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Atoms and Nuclei question

2004 · Shift 0 · Q123
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Atoms and Nuclei question

2004 · Shift 0 · Q123

JEE MainPhysicsAtoms and NucleiMCQ+4 / −1
An α\alphaα-particle of energy 5MeV5MeV5MeV is scattered through 180∘{180^ \circ }180∘ by a fixed uranium nucleus. The distance of closest approach is of the order of
  1. A
    10−12 cm{10^{ - 12}}\,cm10−12cm
  2. B
    10−10 cm{10^{ - 10}}\,cm10−10cm
  3. C
    1A1A1A
  4. D
    10−15a cm{10^{ - 15a}}\,cm10−15acm
View written solutionFree

Correct answer: A

  1. Use energy conservation at closest approach

For head-on scattering of an α\alphaα-particle by a heavy fixed nucleus, the initial kinetic energy is converted into electrostatic potential energy at the distance of closest approach rrr:

E=14πε0Z1Z2e2rE = \frac{1}{4\pi\varepsilon_0}\frac{Z_1 Z_2 e^2}{r}E=4πε0​1​rZ1​Z2​e2​

Here,

  • for the α\alphaα-particle, Z1=2Z_1=2Z1​=2
  • for uranium, Z2=92Z_2=92Z2​=92
  • E=5 MeVE = 5\,\text{MeV}E=5MeV

So,

r=14πε0(2)(92)e25 MeVr = \frac{1}{4\pi\varepsilon_0}\frac{(2)(92)e^2}{5\,\text{MeV}}r=4πε0​1​5MeV(2)(92)e2​
  1. Use the standard nuclear physics constant

We use

e24πε0=1.44 MeV⋅fm\frac{e^2}{4\pi\varepsilon_0} = 1.44\,\text{MeV·fm}4πε0​e2​=1.44MeV⋅fm

Thus,

r=1.44×2×925 fmr = \frac{1.44 \times 2 \times 92}{5}\,\text{fm}r=51.44×2×92​fm r=1.44×1845 fmr = \frac{1.44 \times 184}{5}\,\text{fm}r=51.44×184​fm r=264.965 fm≈52.99 fmr = \frac{264.96}{5}\,\text{fm} \approx 52.99\,\text{fm}r=5264.96​fm≈52.99fm

So,

r≈53 fmr \approx 53\,\text{fm}r≈53fm
  1. Convert to cm

Since

1 fm=10−13 cm1\,\text{fm} = 10^{-13}\,\text{cm}1fm=10−13cm

therefore,

r≈53×10−13 cm=5.3×10−12 cmr \approx 53 \times 10^{-13}\,\text{cm} = 5.3\times 10^{-12}\,\text{cm}r≈53×10−13cm=5.3×10−12cm

This is of the order of

10−12 cm10^{-12}\,\text{cm}10−12cm
  1. Match with options
  • A: 10−12 cm10^{-12}\,\text{cm}10−12cm ✔
  • B: 10−10 cm10^{-10}\,\text{cm}10−10cm ✘
  • C: 1 A˚=10−8 cm1\,\text{\AA} = 10^{-8}\,\text{cm}1A˚=10−8cm ✘
  • D: appears malformed, but clearly not the correct order ✘

Hence, the correct option is A.

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