Dual Nature of Matter

JEE Main level, single correct, medium

15 of 200

Question 15 of 200

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.

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Answer: (D) 1cE2 mp\frac{1}{\mathrm{c}} \sqrt{\frac{\mathrm{E}}{2 \mathrm{~m}_{\mathrm{p}}}}

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
$\begin{aligned}
=\lambda_p= & \frac{h}{P}=\frac{h}{\sqrt{2 m_p E}} \\
\text { Ratio } \frac{\lambda_p}{\lambda} & =\frac{h}{\sqrt{2 m_p E}} \times \frac{E}{h c} \\
& =\frac{1}{c} \sqrt{\frac{E}{2 m_p}}
\end{aligned}$

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