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 find the number of photons striking the metal plate per square meter per second, we can follow these steps:

### Step 1: Calculate the intensity of the light at the distance of 2 meters.
The intensity \( I \) of the light at a distance \( r \) from a point source is given by the formula:

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

Where:
- \( P \) is the power of the light source (1.0 Watt).
- \( r \) is the distance from the source (2 m).

Substituting the values:

\[ I = \frac{1}{4 \pi (2)^2} = \frac{1}{16 \pi} \text{ Watts/m}^2 \]

### Step 2: Calculate the energy of a single photon.
The energy \( E \) of a photon is given by the formula:

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

Where:
- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \).
- \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \).
- \( \lambda \) is the wavelength of the light \( (4.8 \times 10^{-7} \text{ m}) \).

Substituting the values:

\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{4.8 \times 10^{-7}} \approx 4.14 \times 10^{-19} \text{ J} \]

### Step 3: Calculate the number of photons striking the plate per square meter per second.
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:

\[ n = \frac{\frac{1}{16 \pi}}{4.14 \times 10^{-19}} \]

Calculating \( \frac{1}{16 \pi} \):

\[ \frac{1}{16 \pi} \approx 0.019894 \text{ W/m}^2 \]

Now substituting this value into the equation for \( n \):

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

### 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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