Question

A $200\, W$ sodium street lamp emits yellow light of wavelength $0.6 \, \mu m. $ Assuming it to be $25 \% $ efficient in converting electrical energy to light, the number of photons of yellow light it emits per second is

• A. $1.5 \times 10^{20} $
• B. $6 \times 10^{18} $
• C. $62 \times 10^{20} $
• D. $3 \times 10^{19} $

Answer 1

The Correct Option is A

To find the number of photons emitted per second by the sodium street lamp, we need to follow these steps:

1. Determine the amount of energy converted into light per second. Since the sodium lamp has a power of 200 \, \text{W} and is 25\% efficient, the energy converted into light is: E_{\text{light}} = 200 \, \text{W} \times 0.25 = 50 \, \text{W} .
2. Convert the power used for light into energy per second (in Joules), since power is defined as energy per time and 1 W = 1 J/s: E_{\text{light}} = 50 \, \text{J/s} .
3. Calculate the energy of a single photon using the formula: E_{\text{photon}} = \frac{hc}{\lambda}, where:
   • h = 6.63 \times 10^{-34} \, \text{J s} (Planck's constant)
   • c = 3 \times 10^8 \, \text{m/s} (speed of light)
   • \lambda = 0.6 \, \mu\text{m} = 0.6 \times 10^{-6} \, \text{m} (wavelength)
   So, E_{\text{photon}} = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{0.6 \times 10^{-6}} \approx 3.315 \times 10^{-19} \, \text{J} .
4. Calculate the number of photons emitted per second by dividing the total energy by the energy of one photon: \text{Number of photons} = \frac{E_{\text{light}}}{E_{\text{photon}}} = \frac{50}{3.315 \times 10^{-19}} \approx 1.5 \times 10^{20} .

Thus, the number of photons of yellow light emitted per second is 1.5 \times 10^{20}.

The correct answer is: 1.5 \times 10^{20}.

This explains why the other options, such as 6 \times 10^{18}, 62 \times 10^{20}, and 3 \times 10^{19}, are incorrect as they do not match the calculated number of photons emitted.