A small plate of a metal is placed at a distance of 2m from a monochromatic light source of wavelenght `4.8xx10^(-7)m` and power 1.0 Watt. The light falls normally on the plate. Find the number of photons striking the metal plate per square metre per second.

Explanation

To solve the problem step by step, we will follow the outlined approach to find the number of photons striking the metal plate per square meter per second.

### Step 1: Calculate the Intensity of Light at the Plate
The intensity \( I \) of light at a distance \( r \) from a point source is given by the formula:

\[ I = \frac{P}{4 \pi r^2} \]

where \( P \) is the power of the light source. Given:
- \( P = 1.0 \, \text{W} \)
- \( r = 2 \, \text{m} \)

Substituting the values:

\[ I = \frac{1.0}{4 \pi (2)^2} = \frac{1.0}{16 \pi} \, \text{W/m}^2 \]

### Step 2: Find the Energy of Each Photon
The energy \( E \) of a single photon is given by the formula:

\[ E = \frac{hc}{\lambda} \]

where:
- \( h = 6.626 \times 10^{-34} \, \text{J s} \) (Planck's constant)
- \( c = 3.0 \times 10^8 \, \text{m/s} \) (speed of light)
- \( \lambda = 4.8 \times 10^{-7} \, \text{m} \) (wavelength)

Substituting the values:

\[ E = \frac{(6.626 \times 10^{-34})(3.0 \times 10^8)}{4.8 \times 10^{-7}} \]

Calculating this gives:

\[ E \approx 4.14 \times 10^{-19} \, \text{J} \]

### Step 3: Calculate the Number of Photons Striking the Plate
The number of photons \( N \) striking the plate per square meter per second can be calculated using the formula:

\[ N = \frac{I}{E} \]

Substituting the values of intensity \( I \) and energy \( E \):

\[ N = \frac{\frac{1.0}{16 \pi}}{4.14 \times 10^{-19}} \]

Calculating this gives:

\[ N \approx \frac{0.019894}{4.14 \times 10^{-19}} \approx 4.82 \times 10^{16} \, \text{photons/m}^2/\text{s} \]

### Final Answer
The number of photons striking the metal plate per square meter per second is approximately:

\[ N \approx 4.82 \times 10^{16} \, \text{photons/m}^2/\text{s} \]

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