Two electrons are moving with different kinetic energies. If 1st electron is moving with kinetic energy k and the 2nd electron with kinetic energy 4k, the ratio of their de Broglie wavelengths will be respectively:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2 : 1
The de Broglie wavelength of a particle is \( \lambda = h/p \). For a particle of mass \( m \) and kinetic energy \( K \), the momentum is \( p = \sqrt{2mK} \), so
\( \displaystyle \lambda = \frac{h}{\sqrt{2mK}} \;\Rightarrow\; \lambda \propto \frac{1}{\sqrt{K}} \)
Both particles are electrons (same \( m \)), so
\( \displaystyle \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{K_2}{K_1}} = \sqrt{\frac{4k}{k}} = 2 \)
The ratio is \( 2 : 1 \): the slower electron has the longer wavelength.