coulomb potential for the entire trajectory.

Students who fail to recognize that the alpha particle penetrates the sphere will incorrectly use the exterior potential, leading to a spurious quadratic equation and the trap answer $R(\sqrt{5}+1)/4$ .

Furthermore, it rigorously tests the application of the effective potential concept $U_{\text{eff}}(r) = V(r) + L^2/(2mr^2)$ in a piecewise central force field, requiring the synthesis of angular momentum conservation with a harmonic-oscillator-like internal restoring force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Impact Parameter Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free