Answer
See answers.
Work Step by Step
a. Start by calculating the intensity of the light at a distance of R = 1.0 meters.
We divide the power (in the visible spectrum) by the surface area of a sphere of radius R.
$$I=\frac{P}{4\pi R^2}$$
To find the energy entering the eye each second, multiply the intensity by the area of the pupil (assume its diameter is d)
$$P_2=I\pi (d/2)^2=\frac{Pd^2}{16 R^2}$$
Divide this by the energy of a single photon to find the number of visible light photons that enter the eye each second, n.
$$n=\frac{P_2}{hc/ \lambda}=\frac{P\lambda d^2}{16hc R^2}$$
$$n=\frac{(0.030)(100W)(550\times10^{-9}m)(0.004m)^2} {16(6.626\times10^{-34}J s)(3.00\times10^8 m/s) (1.0m)^2}=8.3\times10^{12}photons/s$$
b. Repeat, using R = 1.0 km.
$$n=\frac{(0.030)(100W)(550\times10^{-9}m)(0.004m)^2} {16(6.626\times10^{-34}J s)(3.00\times10^8 m/s) (1.0\times 10^3 m)^2}=8.3\times10^{6}photons/s$$
