Light of wavelength 5000 Å falls on a sensitive surface. If the surface has received ` 10 ^( -7) ` joule of energy, then what is the number of photons falling on the surface ?

Options

A. ` 25 xx 10 ^( 11)`
B. ` 25 xx 10 ^ 2 `
C. ` 0.25 xx 10^(11)`
D. `2.5 xx 10^(11)`

Answer

To solve the problem of finding the number of photons falling on a sensitive surface when light of wavelength 5000 Å (angstroms) delivers an energy of \(10^{-7}\) joules, we can follow these steps:

### Step 1: Convert Wavelength to Meters
First, we need to convert the wavelength from angstroms to meters.
1 angstrom = \(10^{-10}\) meters, so:
\[ \text{Wavelength in meters} = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \]

### Step 2: Calculate the Energy of One Photon
The energy \(E\) of a single photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
Where:
- \(h\) (Planck's constant) = \(6.626 \times 10^{-34} \, \text{J s}\)
- \(c\) (speed of light) = \(3 \times 10^8 \, \text{m/s}\)
- \(\lambda\) = \(5 \times 10^{-7} \, \text{m}\)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{5 \times 10^{-7} \, \text{m}} \]

Calculating the numerator:
\[ E = \frac{1.9878 \times 10^{-25} \, \text{J m}}{5 \times 10^{-7} \, \text{m}} = 3.9756 \times 10^{-19} \, \text{J} \]

### Step 3: Calculate the Number of Photons
Now, we can find the number of photons \(n\) falling on the surface using the total energy received and the energy of one photon:
\[ n = \frac{\text{Total Energy}}{\text{Energy of One Photon}} = \frac{10^{-7} \, \text{J}}{3.9756 \times 10^{-19} \, \text{J}} \]

Calculating \(n\):
\[ n \approx \frac{10^{-7}}{3.9756 \times 10^{-19}} \approx 2.51 \times 10^{11} \]

### Final Answer
Thus, the number of photons falling on the surface is approximately:
\[ n \approx 2.51 \times 10^{11} \]