JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Modern Physics

A proton of mass '

m_{p}

' has same energy as that of a photon of wavelength '

λ

'. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is. (Here

E

denotes the common energy of the proton and the photon.)

Answer: (C)

Solution

Let E be the common energy of the photon and the proton. For the photon,

E = \frac{h c}{λ}

For the non-relativistic proton of mass

m_{p}

, this same energy is purely kinetic:

E = \frac{1}{2} m_{p} v^{2} = \frac{p ^{2}}{2 m _{p}} ⟹ p = 2 m_{p} E

The proton's de Broglie wavelength is

λ_{p r o t o n} = \frac{h}{p} = \frac{h}{2 m _{p} E}

Taking the ratio with the photon's wavelength

λ = h c / E

\frac{λ _{p r o t o n}}{λ} = \frac{h / 2 m _{p} E}{h c / E} = \frac{E}{c 2 m _{p} E} = \frac{1}{c} \frac{E}{2 m _{p}}

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Single correct option