Question 81 markMCQSection A

The 'distance of closest approach' of an alpha-particle is 'd' when it moves with a velocity v head-on towards the target nucleus. If the velocity of alpha particle is halved, the new 'distance of closest approach' will be −

Atoms
Distance of Closest Approach (Rutherford Scattering)

Options

(A)d/2d/2
(B)2d2d
(C)d/4d/4
(D)4d4d

Official answer

Correct option: (D) 4d4d


At closest approach, the alpha-particle's kinetic energy is fully converted to electrostatic potential energy:


  • 12mv2=14πε0(Ze⋅2e)/d\frac{1}{2}mv^2 = \frac{1}{4\pi\varepsilon_0}(Ze\cdot 2e)/d, so d∝1/v2d \propto 1/v^2.
  • Halving v (→ v/2v/2) multiplies d by 1/(1/2)21/(1/2)^2 = 4, giving the new distance = 4d4d.
Answer keywords:distance of closest approachalpha particled proportional to 1/v^2kinetic energyelectrostatic potential energyRutherford scatteringhead-on collision4d

Marking scheme

  1. 1.1 mark: correct option (D) 4d4d.
  2. 2.Key reasoning: d∝1/v2d \propto 1/v^2 from 12mv2=kq1q2d\frac{1}{2}mv^2 = \frac{kq_1q_2}{d}, so halving v gives 4d4d.

Hint

Equate kinetic energy 12mv2\frac{1}{2}mv^2 to Coulomb potential energy — notice d depends on v2v^2, not v.

Quick oral answer

Since the alpha-particle's kinetic energy fully converts to electrostatic potential energy at the closest point, the distance is inversely proportional to the square of the speed, so halving the velocity makes the distance four times larger.

Common mistakes

  1. 1.Using d∝1/vd \propto 1/v (linear) instead of d∝1/v2d \propto 1/v^2, giving 2d2d rather than the correct 4d4d.
  2. 2.Assuming the distance decreases when speed decreases — a slower particle is repelled sooner, so it stops farther away.
  3. 3.Forgetting that at closest approach the velocity is momentarily zero, so all KE=PEKE = PE.

Analysis and explanation

This tests the energy-conservation derivation of the distance of closest approach in Rutherford's alpha-scattering experiment.


Concept: For a head-on collision the alpha-particle momentarily stops when all its kinetic energy becomes potential energy:

  • 12mv2=14πε0⋅2Ze2d  ⟹  d=14πε0⋅4Ze2mv2\frac{1}{2}mv^2 = \frac{1}{4\pi\varepsilon_0}\cdot\frac{2Ze^2}{d} \implies d = \frac{1}{4\pi\varepsilon_0}\cdot\frac{4Ze^2}{mv^2}.
  • Thus d is inversely proportional to v2v^2 (for fixed nucleus and charge).

Why (D) is correct: Replacing v by v/2v/2 gives d′∝1(v/2)2=4v2d' \propto \frac{1}{(v/2)^2} = \frac{4}{v^2}, i.e. d′=4dd' = 4d.


Why the others are wrong:

  • (A) d/2d/2 and (C) d/4d/4 wrongly assume d increases with speed or scales linearly with v.
  • (B) 2d2d comes from taking d∝1/vd \propto 1/v (linear) instead of the correct 1/v21/v^2 dependence.

Exam trap: Students often forget the square in the kinetic-energy term, giving a factor of 2 instead of 4.

Interesting facts

Rutherford's 1911 analysis of the closest approach (a few ×10−14\times 10^{-14} m) first revealed the tiny, dense atomic nucleus, overturning Thomson's plum-pudding model.

The distance of closest approach gives an upper estimate of nuclear size; for typical alpha energies it is about 10410^4 times smaller than the atom itself.

Question details

Q8 of 39 questions

Board
CBSE
Class
Class 12
Subject
Physics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Atoms
Topic
Distance of Closest Approach (Rutherford Scattering)
Bloom's level
apply
Syllabus topic
Unit 8: Atoms — Alpha-particle Scattering and Distance of Closest Approach
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why is the distance of closest approach inversely proportional to v2v^2?

At the closest point the alpha-particle stops, so its entire kinetic energy 12mv2\frac{1}{2}mv^2 equals the Coulomb potential energy kq1q2d\frac{kq_1q_2}{d}. Solving for d gives d=2kq1q2mv2d = \frac{2kq_1q_2}{mv^2}, which is inversely proportional to v².

Does the mass of the alpha-particle affect the distance of closest approach?

Yes. From d=2kq1q2mv2d = \frac{2kq_1q_2}{mv^2}, d is also inversely proportional to the mass m for a given speed; a heavier particle at the same speed approaches closer.