The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2 \lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be:

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Modern Physics

The de Broglie wavelengths of a proton and an

α

particle are

λ

and

2 λ

respectively. The ratio of the velocities of proton and

α

particle will be:

(A) $1 : 8$
(B) $1 : 2$
(C) $4 : 1$
(D) $8 : 1$ ✓ Correct

Answer: (D)

Solution

de Broglie wavelength:

λ = \frac{h}{p} = \frac{h}{m v} ⟹ v = \frac{h}{mλ}

.
For the proton (mass

m_{p}

, wavelength

λ

) and the

α

-particle (mass

4 m_{p}

, wavelength

2 λ

\frac{v _{p}}{v _{α}} = \frac{h / ( m _{p} λ )}{h / ( 4 m _{p} \cdot 2 λ )} = \frac{4 m _{p} \cdot 2 λ}{m _{p} λ} = 8

v_{p} : v_{α} = 8 : 1

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Single correct option

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