Q 12-11-053JEE MainJEE Main 2025 (28 Jan, Shift 1)Medium
A proton of mass $m_p$ has the same energy as that of a photon of wavelength $\lambda$. If the proton is moving at non-relativistic speed, then the ratio of its de Broglie wavelength to the wavelength of the photon is
Answer: (D) $\dfrac{1}{c}\sqrt{\dfrac{E}{2m_p}}$
Let $E$ be the common energy.
Photon: $\lambda = \dfrac{hc}{E}$. Proton (non-relativistic): $\lambda_p = \dfrac{h}{\sqrt{2m_pE}}$.
$$\frac{\lambda_p}{\lambda} = \frac{h}{\sqrt{2m_pE}}\cdot\frac{E}{hc} = \frac{1}{c}\sqrt{\frac{E}{2m_p}}$$
Solution by Sreeraj P, M.Sc Physics