A source `S_(1)` is producing `10^(15)` photons per second of wavelength `5000Ã…`. Another source `S_(2)` is producing `1.02xx10^(15)`
wavelength 5100 A Then, `("Power of" S_(2))//("Power of"S_(1))` is equal to

Options

A. `1.00`
B. 1.02
C. 1.04
D. 0.98

Explanation

To find the ratio of the power of two sources \( S_1 \) and \( S_2 \), we can follow these steps:

### Step 1: Write down the given data
- For source \( S_1 \):
- Number of photons per second, \( n_1 = 10^{15} \)
- Wavelength, \( \lambda_1 = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \)

- For source \( S_2 \):
- Number of photons per second, \( n_2 = 1.02 \times 10^{15} \)
- Wavelength, \( \lambda_2 = 5100 \, \text{Å} = 5100 \times 10^{-10} \, \text{m} \)

### Step 2: Calculate the energy of a single photon for each source
The energy of a photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \)
- \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \)

For source \( S_1 \):
\[ E_1 = \frac{hc}{\lambda_1} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5000 \times 10^{-10}} \]

For source \( S_2 \):
\[ E_2 = \frac{hc}{\lambda_2} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5100 \times 10^{-10}} \]

### Step 3: Calculate the total power for each source
The power produced by each source can be calculated as:
\[ P = n \cdot E \]
Thus, for source \( S_1 \):
\[ P_1 = n_1 \cdot E_1 = n_1 \cdot \frac{hc}{\lambda_1} \]
And for source \( S_2 \):
\[ P_2 = n_2 \cdot E_2 = n_2 \cdot \frac{hc}{\lambda_2} \]

### Step 4: Find the ratio of the powers
To find the ratio of the powers \( \frac{P_2}{P_1} \):
\[ \frac{P_2}{P_1} = \frac{n_2 \cdot E_2}{n_1 \cdot E_1} = \frac{n_2 \cdot \frac{hc}{\lambda_2}}{n_1 \cdot \frac{hc}{\lambda_1}} = \frac{n_2 \cdot \lambda_1}{n_1 \cdot \lambda_2} \]

### Step 5: Substitute the values
Now substituting the values we have:
\[ \frac{P_2}{P_1} = \frac{(1.02 \times 10^{15}) \cdot (5000 \, \text{Å})}{(10^{15}) \cdot (5100 \, \text{Å})} \]

### Step 6: Simplify the expression
\[ = \frac{1.02 \cdot 5000}{5100} \]
\[ = \frac{5100}{5100} = 1 \]

### Final Result
Thus, the ratio of the power of \( S_2 \) to the power of \( S_1 \) is:
\[ \frac{P_2}{P_1} = 1 \]