Problem 30 - College Physics

Question

\(\cdot\) Laser surgery. Using a mixture of \(\mathrm{CO}_{2}, \mathrm{N}_{2},\) and sometimes \(\mathrm{He}, \mathrm{CO}_{2}\) lasers emit a wavelength of 10.6\(\mu \mathrm{m} .\) At power outputs of \(0.100 \mathrm{kW},\) such lasers are used for surgery. How many photons per second does a CO \(_{2}\) laser deliver to the tissue during its use in an operation?

Step-by-Step Solution

Answer

The CO\(_2\) laser delivers approximately \(5.33 \times 10^{21}\) photons per second.

Step 1: Understand the Problem

We need to calculate the number of photons emitted per second by a laser. We know the power of the laser and the wavelength of the emitted light.

Step 2: Recall the Formula for Energy of a Photon

The energy of a single photon is given by 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 \( 3 \times 10^{8} \text{ m/s} \), and \( \lambda \) is the wavelength \( 10.6 \times 10^{-6} \text{ m} \).

Step 3: Calculate the Energy of One Photon

Using the formula, calculate the energy of one photon:\[ E = \frac{(6.626 \times 10^{-34} \text{ J s})(3 \times 10^{8} \text{ m/s})}{10.6 \times 10^{-6} \text{ m}} \approx 1.877 \times 10^{-20} \text{ J} \]

Step 4: Use Power to Find Photons Per Second

The power of the laser is the energy emitted per second. To find the number of photons emitted per second, use the relation:\[ \text{Photon count} = \frac{\text{Power}}{\text{Energy per photon}} \]Substituting the given values, the power \( P = 0.100 \text{ kW} = 100 \text{ J/s} \):\[ \text{Photon count} = \frac{100}{1.877 \times 10^{-20}} \approx 5.33 \times 10^{21} \text{ photons/s} \]

Step 5: Conclusion

The laser delivers approximately \( 5.33 \times 10^{21} \) photons per second during its use.

Key Concepts

Laser Physics

Lasers, or Light Amplification by Stimulated Emission of Radiation, are devices that emit light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. They produce a highly focused beam of light in which all the photons have the same wavelength. This beam can be very powerful and precise, making it ideal for various applications, including medical surgeries.
In the context of laser surgery, the CO extsubscript{2} laser is commonly used due to its high efficiency in cutting and vaporizing tissues. It operates at a wavelength of 10.6 micrometers, which falls in the infrared spectrum, allowing it to be absorbed well by water in tissues. This ability to concentrate energy in a small spot makes CO extsubscript{2} lasers very effective for precision surgical procedures.

• Laser light is monochromatic, meaning it consists of one color or wavelength.
• It is coherent, with all the light waves moving in unison.
• Lasers can target specific tissues, minimizing damage to surrounding areas.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics denoted by the symbol \( h \). Its value is approximately \( 6.626 \times 10^{-34} \text{ Joule seconds} \). This constant is crucial to understanding the relationship between the energy of a photon and its frequency.
Planck's constant represents the proportionality factor between the energy \( E \) of a photon and its frequency \( f \), as defined by the equation \( E = hf \). This concept is foundational in quantum theory and highlights the particle-like behavior of light, where energy is quantized and can only exist in discrete amounts.

• Planck's constant is essential for calculating the energy carried by photons.
• It underscores the quantum nature of light and matter interactions.
• Planck's work helped lay the groundwork for the development of quantum mechanics.

Wavelength and Frequency

Wavelength and frequency are two of the key components that define the characteristics of light waves. The wavelength, denoted by \( \lambda \), is the distance between successive peaks of a wave, and is usually measured in meters or micrometers. In the case of the CO extsubscript{2} laser, the wavelength is 10.6 micrometers.
Frequency, denoted by \( f \), is the number of wave cycles that pass a given point per second, measured in Hertz (Hz). The speed of light \( c \), which is approximately \( 3 \times 10^8 \text{ meters/second} \), is related to both wavelength and frequency by the equation \( c = \lambda f \).

• Higher frequency means more waves per second, typically translating to higher energy photons.
• Wavelength is inversely related to frequency; shorter wavelengths correspond to higher frequencies.
• The speed of light equation ties wavelength and frequency together, providing a foundation for photon energy calculations.

Power and Energy Conversion

Power is the rate at which energy is transferred or converted. In the context of lasers, power is measured in watts (W), where one watt equals one joule per second. The power output of the CO extsubscript{2} laser in our example is 0.100 kW, which equates to 100 joules per second.
To calculate how many photons a laser emits per second, one must first understand how energy is distributed among the photons. This begins with calculating the energy of a single photon using the equation \(E = \frac{hc}{\lambda} \), which requires Planck's constant, the speed of light, and the specific wavelength.
Once the energy of a single photon is known, the total number of photons emitted per second can be calculated using the formula \(\text{Photon count} = \frac{\text{Power}}{\text{Energy per photon}}\). This conversion allows us to determine the photon delivery rate of the laser.

• Understanding laser power helps determine the intensity and effectiveness of laser applications.
• Power calculations are essential for assessing the energy transfer during operations like laser surgery.
• Converting energy to photon count is crucial for understanding how lasers interact with materials at a quantum level.