Question

A star which can be seen with naked eye from Earth has intensity $1.6 \times 10^{-9} \mathrm{Wm}^{-2}\( on Earth. If the corresponding wavelength is \)560 \mathrm{~nm}\(, and the radius of the human eye is \)2.5 \times 10^{-3} \mathrm{~m}\(, the number of photons entering in our in \)1 \mathrm{~s}$ is..... (A) \(7.8 \times 10^{4} \mathrm{~s}^{-1}\) (B) \(8.85 \times 10^{4} \mathrm{~s}^{-1}\) (C) \(7.85 \times 10^{5} \mathrm{~s}^{-1}\) (D) \(8.85 \times 10^{5} \mathrm{~s}^{-1}\)

Short answer

The number of photons entering the human eye in 1 second is approximately \(7.8 \times 10^{4} \mathrm{~s}^{-1}\) (Option A).

Step-by-step solution

Calculate the energy of a single photon

To calculate the energy of a single photon, we will use the equation: \(E = \dfrac{hc}{\lambda}\) where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), \(c\) is the speed of light (\(2.998 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the wavelength of the light (\(5.6 \times 10^{-7} \mathrm{m}\)). Using these values, we can find the energy of a single photon: \(E = \dfrac{(6.626 \times 10^{-34} \mathrm{Js})(2.998 \times 10^{8} \mathrm{m/s})}{5.6 \times 10^{-7} \mathrm{m}} \)

Calculate the total energy entering the eye

To find the total energy entering the eye, we will multiply the intensity of the light by the area of the aperture of the human eye (which can be approximated as a circle): \(\mathrm{Total~Energy} = \mathrm{Intensity} \times \mathrm{Area} \times \mathrm{Time}\) The area of the aperture can be calculated using the formula \(A = \pi r^2\), where \(A\) is the area and \(r\) is the radius of the eye (given as \(2.5 \times 10^{-3} \mathrm{m}\)). First, calculate the area of the aperture: \(A = \pi (2.5 \times 10^{-3} \mathrm{m})^2\) Now, calculate the total energy entering the eye in 1 second: \(\mathrm{Total~Energy} = (1.6 \times 10^{-9} \mathrm{Wm}^{-2}) (\pi (2.5 \times 10^{-3} \mathrm{m})^2)(1 \mathrm{s})\)

Determine the number of photons entering the eye per second

Lastly, we will find the number of photons entering the eye per second by dividing the total energy by the energy of a single photon: \(\mathrm{Number~of~Photons\, =\, \dfrac{Total\,~Energy}{Energy\,~of\,~a\,~Single\,~Photon}}\) \(\mathrm{Number~of ~Photons~ in ~1s} = \dfrac{(1.6 \times 10^{-9} \mathrm{Wm}^{-2}) (\pi (2.5 \times 10^{-3} \mathrm{m})^2)(1 \mathrm{s})}{\dfrac{(6.626 \times 10^{-34} \mathrm{Js})(2.998 \times 10^{8} \mathrm{m/s})}{5.6 \times 10^{-7} \mathrm{m}}}\) After calculating the number of photons, we find the closest answer is: (A) \(7.8 \times 10^{4} \mathrm{~s}^{-1}\)