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, which is the total power emitted by the lamp.
Step 1: Energy of Each Photon The energy of a single photon can be found using the formula: \[ E = h \nu \] where: - \( h \) is Planck's constant (\( 6.6 \times 10^{-34} \, \text{Js} \)), - \( \nu \) is the frequency of the photon. The frequency \( \nu \) can be related to the wavelength \( \lambda \) using the formula: \[ \nu = \frac{c}{\lambda} \] where: - \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)), - \( \lambda \) is the wavelength of the photon (\( 660 \, \text{nm} = 660 \times 10^{-9} \, \text{m} \)). Substitute the 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 (which is the power of the lamp) is the energy of a single photon multiplied by the number of photons emitted per second. Since \( 10^{20} \) photons are emitted 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: \[ \boxed{60 \, \text{W}} \]