A source `S_(1)` is producing `10^(15)` photons//s of wavelength `5000 Å` Another source `S_(2)` is producing `1.02 xx 10^(15)` photons per second of wavelength `5100 Å`. 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 source \( S_2 \) to the power of source \( S_1 \), we can follow these steps:

### Step 1: Understand the relationship between power and photons
The power \( P \) of a source emitting photons can be expressed as:
\[ P = \frac{nE}{T} \]
where:
- \( n \) is the number of photons emitted per second,
- \( E \) is the energy of one photon,
- \( T \) is the time (which we can consider as 1 second for our calculation).

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

### Step 3: Calculate the energy for both sources
Convert the wavelengths from angstroms to meters:
- \( \lambda_1 = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \)
- \( \lambda_2 = 5100 \, \text{Å} = 5100 \times 10^{-10} \, \text{m} = 5.1 \times 10^{-7} \, \text{m} \)

Now calculate the energy for each source:
- For \( S_1 \):
\[ E_1 = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5 \times 10^{-7}} = \frac{1.9878 \times 10^{-25}}{5 \times 10^{-7}} = 3.9756 \times 10^{-19} \, \text{J} \]

- For \( S_2 \):
\[ E_2 = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5.1 \times 10^{-7}} = \frac{1.9878 \times 10^{-25}}{5.1 \times 10^{-7}} = 3.895 \times 10^{-19} \, \text{J} \]

### Step 4: Calculate the power for both sources
Using the number of photons emitted:
- For \( S_1 \):
\[ P_1 = n_1 E_1 = (10^{15})(3.9756 \times 10^{-19}) = 3.9756 \times 10^{-4} \, \text{W} \]

- For \( S_2 \):
\[ P_2 = n_2 E_2 = (1.02 \times 10^{15})(3.895 \times 10^{-19}) = 3.9739 \times 10^{-4} \, \text{W} \]

### Step 5: Calculate the ratio of powers
Now we can find the ratio of the powers:
\[ \frac{P_2}{P_1} = \frac{3.9739 \times 10^{-4}}{3.9756 \times 10^{-4}} = 0.99957 \approx 1 \]

### Conclusion
Thus, the ratio of the power of \( S_2 \) to the power of \( S_1 \) is approximately equal to \( 1 \).

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