Monochromatic light of wavelength `3000 Å` is incident on a surface area `4 cm^(2)`. If intensity of light is `150 m W//m^(2)` , then rate at which photons strike the target is

Options

A. `3 xx 10^(10) //sec`
B. `9 xx 10^(13)//sec`
C. `7 xx 10^(15)//sec`
D. `6 xx 10^(19)//sec`

Explanation

To find the rate at which photons strike the target when monochromatic light of wavelength \(3000 \, \text{Å}\) is incident on a surface area of \(4 \, \text{cm}^2\) with an intensity of \(150 \, \text{mW/m}^2\), we can follow these steps:

### Step 1: Convert the Area from cm² to m²
The area given is \(4 \, \text{cm}^2\). We need to convert this to square meters:
\[ 4 \, \text{cm}^2 = 4 \times 10^{-4} \, \text{m}^2 \]

### Step 2: Convert the Intensity from mW/m² to W/m²
The intensity is given as \(150 \, \text{mW/m}^2\). We convert this to watts:
\[ 150 \, \text{mW/m}^2 = 150 \times 10^{-3} \, \text{W/m}^2 = 0.150 \, \text{W/m}^2 \]

### Step 3: Calculate the Power Incident on the Surface
Using the formula for power \(P\) based on intensity \(I\) and area \(A\):
\[ P = I \times A \]
Substituting the values:
\[ P = 0.150 \, \text{W/m}^2 \times 4 \times 10^{-4} \, \text{m}^2 = 0.150 \times 4 \times 10^{-4} = 6 \times 10^{-5} \, \text{W} \]

### Step 4: Calculate the Energy of One Photon
The energy \(E\) of one photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
Where:
- \(h = 6.626 \times 10^{-34} \, \text{J s}\) (Planck's constant)
- \(c = 3 \times 10^8 \, \text{m/s}\) (speed of light)
- \(\lambda = 3000 \, \text{Å} = 3000 \times 10^{-10} \, \text{m}\)

Calculating \(E\):
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{3000 \times 10^{-10} \, \text{m}} = \frac{1.9878 \times 10^{-25}}{3 \times 10^{-7}} = 6.626 \times 10^{-19} \, \text{J} \]

### Step 5: Calculate the Rate of Photons Striking the Target
The rate at which photons strike the target \(N/T\) can be calculated using the formula:
\[ \frac{N}{T} = \frac{P}{E} \]
Substituting the values:
\[ \frac{N}{T} = \frac{6 \times 10^{-5} \, \text{W}}{6.626 \times 10^{-19} \, \text{J}} \approx 9.05 \times 10^{13} \, \text{photons/s} \]

### Conclusion
The rate at which photons strike the target is approximately:
\[ \frac{N}{T} \approx 9 \times 10^{13} \, \text{photons/s} \]