Two sources of light emit with a power of 200 W. The ratio of number of photons of visible light emitted by each source having wavelengths 300 nm and 500 nm respectively, will be :

Options

A. `1 : 5`
B. `1 : 3`
C. `5 : 3`
D. `3 : 5`

Answer

To find the ratio of the number of photons emitted by two sources of light with wavelengths 300 nm and 500 nm, we can follow these steps:

### Step 1: Understand the relationship between power, energy, and number of photons
The energy of a single photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of one photon,
- \( h \) 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 of the light.

### Step 2: Calculate the energy of photons for each source
For the first source (wavelength \( \lambda_1 = 300 \, \text{nm} = 300 \times 10^{-9} \, \text{m} \)):
\[ E_1 = \frac{hc}{\lambda_1} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{300 \times 10^{-9}} \]

For the second source (wavelength \( \lambda_2 = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)):
\[ E_2 = \frac{hc}{\lambda_2} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{500 \times 10^{-9}} \]

### Step 3: Calculate the number of photons emitted by each source
The number of photons emitted by a source can be calculated using the formula:
\[ n = \frac{P \cdot t}{E} \]
where:
- \( n \) is the number of photons,
- \( P \) is the power of the source,
- \( t \) is the time duration (which will cancel out in the ratio).

Since both sources have the same power \( P = 200 \, \text{W} \) and we can assume the same time \( t \), we can express the number of photons for each source as:
\[ n_1 = \frac{P \cdot t}{E_1} \]
\[ n_2 = \frac{P \cdot t}{E_2} \]

### Step 4: Find the ratio of the number of photons
The ratio of the number of photons emitted by the two sources can be expressed as:
\[ \frac{n_1}{n_2} = \frac{E_2}{E_1} \]

Substituting the expressions for \( E_1 \) and \( E_2 \):
\[ \frac{n_1}{n_2} = \frac{\frac{hc}{\lambda_2}}{\frac{hc}{\lambda_1}} = \frac{\lambda_1}{\lambda_2} \]

### Step 5: Substitute the values of wavelengths
Now substituting the values of \( \lambda_1 \) and \( \lambda_2 \):
\[ \frac{n_1}{n_2} = \frac{300 \, \text{nm}}{500 \, \text{nm}} = \frac{300}{500} = \frac{3}{5} \]

### Final Answer
Thus, the ratio of the number of photons emitted by the two sources is:
\[ n_1 : n_2 = 3 : 5 \]

---