Question

A laser beam of frequency \( 3.0 \times 10^{14} \) Hz produces average power of 9 mW. Find (i) the energy of a photon of the beam, and (ii) the number of photons emitted per second on an average by the source.

Hint:

Use \( E = h\nu \) for single photon energy, and divide power by energy per photon to get the photon emission rate. Convert mW to W when using SI units.

Answer 1

(i) Photon Energy:
The energy of a photon is calculated using the formula:
\[ E = h u \]
Where:
\( h = 6.63 \times 10^{-34}\ \text{J·s} \) (Planck's constant)
\( u = 3.0 \times 10^{14}\ \text{Hz} \)

\[ E = 6.63 \times 10^{-34} \times 3.0 \times 10^{14} = 1.989 \times 10^{-19}\ \text{J} \]

(ii) Photon Emission Rate:
Power represents the energy emitted per second.
Given:
\[ P = 9\ \text{mW} = 9 \times 10^{-3}\ \text{W} \]
\[ \text{Number of photons per second} = \frac{P}{E} = \frac{9 \times 10^{-3}}{1.989 \times 10^{-19}} \approx 4.52 \times 10^{16} \]

Summary:
- Energy per photon: \( 1.989 \times 10^{-19} \) J
- Photons emitted per second: \( 4.52 \times 10^{16} \)