A proton, a neutron, an electron and an α particle have same energy. If λₚ, λₙ, λₑ and λₐ are the de Broglie's wavelengths of proton, neutron, electron and α particle respectively, then choose the correct relation from the following :

Solution

Physics sir · 2 steps

Same kinetic energy means λ ∝ 1/√m, so just line the four particles up by mass.

1. Link λ to energy. Kinetic energy is E = p²/2m, so the momentum is p = √2mE. Put this into de Broglie's relation λ = h/p.

   λ = h/√2mE ⇒ λ ∝ 1/√m (same E)

   Particle	Mass	Wavelength
   electron	mₑ ≈ mₚ/1836, lightest	≈ 43 λₚ, longest
   proton	mₚ	λₚ
   neutron	mₙ, slightly more than mₚ	slightly less than λₚ
   α particle	≈ 4mₚ, heaviest	≈ λₚ/2, shortest

2. So. Heavier means shorter wavelength, which gives λ_α < λₙ < λₚ < λₑ. That is option (b).

The idea to keep. For equal kinetic energy, λ = h/√2mE makes the lightest particle the longest wave. Do not forget the neutron is a little heavier than the proton.

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