An electron of mass mₑ and a proton of mass mₚ = 1836 mₑ are moving with the same speed. The ratio of their de Broglie wavelength (^λ electron)/(^λ proton) will be :

Solution

Physics sir · 3 steps

Same speed means momentum goes with mass, and wavelength goes the other way: the lighter electron wins.

1. de Broglie relation. A particle of mass m moving with speed v has momentum p = mv. Its de Broglie wavelength is λ = h/p = h/mv.
2. Same speed. Both particles have the same v, and h is a constant. So λ depends only on mass, and it is inversely proportional to it.

   λₑ/λₚ = (h/(mₑ v))/(h/(mₚ v)) = mₚ/mₑ = (1836 mₑ)/mₑ

3. So. λₑ/λₚ = 1836. The electron, being 1836 times lighter, has a wavelength 1836 times longer, which is option (b).

The idea to keep. At equal speed, λ ∝ 1/m. At equal kinetic energy, λ ∝ 1/√m. Always check what is held equal before you write the ratio.

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