The pupil allows light to pass through it and has a diameter of `2xx10^(-3)m.` If minimum light intensity that should enter the eye is `1.8xx10^(-10) W//m^(-2)`, then find the number of photon in visible range must enter the pupil to have vision. Wavelength of photon in visible range can be taken as `5,500Å` .

Answer

To solve the problem, we need to find the number of photons that must enter the pupil of the eye to achieve the minimum light intensity required for vision. Let's break this down step by step.

### Step 1: Calculate the Area of the Pupil
The diameter of the pupil is given as \(2 \times 10^{-3} \, \text{m}\). The radius \(r\) is half of the diameter:
\[ r = \frac{2 \times 10^{-3}}{2} = 1 \times 10^{-3} \, \text{m} \]
The area \(A\) of the pupil can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 = \pi (1 \times 10^{-3})^2 = \pi \times 10^{-6} \, \text{m}^2 \]

### Step 2: Calculate the Total Power (Energy per Unit Time) Entering the Pupil
The intensity \(I\) is given as \(1.8 \times 10^{-10} \, \text{W/m}^2\). The total power \(P\) entering the pupil can be calculated using the formula:
\[ P = I \times A \]
Substituting the values:
\[ P = 1.8 \times 10^{-10} \times \pi \times 10^{-6} \]

### Step 3: Calculate the Energy of a Single Photon
The wavelength of the photon is given as \(5500 \, \text{Å}\) (which is \(5500 \times 10^{-10} \, \text{m}\)). The energy \(E\) of a single photon can be calculated using Planck's equation:
\[ E = \frac{hc}{\lambda} \]
Where:
- \(h = 6.626 \times 10^{-34} \, \text{J s}\) (Planck's constant)
- \(c = 3 \times 10^8 \, \text{m/s}\) (speed of light)
- \(\lambda = 5500 \times 10^{-10} \, \text{m}\)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5500 \times 10^{-10}} \]

### Step 4: Calculate the Number of Photons Required
The number of photons \(N\) required can be calculated by dividing the total power \(P\) by the energy of a single photon \(E\):
\[ N = \frac{P}{E} \]

### Step 5: Substitute and Calculate
Now, we can substitute the calculated values into the equations to find \(N\).

1. Calculate the area \(A\):
\[ A \approx 3.14 \times 10^{-6} \, \text{m}^2 \]

2. Calculate the total power \(P\):
\[ P \approx 1.8 \times 10^{-10} \times 3.14 \times 10^{-6} \approx 5.65 \times 10^{-16} \, \text{W} \]

3. Calculate the energy of a single photon \(E\):
\[ E \approx \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5500 \times 10^{-10}} \approx 3.61 \times 10^{-19} \, \text{J} \]

4. Calculate the number of photons \(N\):
\[ N \approx \frac{5.65 \times 10^{-16}}{3.61 \times 10^{-19}} \approx 1565 \]

Thus, the number of photons that must enter the pupil to have vision is approximately \(N \approx 1565\).