There are two sources of light, each emitting with a power of `100W`. One emits X-rays of wavelength `1nm` and the other visible light at `500nm`. Find the ratio of number of photons of X-rays to the photons of visible light of the given wavelength?

Options

A. `1:500`
B. `1:400`
C. `1:300`
D. `1:200`

Explanation

To find the ratio of the number of photons of X-rays to the number of photons of visible light, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Power of both sources, \( P = 100 \, \text{W} \)
- Wavelength of X-rays, \( \lambda_1 = 1 \, \text{nm} = 1 \times 10^{-9} \, \text{m} \)
- Wavelength of visible light, \( \lambda_2 = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)

2. **Relate Power to Number of Photons:**
The power emitted by a source can be expressed in terms of the number of photons emitted per second and the energy of each photon:
\[ P = n \cdot E \]
where \( n \) is the number of photons and \( E \) is the energy of a single photon.

3. **Calculate the Energy of a Photon:**
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)) and \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).

4. **Calculate the Number of Photons for X-rays:**
For X-rays (wavelength \( \lambda_1 \)):
\[ E_1 = \frac{hc}{\lambda_1} = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{1 \times 10^{-9} \, \text{m}} \]
\[ E_1 = 1.9878 \times 10^{-14} \, \text{J} \]
Now, using the power to find the number of photons:
\[ n_1 = \frac{P}{E_1} = \frac{100 \, \text{W}}{1.9878 \times 10^{-14} \, \text{J}} \approx 5.03 \times 10^{15} \, \text{photons/s} \]

5. **Calculate the Number of Photons for Visible Light:**
For visible light (wavelength \( \lambda_2 \)):
\[ E_2 = \frac{hc}{\lambda_2} = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{500 \times 10^{-9} \, \text{m}} \]
\[ E_2 = 3.976 \times 10^{-19} \, \text{J} \]
Now, using the power to find the number of photons:
\[ n_2 = \frac{P}{E_2} = \frac{100 \, \text{W}}{3.976 \times 10^{-19} \, \text{J}} \approx 2.51 \times 10^{20} \, \text{photons/s} \]

6. **Find the Ratio of Number of Photons:**
Now we can find the ratio of the number of photons of X-rays to the number of photons of visible light:
\[ \text{Ratio} = \frac{n_1}{n_2} = \frac{5.03 \times 10^{15}}{2.51 \times 10^{20}} \approx \frac{1}{500} \]

### Final Answer:
The ratio of the number of photons of X-rays to the number of photons of visible light is \( 1:500 \).

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