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

The Correct Option is D

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).