Problem 53
Question
A helium-neon laser emits laser light at a wavelength of \(632.8 \mathrm{~nm}\) and a power of \(2.3 \mathrm{~mW}\). At what rate are photons emitted by this device?
Step-by-Step Solution
Verified Answer
The laser emits photons at a rate of approximately \(7.32 \times 10^{15}\) photons per second.
1Step 1: Calculate Energy of a Single Photon
The energy of a single photon can be calculated using the formula: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \), \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength \( 632.8 \text{ nm} = 632.8 \times 10^{-9} \text{ m} \). Thus, \( E = \frac{(6.63 \times 10^{-34})(3 \times 10^8)}{632.8 \times 10^{-9}} \approx 3.14 \times 10^{-19} \text{ J} \).
2Step 2: Calculate Number of Photons Emitted Per Second
The rate at which photons are emitted is the power of the laser divided by the energy per photon. Given the power \( P = 2.3 \text{ mW} = 2.3 \times 10^{-3} \text{ W} \), the rate of photon emission is \( n = \frac{P}{E} = \frac{2.3 \times 10^{-3}}{3.14 \times 10^{-19}} \approx 7.32 \times 10^{15} \text{ photons/second} \).
Key Concepts
Helium-Neon LaserPhoton Energy CalculationWavelength and FrequencyPhoton Emission Rate
Helium-Neon Laser
Helium-Neon lasers, often abbreviated as He-Ne lasers, are commonly used in various applications such as holography, barcode scanning, and in educational demonstrations of laser technology.
These lasers operate by exciting helium and neon ions in a gas discharge tube, where electrical energy is used to stimulate the atoms. This excitation causes the helium atoms to collide with neon atoms, transferring energy and placing the neon atoms in an excited state. When the neon atoms return to their ground state, they emit light at a wavelength of 632.8 nm, which is visible red light.
He-Ne lasers are valued for their ability to consistently produce coherent, monochromatic light with a very stable wavelength, making them ideal for precision measurement tasks.
These lasers operate by exciting helium and neon ions in a gas discharge tube, where electrical energy is used to stimulate the atoms. This excitation causes the helium atoms to collide with neon atoms, transferring energy and placing the neon atoms in an excited state. When the neon atoms return to their ground state, they emit light at a wavelength of 632.8 nm, which is visible red light.
He-Ne lasers are valued for their ability to consistently produce coherent, monochromatic light with a very stable wavelength, making them ideal for precision measurement tasks.
Photon Energy Calculation
The energy of a single photon, which is a particle of light, is found by using the formula: \[ E = \frac{hc}{\lambda} \]Here, \( h \) represents Planck's constant, \( 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \), and \( c \), the speed of light, is \( 3 \times 10^8 \text{ m/s} \).
\( \lambda \) symbolizes the wavelength, given in the problem as \( 632.8 \text{ nm} \), which we convert to meters by multiplying by \( 10^{-9} \).
This calculation provides the energy each photon carries: approximately \( 3.14 \times 10^{-19} \text{ J} \). This small energy value aligns with photons being fundamental units of light that carry quantized energy higher than the classical particles.
\( \lambda \) symbolizes the wavelength, given in the problem as \( 632.8 \text{ nm} \), which we convert to meters by multiplying by \( 10^{-9} \).
This calculation provides the energy each photon carries: approximately \( 3.14 \times 10^{-19} \text{ J} \). This small energy value aligns with photons being fundamental units of light that carry quantized energy higher than the classical particles.
Wavelength and Frequency
Wavelength and frequency are intrinsic properties of waves, including light waves such as those emitted by a helium-neon laser. While wavelength is the distance between two consecutive peaks of the wave and is measured in meters (or nanometers), frequency is the number of wave cycles that pass a point per second, measured in Hertz (Hz).
The relationship between wavelength \( \lambda \) and frequency \( u \) can be expressed using the equation:\[ c = \lambda u \] where \( c \) is the speed of light. This relationship implies that as the wavelength decreases, the frequency increases, assuming the speed of light remains constant.
The relationship between wavelength \( \lambda \) and frequency \( u \) can be expressed using the equation:\[ c = \lambda u \] where \( c \) is the speed of light. This relationship implies that as the wavelength decreases, the frequency increases, assuming the speed of light remains constant.
- For the He-Ne laser, the long wavelength of 632.8 nm corresponds to its specific red color observed.
- This characteristic wavelength signifies how laser light is differentiated from other forms, such as ultraviolet or infrared.
Photon Emission Rate
The rate of photon emission describes how many photons a light source, such as a helium-neon laser, emits in a given second. It is a crucial factor in determining the intensity of the laser, directly proportional to its power output.
This rate can be calculated by dividing the laser's power by the energy of a single photon:\[ n = \frac{P}{E} \]Using the given power of the laser, \( 2.3 \text{ mW} \), converted to watts (\( 2.3 \times 10^{-3} \text{ W} \)), over the photon's energy (\( 3.14 \times 10^{-19} \text{ J} \)), results in a photon emission rate of approximately \( 7.32 \times 10^{15} \text{ photons/second} \).
This calculation helps visualize how intensely the He-Ne laser operates, providing insight into both its power usage and beam intensity.
This rate can be calculated by dividing the laser's power by the energy of a single photon:\[ n = \frac{P}{E} \]Using the given power of the laser, \( 2.3 \text{ mW} \), converted to watts (\( 2.3 \times 10^{-3} \text{ W} \)), over the photon's energy (\( 3.14 \times 10^{-19} \text{ J} \)), results in a photon emission rate of approximately \( 7.32 \times 10^{15} \text{ photons/second} \).
This calculation helps visualize how intensely the He-Ne laser operates, providing insight into both its power usage and beam intensity.
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