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 ascertain the ratio of photons emitted per source, the relationship between light energy, photon count, and wavelength must be established. The energy \(E\) of an individual photon is defined by the equation:

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

where:

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

The power \(P\) of a light source is proportional to the number of photons emitted per second \(N\), as follows:

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

Given that both sources exhibit a power of 200 W, the equations for each source are:

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

and

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

The objective is to compute the ratio \(\frac{N_{1}}{N_{2}}\). The procedure involves:

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

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

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

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

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

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

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

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

Consequently, the photon emission ratio for the two light sources is 3:5, confirming the answer as 3:5.