A proton of mass ' m_P ' has same energy as that of a photon of wavelength ' λ '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is.

Solution

Physics sir · 3 steps

Write both wavelengths in terms of the same energy E, then divide. The h cancels and the answer drops out.

1. Proton wavelength. At non-relativistic speed, E = p²/2mₚ, so p = √2mₚE. The de Broglie wavelength is λₚ = h/p = h/√2mₚE.
2. Photon wavelength. For a photon, E = hc/λ, so λ = hc/E.

   λₚ/λ = h/√2mₚE × E/hc = E/c√2mₚE = 1/c√(E/2mₚ)

3. So. The factor 2 stays inside the root, in the denominator, which is option (d).

The idea to keep. For a slow particle, use λ = h/√2mE. For a photon, use λ = hc/E. Never use hc/E for a particle with mass.

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