JEE Main 2025 Jan 28 Shift 1, Physics Q18: Wave Nature of Matter

JEE Main2025Jan 28, Shift 1Physics

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.

A
$\frac{1}{c} \frac{2 E}{m _{p}}$
B
$\frac{1}{c} \frac{E}{m _{p}}$
C
$\frac{1}{c} \frac{E}{2 m _{p}}$
D
$\frac{1}{2 c} \frac{E}{m _{p}}$

Solution

Let $E$ denote the common energy of the photon and the proton.

Photon: $E = \frac{h c}{λ _{photon}} \Rightarrow λ_{photon} = \frac{h c}{E}$ .

Proton (non-relativistic): $E = \frac{1}{2} m_{p} v^{2} \Rightarrow p = m_{p} v = 2 m_{p} E$ . Then $λ_{proton} = \frac{h}{p} = \frac{h}{2 m _{p} E}$ .

Take the ratio:

$\frac{λ _{proton}}{λ _{photon}} = \frac{h / 2 m _{p} E}{h c / E} = \frac{E}{c 2 m _{p} E} = \frac{1}{c} \frac{E}{2 m _{p}}$ .

$∴$ Option (3).

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JEE Main 2025 Jan 28 Shift 1, Physics Q18: Wave Nature of Matter

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