How many photons per second are emitted by a $7.50$-mW CO₂ laser that has a wavelength of $10.6$ mm?

Verified step by step guidance

1. Determine the energy of a single photon using the formula: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \ \text{J·s} \)), \( c \) is the speed of light (\( 3.00 \times 10^8 \ \text{m/s} \)), and \( \lambda \) is the wavelength of the laser (\( 10.6 \ \text{mm} \) or \( 10.6 \times 10^{-3} \ \text{m} \)).
2. Convert the power of the laser from milliwatts to watts: \( 7.50 \ \text{mW} = 7.50 \times 10^{-3} \ \text{W} \). Power is the energy emitted per second.
3. Calculate the total energy emitted per second by the laser using the given power: \( P = \frac{E_{\text{total}}}{t} \), where \( P \) is the power and \( t \) is time (1 second in this case). Rearrange to find \( E_{\text{total}} = P \cdot t \).
4. Determine the number of photons emitted per second by dividing the total energy emitted per second by the energy of a single photon: \( N = \frac{E_{\text{total}}}{E} \), where \( N \) is the number of photons, \( E_{\text{total}} \) is the total energy emitted per second, and \( E \) is the energy of a single photon.
5. Substitute all known values into the equations and simplify to find the number of photons emitted per second. Ensure consistent units throughout the calculation.

Key Concepts

Photon Energy

The energy of a photon is determined by its wavelength and can be calculated using the formula E = hc/λ, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength in meters. For a CO2 laser with a wavelength of 10.6 mm, this relationship is crucial for determining how much energy each emitted photon carries.

Power and Energy Relationship

Power is defined as the rate at which energy is emitted or consumed, measured in watts (W). In this context, the power of the laser (7.50 mW) indicates how much energy is emitted per second. To find the number of photons emitted per second, one must relate the total power output to the energy of individual photons.

Calculating Photon Emission Rate

To find the number of photons emitted per second by the laser, divide the total power output by the energy of a single photon. This can be expressed mathematically as N = P/E, where N is the number of photons, P is the power in watts, and E is the energy of one photon. This calculation provides the rate of photon emission, which is essential for understanding the laser's performance.