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A light-bulb filament with surface area 64 mm² behaves as a black body at 2500 K and, seen from far away, acts essentially as a point source. The bulb is viewed at night from 100 m. The observer's pupil is circular with radius 3 mm. Which of the following statements is/are correct? (Use Stefan-Boltzmann constant = 5.67*10⁻⁸ W m⁻² K⁻⁴, Wien's constant = 2.90*10⁻³ m K, h = 6.63*10⁻³⁴ J s, c = 3.00*10⁸ m/s) (A) the filament radiates a total power between 642 W and 645 W (B) the radiated power that enters one of the observer's eyes lies between 3.15*10⁻⁸ W and 3.25*10⁻⁸ W (C) the wavelength of peak emission intensity is 1160 nm (D) treating the mean emitted wavelength as 1740 nm, the number of photons entering one eye per second is between 2.75*10¹¹ and 2.85*10¹¹

1. the filament radiates a total power between 642 W and 645 W
2. the radiated power that enters one of the observer's eyes lies between 3.15*10⁻⁸ W and 3.25*10⁻⁸ W
3. the wavelength of peak emission intensity is 1160 nm
4. treating the mean emitted wavelength as 1740 nm, the number of photons entering one eye per second is between 2.75*10¹¹ and 2.85*10¹¹

Correct answer: treating the mean emitted wavelength as 1740 nm, the number of photons entering one eye per second is between 2.75*10¹¹ and 2.85*10¹¹

Solution

This is a multiple-correct question. Total power P = sigma*A*T⁴ = 5.67e-8 * 64e-6 * (2500)⁴ = 141.75 W, NOT 642-645 W, so (A) is wrong. Wien's law gives lambdaₘ = 2.90e-3/2500 = 1160 nm, so (C) is correct. Power entering eye = P * (pi*(3e-3)²)/(4*pi*(100)²) = 141.75 * (9e-6)/(4e4) = 3.19e-8 W, so (B) is correct. Photons/s = (3.19e-8)/(h*c/1740e-9) = (3.19e-8)/(1.143e-19) = 2.79e11, so (D) is correct. Correct options are B, C, D; this single-answer record lists D as the canonical key.

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