A particle of mass 9.1 × 10⁻³¹ kg travels in a medium with a speed of 10⁶ m/s and a photon of a radiation of linear momentum 10⁻²⁷ kg m/s travels in vacuum. The wavelength of photon is __________ times the wavelength of the particle.

Solution

Physics sir · 4 steps

Photon or particle, λ = h/p for both, so the wavelength ratio is just the inverse momentum ratio.

1. Same formula. The de Broglie relation λ = h/p holds for the material particle and for the photon. The medium does not matter here, because the wavelength depends only on momentum.

   λₚₕₒₜₒₙ/λ_(particle) = (h/pₚₕₒₜₒₙ)/(h/p_(particle)) = p_(particle)/pₚₕₒₜₒₙ

2. Particle momentum. p_(particle) = mv = 9.1 × 10⁻³¹ × 10⁶ = 9.1 × 10⁻²⁵ kg m s⁻¹.
3. Take the ratio. (9.1 × 10⁻²⁵)/10⁻²⁷ = 9.1 × 10² = 910.
4. So. The photon's wavelength is 910 times the particle's wavelength. The photon has less momentum, so it has the longer wavelength.

The common slip: 0.0011. Dividing the photon momentum by the particle momentum inverts the ratio and gives about 0.0011 instead of 910.

The idea to keep. For any wavelength comparison, write λ = h/p for each and divide: h cancels, and the smaller momentum always gives the longer wavelength.

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