Question

The ratio of the wavelength of two particles with energy $ E $ and $ 3E $ respectively, is:

• A. \( 1: \sqrt{3} \)
• B. \( \sqrt{3}:1 \)
• C. \( 1:3 \)
• D. \( 3:1 \)

Hint:

For particles with energies in the ratio \( E:3E \), the ratio of their de Broglie wavelengths is \( 1: \sqrt{3} \).

Answer 1

The Correct Option is A

The de Broglie wavelength \( \lambda \) of a particle is related to its momentum \( p \) by: \[ \lambda = \frac{h}{p} \] where \( h \) is Planck's constant. For a particle with kinetic energy \( E \), its momentum is: \[ p = \sqrt{2mE} \]
Thus, the de Broglie wavelength \( \lambda \) is: \[ \lambda = \frac{h}{\sqrt{2mE}} \] Now, consider two particles, one with energy \( E \) and the other with energy \( 3E \). The ratio of the wavelengths is: \[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{\sqrt{2mE}}}{\frac{h}{\sqrt{2m(3E)}}} = \frac{\sqrt{3}}{1} \]
Thus, the ratio of wavelengths is \( 1: \sqrt{3} \).