An isotropic ultraviolet source emits 4.0 W entirely in photons of energy 20 eV. A small gas cell of effective collecting area 5.0 cm² is located 2.0 m from the source. Assume that 2% of all photons entering the cell produce one photoionisation event. Take 1 eV = 1.6×10⁻¹⁹ J. If the photoionisation rate is approximately N×10¹⁰ s⁻¹, find N.
Physics · Photoelectric effect and dual nature of matter · question based on photoelectric effect
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Question

An isotropic ultraviolet source emits 4.0 W entirely in photons of energy 20 eV. A small gas cell of effective collecting area 5.0 cm² is located 2.0 m from the source. Assume that 2% of all photons entering the cell produce one photoionisation event. Take 1 eV = 1.6×10⁻¹⁹ J. If the photoionisation rate is approximately N×10¹⁰ s⁻¹, find N.

Physics · Photoelectric effect and dual nature of matter · question based on photoelectric effect

G1 Physics · Accepted Answer

Energy of one photon: Eγ = 20×1.6×10⁻¹⁹ = 3.2×10⁻¹⁸ J. Total photon production rate of the source: Ṅ(source) = 4.0 ÷ 3.2×10⁻¹⁸ = 1.25×10¹⁸ s⁻¹. At distance r = 2.0 m, the photons are distributed over area 4πr² = 16π m². Cell area: A = 5.0 cm² = 5.0×10⁻⁴ m². Fraction of photons entering the cell: A ÷ (4πr²) = 5.0×10⁻⁴ ÷ (16π). Therefore the photon rate entering the cell is approximately 1.24×10¹³ s⁻¹. Only 2% cause photoionisation: Ṅ(ion) = 0.02×1.24×10¹³ ≈ 2.49×10¹¹ s⁻¹. Thus, Ṅ(ion) ≈ 24.9×…