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JEE Main 202528 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
28 January
Shift
Morning Shift
Subject
Physics
Chapter
Dual Nature of Matter

JEE Main 2025 · 28 January · Morning Shift · Physics · Dual Nature of Matter

A proton of mass ' m_ p ' has same energy as that of a photon of...

A proton of mass ' mpm_{\mathrm{p}} ' has same energy as that of a photon of wavelength ' λ\lambda '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is.

Options

  1. A1cEmp\frac{1}{\mathrm{c}} \sqrt{\frac{\mathrm{E}}{\mathrm{m}_{\mathrm{p}}}}
  2. B1c2Emp\frac{1}{\mathrm{c}} \sqrt{\frac{2 \mathrm{E}}{\mathrm{m}_{\mathrm{p}}}}
  3. C12cEmp\frac{1}{2 c} \sqrt{\frac{E}{m_p}}
  4. D1cE2 mp\frac{1}{\mathrm{c}} \sqrt{\frac{\mathrm{E}}{2 \mathrm{~m}_{\mathrm{p}}}}

Correct answer

D. 1cE2 mp\frac{1}{\mathrm{c}} \sqrt{\frac{\mathrm{E}}{2 \mathrm{~m}_{\mathrm{p}}}}

Step-by-step solution

Energy of photon =E=hcλ⇒=E=\frac{h c}{\lambda} \Rightarrow Wavelength of photon =λ=hcE=\lambda=\frac{h c}{E} Energy of proton =E=12mpv2=P22mp⇒=E=\frac{1}{2} m_p v^2=\frac{P^2}{2 m_p} \Rightarrow Linear momentum of proton =P=2mpE=P=\sqrt{2 m_p E} Or de-Broglie wavelength of proton =λp=hP=h2mpE Ratio λpλ=h2mpE×Ehc=1cE2mp\begin{aligned} =\lambda_p= & \frac{h}{P}=\frac{h}{\sqrt{2 m_p E}} \\ \text{ Ratio } \frac{\lambda_p}{\lambda} & =\frac{h}{\sqrt{2 m_p E}} \times \frac{E}{h c} \\ & =\frac{1}{c} \sqrt{\frac{E}{2 m_p}} \end{aligned}
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