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BITSAT 2023Paper 2
Exam
BITSAT
Year
2023
Paper
Paper 2
Subject
Physics
Chapter
Dual Nature of Matter

BITSAT 2023 · Paper 2 · Physics · Dual Nature of Matter

A proton and an α -particle are accelerated from rest by 2 V and 4...

A proton and an α\alpha-particle are accelerated from rest by 2V2 \mathrm{ V} and 4V4 \mathrm{ V} potentials, respectively. The ratio of their de-Broglie wavelength is:

Options

  1. A
    4 : 1
  2. B
    2 : 1
  3. C
    8 : 1
  4. D
    16 : 1

Correct answer

A.
4 : 1

Step-by-step solution

De-Broglie wavelength,λ=hmv=h2mK=h2mqΔV∴λαλp=2mpVpqp2mαVαqα∵Vp=2V and Vα=4V (given) mα=4mp and qα=2qp\begin{aligned} \\ & \lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mK}}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mq} \Delta \mathrm{V}}} \\ \\ & \therefore \frac{\lambda_\alpha}{\lambda_{\mathrm{p}}}=\sqrt{\frac{2 \mathrm{ m}_{\mathrm{p}} \mathrm{V}_{\mathrm{p}} \mathrm{q}_{\mathrm{p}}}{2 \mathrm{ m}_\alpha \mathrm{V}_\alpha \mathrm{q}_\alpha}} \\ \\ & \because \mathrm{V}_{\mathrm{p}}=2 \mathrm{ V} \text{ and } \mathrm{V}_\alpha=4 \mathrm{ V} \text{ (given) } \\ \\ & \mathrm{m}_\alpha=4 \mathrm{ m}_{\mathrm{p}} \text{ and } \mathrm{q}_\alpha=2 \mathrm{q}_{\mathrm{p}} \\ \end{aligned}\therefore \frac{\lambda_\alpha}{\lambda_p}=\sqrt{\frac{1 \times 2 \times 1}{4 \times 4 \times 2}}=\frac{1}{4} \Rightarrow \lambda_p: \lambda_\alpha=4: 1$
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