A proton and a triton are accelerated from rest through a potential V. The ratio of their de Broglie wavelength is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \sqrt{3} : 1 \)
A charge \( q \) accelerated from rest through a potential \( V \) gains kinetic energy \( qV \), so its momentum is \( p = \sqrt{2mqV} \) and its de Broglie wavelength is
\( \displaystyle \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mqV}} \)
The proton and the triton carry the same charge \( e \) and are accelerated through the same \( V \). A triton (one proton and two neutrons) has three times the mass of a proton: \( m_t = 3m_p \).
\( \displaystyle \lambda \propto \frac{1}{\sqrt{m}} \)
\( \displaystyle \frac{\lambda_p}{\lambda_t} = \sqrt{\frac{m_t}{m_p}} = \sqrt{\frac{3m_p}{m_p}} = \sqrt{3} \)
So \( \lambda_p : \lambda_t = \sqrt{3} : 1 \).