Question

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 :

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

Answer 1

The Correct Option is D

To determine the ratio of the number of photons emitted by each source, we need to first understand how the energy of light is related to the number of photons and their wavelength. 

The energy \(E\) of a single photon is given by the equation:

\(E = \frac{hc}{\lambda}\)

where:

• \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \text{ J s}\)
• \(c\) is the speed of light, approximately \(3 \times 10^{8} \text{ m/s}\)
• \(\lambda\) is the wavelength of the light

The power \(P\) of the light source is related to the number of photons emitted per second \(N\) by:

\(P = N \times E = N \times \frac{hc}{\lambda}\)

Since both sources emit light with a power of 200 W, the equation for each source is:

\(200 = N_{1} \times \frac{hc}{300 \times 10^{-9}}\)

and

\(200 = N_{2} \times \frac{hc}{500 \times 10^{-9}}\)

We are tasked with finding the ratio \(\frac{N_{1}}{N_{2}}\):

1. Solve the first equation for \(N_{1}\):

\(N_{1} = \frac{200 \cdot 300 \times 10^{-9}}{hc}\)

1. Solve the second equation for \(N_{2}\):

\(N_{2} = \frac{200 \cdot 500 \times 10^{-9}}{hc}\)

1. Calculate the ratio \(\frac{N_{1}}{N_{2}\):

\(\frac{N_{1}}{N_{2}} = \left(\frac{200 \cdot 300 \times 10^{-9}}{hc}\right) \div \left(\frac{200 \cdot 500 \times 10^{-9}}{hc}\right)\)

\(\frac{N_{1}}{N_{2}} = \frac{300}{500}\)

\(\frac{N_{1}}{N_{2}} = \frac{3}{5}\)

Thus, the ratio of the number of photons emitted by the two light sources is 3:5. Therefore, the correct answer is 3:5.

Answer 2

Let \( n_1 \) and \( n_2 \) be the number of photons emitted by the sources with wavelengths \( \lambda_1 = 300 \, \text{nm} \) and \( \lambda_2 = 500 \, \text{nm} \), respectively.

Step 1. Calculate the energy of photons for each wavelength:  The power emitted by each source is given as 200 W, so:  
 
  \(n_1 \times \frac{hc}{\lambda_1} = 200\)
  
 \(n_2 \times \frac{hc}{\lambda_2} = 200\)
  
Step 2. Formulate the ratio:  

  \(\frac{n_1}{n_2} = \frac{\lambda_2}{\lambda_1}\)

  Substituting values:  

  \(\frac{n_1}{n_2} = \frac{300}{500}\)
 

  Simplifying, we get:  

  \(\frac{n_1}{n_2} = \frac{3}{5}\)
 

Therefore, the ratio of the number of photons emitted by each source is 3 : 5.

The Correct Answer is: 3:5