Question

Ratio of de-Broglie wavelengths of a proton and an alpha particle accelerated through the same potential is:

• A. \( 1:2 \)
• B. \( 2\sqrt{2} : 1 \)
• C. \( 2:1 \)
• D. \( \sqrt{8} : 1 \)

Hint:

For particles accelerated through the same potential, \[ \lambda \propto \frac{1}{\sqrt{mq}} \] Heavier mass or higher charge means shorter de-Broglie wavelength.

Answer 1

The Correct Option is D

Step 1: The de-Broglie wavelength of a particle accelerated through a potential \( V \) is: \[ \lambda = \frac{h}{\sqrt{2mqV}} \]
Step 2: For a proton: \[ m_p = m, \quad q_p = e \] \[ \lambda_p \propto \frac{1}{\sqrt{me}} \]
Step 3: For an alpha particle: \[ m_\alpha = 4m, \quad q_\alpha = 2e \] \[ \lambda_\alpha \propto \frac{1}{\sqrt{4m \cdot 2e}} = \frac{1}{\sqrt{8me}} \]
Step 4: Ratio of wavelengths: \[ \frac{\lambda_p}{\lambda_\alpha} = \sqrt{8} : 1 \]