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Question #43635

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and positive charge $q$ is fired from infinity with an initial velocity $v_0$ towards a fixed, non-conducting solid sphere of radius $R$ having a uniform total positive charge $Q$ . The impact parameter of the particle is $b$ , where $b < R$ . Assume that $v_0$ is sufficiently large such that the deflection angle $\theta$ is very small, allowing the trajectory to be approximated as a straight line. Which of the following expressions correctly represents the small deflection angle $\theta$ ? (Here $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{qQ}{2\pi\epsilon_0 m v_0^2 b} \left[ 1 - \left(1 - \frac{b^2}{R^2}\right)^{3/2} \right]$
B.
$\frac{qQ}{2\pi\epsilon_0 m v_0^2 b}$
C.
$\frac{qQ}{2\pi\epsilon_0 m v_0^2 b} \left[ \frac{b^2}{R^2} \left(1 - \frac{b^2}{R^2}\right)^{1/2} \right]$
D.
$\frac{qQ}{2\pi\epsilon_0 m v_0^2 b} \left[ 1 - \left(1 - \frac{b^2}{R^2}\right)^{1/2} \right]$

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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|Non-conducting Charged Sphere|Conservation of Angular Momentum

Why This Question Is Tricky

Common Mistakes

Students incorrectly apply the standard Rutherford scattering formula (which assumes a pure Coulomb $1/r^2$ potential everywhere) to a trajectory that penetrates a non-conducting sphere.

They fail to recognize that inside the sphere, the electric field scales linearly with $r$ ( $E \propto r$ ), fundamentally altering the transverse impulse.

This trap tests the ability to abandon memorized formulas and perform a rigorous piecewise integration of the transverse force using the impulse-momentum theorem under the small-deflection approximation.

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