Given two points sources having same power of 200W.One source is emitting photons of `lambda_1 = 500 nm ` and other emitting X-ray photons of `lambda_2 = 1nm`. Find ratio of photon emitted from both sources.

Options

A. 200
B. 500
C. 250
D. 0.4

Answer

To solve the problem of finding the ratio of photons emitted from two point sources with the same power, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Both sources have the same power: \( P = 200 \, W \).
- Wavelength of the first source: \( \lambda_1 = 500 \, nm = 500 \times 10^{-9} \, m \).
- Wavelength of the second source: \( \lambda_2 = 1 \, nm = 1 \times 10^{-9} \, m \).

2. **Energy of a Photon**:
- The energy of a single photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, J \cdot s \)) and \( c \) is the speed of light (\( 3 \times 10^8 \, m/s \)).

3. **Power and Number of Photons**:
- The power can also be expressed in terms of the number of photons emitted per second (\( n \)):
\[ P = \frac{dE}{dt} = n \cdot E \]
- Therefore, for each source, we can write:
\[ P = n \cdot \frac{hc}{\lambda} \]
- Rearranging gives:
\[ n = \frac{P \cdot \lambda}{hc} \]

4. **Calculate Number of Photons for Each Source**:
- For the first source (\( n_1 \)):
\[ n_1 = \frac{200 \cdot (500 \times 10^{-9})}{hc} \]
- For the second source (\( n_2 \)):
\[ n_2 = \frac{200 \cdot (1 \times 10^{-9})}{hc} \]

5. **Find the Ratio of Photons**:
- The ratio of the number of photons emitted by the two sources is:
\[ \frac{n_1}{n_2} = \frac{\frac{200 \cdot (500 \times 10^{-9})}{hc}}{\frac{200 \cdot (1 \times 10^{-9})}{hc}} = \frac{500 \times 10^{-9}}{1 \times 10^{-9}} = 500 \]

6. **Final Answer**:
- Thus, the ratio of the number of photons emitted from the two sources is:
\[ \frac{n_1}{n_2} = 500 \]