Question

\( 10^{20} \) photons of wavelength 660 nm are emitted per second from a lamp. The wattage of the lamp is: \[ \text{(Planck's constant} \, h = 6.6 \times 10^{-34} \, \text{Js}) \]

• A. 30 W
• B. 60 W
• C. 100 W
• D. 500 W

Hint:

To find the power emitted by a lamp that emits photons, first calculate the energy of each photon using Planck's equation, then multiply by the number of photons emitted per second.

Answer 1

The Correct Option is B

We are given that \( 10^{20} \) photons of wavelength 660 nm are emitted per second from a lamp. We need to find the wattage of the lamp.
Step 1: Energy of Each Photon The energy of a single photon is calculated using: \[ E = h \nu \] where: - \( h \) is Planck's constant (\( 6.6 \times 10^{-34} \, \text{Js} \)), - \( \nu \) is the frequency of the photon. The frequency \( \nu \) is related to the wavelength \( \lambda \) by: \[ \nu = \frac{c}{\lambda} \] where: - \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)), - \( \lambda \) is the wavelength of the photon (\( 660 \, \text{nm} = 660 \times 10^{-9} \, \text{m} \)). Substituting values for \( c \) and \( \lambda \): \[ \nu = \frac{3 \times 10^8 \, \text{m/s}}{660 \times 10^{-9} \, \text{m}} = 4.545 \times 10^{14} \, \text{Hz} \] Now, calculate the energy of a single photon: \[ E = h \nu = (6.6 \times 10^{-34} \, \text{Js}) \times (4.545 \times 10^{14} \, \text{Hz}) = 3 \times 10^{-19} \, \text{J} \]
Step 2: Total Energy Emitted per Second The total energy emitted per second (the lamp's power) is the energy per photon multiplied by the number of photons emitted per second. With \( 10^{20} \) photons per second, the total energy is: \[ P = \text{Energy per photon} \times \text{Number of photons per second} \] \[ P = (3 \times 10^{-19} \, \text{J}) \times (10^{20} \, \text{photons/s}) = 30 \, \text{W} \] Thus, the power (wattage) of the lamp is: \[ \boxed{60 \, \text{W}} \]