Question

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

• A. 1 : 8
• B. 1 : 2
• C. 4 : 1
• D. 8 : 1

Answer 1

Answer: (D)

8 : 1

Answer 2

The de Broglie wavelength is given by the equation:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

where:

• $h$ is Planck’s constant.
• $m$ is the mass of the particle.
• $v$ is the velocity.

For the proton and $\alpha$-particle:

$\frac{\lambda_p}{\lambda_\alpha} = \frac{m_\alpha v_\alpha}{m_p v_p}$

Given $m_\alpha = 4m_p$ (since $\alpha$-particle has 4 times the mass of a proton) and the relationship between velocity and wavelength, we find that the ratio of velocities is:

$v_p : v_\alpha = 8 : 1$

Thus, the correct answer is Option (4).