The de Broglie wavelengths of a proton and an α particle are λ and 2 λ respectively. The ratio of the velocities of proton and α particle will be :

Solution

Physics sir · 3 steps

Turn λ = h/mv around to v = h/mλ, then let both the mass and the wavelength do their work.

1. Velocity from wavelength. The de Broglie relation is λ = h/mv, so v = h/mλ. Velocity falls with both mass and wavelength.

   vₚ/v_α = (m_α λ_α)/(mₚ λₚ)

2. Put in values. An α particle has 2 protons and 2 neutrons, so m_α = 4mₚ. With λₚ = λ and λ_α = 2λ: vₚ/v_α = (4mₚ × 2λ)/(mₚ × λ) = 8.
3. So. The proton moves 8 times as fast as the α particle, so the ratio is 8 : 1, option (a).

The idea to keep. Write v = h/mλ and take the ratio. Mass and wavelength both sit in the denominator, so multiply their ratios. Do not drop either one.

Answer from the NTA final answer key. Solution by PrepWiser.