Question

How many photons per second are emitted by a 7.50-mW CO\(_2\) laser that has a wavelength of 10.6 \(\mu\)m?

Short answer

The laser emits approximately \( 3.99 \times 10^{16} \) photons per second.

Step-by-step solution

Identify and Note Down the Given Information

Firstly, we need to note the given values. The power of the laser is \( P = 7.50 \, \text{mW} = 7.50 \times 10^{-3} \, \text{W} \). The wavelength of the laser is \( \lambda = 10.6 \, \mu\text{m} = 10.6 \times 10^{-6} \, \text{m} \).

Calculate the Energy of a Single Photon

The energy \( E \) of a single photon can be calculated using the formula: \( E = \frac{hc}{\lambda} \), where \( h = 6.63 \times 10^{-34} \, \text{J s} \) is Planck's constant and \( c = 3.00 \times 10^{8} \, \text{m/s} \) is the speed of light. Substituting the values, we get \( E = \frac{6.63 \times 10^{-34} \times 3.00 \times 10^{8}}{10.6 \times 10^{-6}} \approx 1.88 \times 10^{-20} \, \text{J} \).

Calculate the Number of Photons Emitted Per Second

The number of photons emitted per second is calculated by dividing the power of the laser by the energy of a single photon: \( N = \frac{P}{E} = \frac{7.50 \times 10^{-3}}{1.88 \times 10^{-20}} \approx 3.99 \times 10^{16} \).

Conclusion

The laser emits approximately \( 3.99 \times 10^{16} \) photons per second.

Key concepts

Planck's Constant

Planck's Constant is a fundamental constant used in quantum mechanics that relates the energy of a photon to its frequency. Denoted by the symbol \( h \), its value is approximately \( 6.63 \times 10^{-34} \, \text{J s} \). This tiny value highlights the minute energies involved in individual photons compared to everyday objects.
In the formula \( E = \frac{hc}{\lambda} \), Planck's Constant \( h \) is used along with the speed of light \( c \) and the wavelength \( \lambda \) to determine the energy \( E \) of a single photon. The interplay of these constants allows us to translate the seemingly abstract concept of wave properties into tangible energy values. Understanding this relationship is crucial to grasp topics like the photoelectric effect and quantum theory.
Key points about Planck's Constant include:

• Bridges a gap between the energy and frequency of electromagnetic waves.
• Essential in explaining how energy is quantized in quantum mechanics.
• Integral in calculations involving photon emissions in lasers and other devices.

Understanding and correctly applying Planck's Constant can profoundly impact how we compute and conceptualize energy transformations at the microscopic level.

Photon Energy

Photon Energy refers to the energy carried by a single photon, the basic unit of light and other electromagnetic radiation. It is a direct result of the frequency and wavelength associated with that photon.
The relationship between these properties is given by the formula \( E = \frac{hc}{\lambda} \). By substituting the constants for Planck's Constant \( h \) and the speed of light \( c \), as well as the wavelength \( \lambda \), you can calculate the energy \( E \) of a photon in joules. This formula highlights how the energy of a photon is inversely proportional to its wavelength. Therefore, shorter wavelengths imply higher energy photons.
Important aspects of calculating Photon Energy include:

• The equation's dependence on accurate measurements of the listed constants.
• How varying wavelengths affect energy levels.
• Its role in technologies like lasers and solar panels, where precise energy levels are crucial.

Mastering the understanding of photon energy is fundamental in both theoretical physics and applied technologies, making it a key concept for students and scientists alike.

Laser Power

Laser Power is a measure of the total energy output rate of a laser, typically given in watts (W) or milliwatts (mW). It indicates how much energy is delivered over a period, often per second.
In the context of photon emission, laser power directly relates to how many photons are emitted by the laser each second. By knowing the laser power and the energy per photon, one can determine the emission rate using the formula \( N = \frac{P}{E} \), where \( N \) is the number of photons per second, \( P \) is the power of the laser, and \( E \) is the energy of a single photon.
Considerations when dealing with Laser Power:

• Specified in either watts or a fraction thereof, such as milliwatts, for precision in measurement.
• Critical in applications like communications, manufacturing, and medical treatments.
• Influences the capability of a laser to perform diverse functions based on the number of photons it emits.

Understanding laser power helps clarify how laser systems work and optimize their usage in both experimental and practical applications, emphasizing its significance in both scientific inquiry and industry practices.

Wavelength

Wavelength is the distance between successive crests of a wave, especially points in a sound wave or electromagnetic wave. For photons, and particularly in the case of laser emissions, wavelength is inherently tied to both the energy formula and photon emission rates.
The wavelength of a laser, noted usually in micrometers (\( \mu m \)) or meters (\( m \)), provides critical details about the laser's color and the energy each photon carries. A laser's wavelength directly determines the energy per photon through the equation \( E = \frac{hc}{\lambda} \), illustrating the inverse relationship between wavelength and energy.
Central points regarding wavelength include:

• A shorter wavelength corresponds to a higher energy photon.
• Essential in determining a laser's efficiency, effectiveness, and specific application scope.
• Varies across regions like ultraviolet, visible, infrared, impacting its practical applications.

Grasping how wavelength influences photon properties is vital for interpreting the behavior of light in numerous scientific and technological contexts, strengthening one's understanding of electromagnetic theories and practices.