A proton and an electron have the same de Broglie wavelength. If Kₚ and Kₑ be the kinetic energies of proton and electron respectively, then choose the correct relation :

Solution

Physics sir · 4 steps

Same wavelength means same momentum, and with equal momentum the heavier particle carries less kinetic energy.

1. Same momentum. The de Broglie wavelength is λ = h/p. If the proton and the electron have the same λ, they must have the same momentum p.
2. Link K to p. Kinetic energy is 1/2mv² and momentum is mv. Put v = p/m into the energy and you get kinetic energy in terms of momentum.

   K = p²/2m ⇒ Kₚ/Kₑ = mₑ/mₚ

3. Compare masses. The proton is about 1836 times heavier than the electron, so Kₚ/Kₑ = mₑ/mₚ ≈ 1/1836, which is far less than 1.
4. So. Kₚ < Kₑ, which is option (c).

The idea to keep. Equal de Broglie wavelengths always mean equal momenta. Then use K = p²/2m: at the same momentum, kinetic energy goes as 1/m, so the lighter particle has more.

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