Question

The minimum intensity of white light that our eyes can perceive is about \(0.1 \, nWm}^{-2}\). Calculate the number of photons of this light entering our pupil (area \(0.4 \, cm}^2\)) per second.
(Take average wavelength of white light \(= 500 \, nm}\) and Planck’s constant \(= 6.6 \times 10^{-34} \, Js}\).) % Correct answer Correct Answer: To find the number of photons, we first calculate the energy per photon using the equation for photon energy: \[ E = \frac{hc}{\lambda} \] where \( h = 6.6 \times 10^{-34} \, Js} \) (Planck's constant), \( c = 3 \times 10^8 \, m/s} \) (speed of light), and \( \lambda = 500 \, nm} = 500 \times 10^{-9} \, m} \).

Hint:

When calculating the number of photons based on intensity, it's essential to correctly convert all units to the standard SI units to ensure accurate calculations.

Answer 1

1. Calculate the energy of one photon:
The energy \(E\) of a photon is given by: \[ E = \frac{hc}{\lambda} \] where:
\begin{itemize} \item \(h = 6.6 \times 10^{-34} \, Js}\) (Planck's constant),
\item \(c = 3 \times 10^8 \, m/s}\) (speed of light),
\item \(\lambda = 500 \, nm} = 500 \times 10^{-9} \, m}\) (wavelength).
\end{itemize} Substituting the values:
\[ E = \frac{6.6 \times 10^{-34} \times 3 \times 10^8}{500 \times 10^{-9}} \] \[ E = \frac{19.8 \times 10^{-26}}{500 \times 10^{-9}} \] \[ E = \frac{19.8 \times 10^{-26}}{5 \times 10^{-7}} \] \[ E = 3.96 \times 10^{-19} \, J} \] 2. Calculate the total energy entering the pupil per second:
The intensity \(I\) is given as \(0.1 \, nWm}^{-2} = 0.1 \times 10^{-9} \, Wm}^{-2}\).
The area \(A\) of the pupil is \(0.4 \, cm}^2 = 0.4 \times 10^{-4} \, m}^2\).
The power \(P\) entering the pupil is:
\[ P = I \times A \] \[ P = 0.1 \times 10^{-9} \times 0.4 \times 10^{-4} \] \[ P = 4 \times 10^{-15} \, W} \] Since \(1 \, W} = 1 \, J/s}\), the energy per second is:
\[ E_{total}} = 4 \times 10^{-15} \, J/s} \] 3. Calculate the number of photons per second:
The number of photons \(N\) per second is given by: \[ N = \frac{E_{total}}}{E} \] \[ N = \frac{4 \times 10^{-15}}{3.96 \times 10^{-19}} \] \[ N \approx 1.01 \times 10^4 \] Final Answer:
\[ \boxed{1.01 \times 10^4} \]