Given two points sources having same power of 200W.One source is emitting photons of `lambda_1 = 500 nm ` and other emitting X-rey 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 1: Understand the relationship between power, energy, and photons
Power (P) is defined as the rate of energy emitted per unit time. According to Planck's quantum theory, the energy (E) of a single photon can be expressed as:
\[ E = \frac{hc}{\lambda} \]
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 of the emitted photon.

### Step 2: Relate power to the number of photons emitted
The total power emitted by a source can be expressed in terms of the number of photons emitted per second (\( n \)):
\[ P = n \cdot E \]
Substituting the expression for energy:
\[ P = n \cdot \frac{hc}{\lambda} \]
From this, we can express the number of photons emitted per second as:
\[ n = \frac{P \cdot \lambda}{hc} \]

### Step 3: Calculate the number of photons for both sources
For both sources, we know the power is the same (200 W). We can calculate the number of photons emitted per second for each source using their respective wavelengths.

1. For the first source (\( \lambda_1 = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)):
\[ n_1 = \frac{200 \cdot (500 \times 10^{-9})}{(6.626 \times 10^{-34})(3 \times 10^8)} \]

2. For the second source (\( \lambda_2 = 1 \, \text{nm} = 1 \times 10^{-9} \, \text{m} \)):
\[ n_2 = \frac{200 \cdot (1 \times 10^{-9})}{(6.626 \times 10^{-34})(3 \times 10^8)} \]

### Step 4: Find the ratio of photons emitted
To find the ratio of photons emitted from both sources, we can simplify:
\[ \frac{n_1}{n_2} = \frac{\frac{200 \cdot (500 \times 10^{-9})}{(6.626 \times 10^{-34})(3 \times 10^8)}}{\frac{200 \cdot (1 \times 10^{-9})}{(6.626 \times 10^{-34})(3 \times 10^8)}} \]
The constants and power cancel out:
\[ \frac{n_1}{n_2} = \frac{500 \times 10^{-9}}{1 \times 10^{-9}} = 500 \]

### Final Result
The ratio of the number of photons emitted from the first source to the second source is:
\[ \frac{n_1}{n_2} = 500 \]