A 160 watt light source is radiating light of wavelength `6200 Å` uniformly in all directions. The photon flux at a distance of `1.8 m` is of the order of (Plank's constant `6.62 xx 10^(-34) J-s`)

Options

A. `10^(2) m^(-2)s^(-1)`
B. `10^(12) m^(-2)s^(-1)`
C. `10^(19) m^(-2) s^(-1)`
D. `10^(25) m^(-2) s^(-1)`

Explanation

To solve the problem of finding the photon flux at a distance of 1.8 m from a 160 watt light source emitting light of wavelength 6200 Å, we can follow these steps:

### Step 1: Calculate the Intensity of the Light Source
The intensity \( I \) of the light at a distance \( d \) from a point source is given by the formula:
\[ I = \frac{P}{4 \pi d^2} \]
where \( P \) is the power of the light source.

Given:
- \( P = 160 \, \text{W} \)
- \( d = 1.8 \, \text{m} \)

Substituting the values:
\[ I = \frac{160}{4 \pi (1.8)^2} \]

### Step 2: Calculate \( d^2 \)
First, calculate \( d^2 \):
\[ d^2 = (1.8)^2 = 3.24 \]

### Step 3: Substitute \( d^2 \) into the Intensity Formula
Now substitute \( d^2 \) back into the intensity formula:
\[ I = \frac{160}{4 \pi \times 3.24} \]

### Step 4: Calculate \( 4 \pi \)
Calculate \( 4 \pi \):
\[ 4 \pi \approx 12.5664 \]

### Step 5: Calculate the Intensity
Now substitute \( 4 \pi \) into the equation:
\[ I = \frac{160}{12.5664 \times 3.24} \approx \frac{160}{40.785} \approx 3.92 \, \text{W/m}^2 \]

### Step 6: Calculate the Photon Flux
The photon flux \( \Phi \) can be calculated using the formula:
\[ \Phi = \frac{I \cdot \lambda}{hc} \]
where:
- \( h = 6.62 \times 10^{-34} \, \text{J s} \) (Planck's constant)
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light)
- \( \lambda = 6200 \, \text{Å} = 6200 \times 10^{-10} \, \text{m} \)

### Step 7: Substitute Values into the Photon Flux Formula
Substituting the values:
\[ \Phi = \frac{3.92 \times 6200 \times 10^{-10}}{6.62 \times 10^{-34} \times 3 \times 10^8} \]

### Step 8: Calculate the Numerator and Denominator
Calculate the numerator:
\[ 3.92 \times 6200 \times 10^{-10} \approx 2.43 \times 10^{-6} \]
Calculate the denominator:
\[ 6.62 \times 10^{-34} \times 3 \times 10^8 \approx 1.986 \times 10^{-25} \]

### Step 9: Calculate the Photon Flux
Now calculate \( \Phi \):
\[ \Phi \approx \frac{2.43 \times 10^{-6}}{1.986 \times 10^{-25}} \approx 1.22 \times 10^{19} \, \text{photons/m}^2/\text{s} \]

### Final Answer
Thus, the photon flux at a distance of 1.8 m is approximately:
\[ \Phi \approx 1.22 \times 10^{19} \, \text{photons/m}^2/\text{s} \]