Question

An electron and proton are accelerated through the same potential difference. The ratio of their de-Broglie wavelengths \( \lambda_p \) to \( \lambda_e \) is (\( m_e \) = mass of electron, \( m_p \) = mass of proton)

• A. \( \left(\dfrac{m_p}{m_e}\right)^{1/2} \)
• B. \( \left(\dfrac{m_e}{m_p}\right)^{1/2} \)
• C. \( \dfrac{m_e}{m_p} \)
• D. \( \dfrac{m_p}{m_e} \)

Hint:

For same accelerating potential, de-Broglie wavelength varies inversely as square root of mass.

Answer 1

The Correct Option is B

Step 1: Write de-Broglie wavelength formula.
For a particle accelerated through potential difference \( V \): \[ \lambda = \frac{h}{\sqrt{2m e V}} \]
Step 2: Write wavelengths for electron and proton.
\[ \lambda_e = \frac{h}{\sqrt{2m_e eV}}, \quad \lambda_p = \frac{h}{\sqrt{2m_p eV}} \]
Step 3: Take ratio.
\[ \frac{\lambda_p}{\lambda_e} = \sqrt{\frac{m_e}{m_p}} \]
Step 4: Conclusion.
The required ratio is \( \left(\dfrac{m_e}{m_p}\right)^{1/2} \).