Assuming the sun is directly overhead, the surface of the earth receives sunlight at 1.4 10^3 W m ⁻² . Treat the light as monochromatic with an average wavelength of 500 nm , and assume no light is absorbed between the sun and the earth's surface. The distance from the sun to the earth is 1.5 10¹¹ m . How many photons are present in each cubic metre near the earth's surface at any instant?

Step-by-step solution

The intensity of sunlight at the earth's surface is given by I = 1.4 10^3 W m ⁻² . The energy of a single photon is given by E = hc . The number of photons crossing a unit area per unit time is n = I E = I hc . Since these photons travel at the speed of light c , the number of photons present per unit volume (photon density) is N = n c = I hc^2 . Substituting the given values: N = 1.4 10^3 500 10⁻⁹ 6.63 10⁻³⁴ (3 10^8)^2 N = 7 10⁻⁴ 6.63 10⁻³⁴ 9 10¹⁶ N = 7 10⁻⁴ 59.67 10⁻¹⁸ N 1.17 10¹³ m ⁻³ Rounding to appropriate sig