A photon and a non-relativistic electron have the same wavelength of 0.10 nm. Take h = 6.63×10⁻³⁴ J·s, c = 3.0×10⁸ m·s⁻¹ and mₑ = 9.1×10⁻³¹ kg. If the photon energy divided by the electron kinetic energy is approximately N, find N to the nearest integer.
Physics · Photoelectric effect and dual nature of matter · debroglie duel nature of light
JEE Mains NEET JEE Advanced

Question

A photon and a non-relativistic electron have the same wavelength of 0.10 nm. Take h = 6.63×10⁻³⁴ J·s, c = 3.0×10⁸ m·s⁻¹ and mₑ = 9.1×10⁻³¹ kg. If the photon energy divided by the electron kinetic energy is approximately N, find N to the nearest integer.

Physics · Photoelectric effect and dual nature of matter · debroglie duel nature of light

G1 Physics · Accepted Answer

Equal wavelength means equal momentum magnitude: p = h ÷ λ. For the photon, Eγ = pc. For the non-relativistic electron, Kₑ = p² ÷ (2mₑ). Therefore, Eγ ÷ Kₑ = pc ÷ [p² ÷ (2mₑ)] = 2mₑc ÷ p. Since p = h ÷ λ, Eγ ÷ Kₑ = 2mₑcλ ÷ h. Substitute λ = 0.10 nm = 1.0×10⁻¹⁰ m: Eγ ÷ Kₑ = 2×9.1×10⁻³¹×3.0×10⁸×1.0×10⁻¹⁰ ÷ 6.63×10⁻³⁴ ≈ 82.35. To the nearest integer, N = 82.