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A laser used to weld detached retinas emits light with a wavelength of 652 nm in pulses that are 20.0 ms in duration. The average power during each pulse is 0.600 W. (a) How much energy is in each pulse in joules? In electron volts? (b) What is the energy of one photon in joules? In electron volts? (c) How many photons are in each pulse?

Short Answer

Expert verified
(a) 0.012 J, 7.49e+16 eV. (b) 3.05e-19 J, 1.91 eV. (c) 3.93e+16 photons.

Step by step solution

01

Understand Power and Energy Relationship

The average power during each pulse is given as 0.600 W, and the pulse duration is 20.0 ms. To calculate energy, use the equation \( E = P \times t \), where \( E \) is energy (in joules), \( P \) is power (in watts), and \( t \) is time (in seconds). Convert time from milliseconds to seconds: \( 20.0 \text{ ms} = 0.020 \text{ s} \).
02

Calculate Energy in Joules

Using the formula \( E = P \times t \), substitute \( P = 0.600 \text{ W} \) and \( t = 0.020 \text{ s} \) to find \( E \). \[ E = 0.600 \times 0.020 = 0.012 \text{ J} \].
03

Convert Joules to Electron Volts

To convert energy from joules to electron volts, use the conversion factor: \( 1 \text{ J} = 6.242 \times 10^{18} \text{ eV} \). \[ E = 0.012 \times 6.242 \times 10^{18} = 7.4904 \times 10^{16} \text{ eV} \].
04

Determine Energy of One Photon in Joules

Use the equation for the energy of a photon, \( E_{\text{photon}} = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \) (Planck's constant), \( c = 3.00 \times 10^8 \text{ m/s} \) (speed of light), and \( \lambda = 652 \text{ nm} = 652 \times 10^{-9} \text{ m} \) (wavelength). \[ E_{\text{photon}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{652 \times 10^{-9}} \approx 3.05 \times 10^{-19} \text{ J} \].
05

Convert Energy of One Photon to Electron Volts

Use the conversion \( 1 \text{ J} = 6.242 \times 10^{18} \text{ eV} \) to find the energy in electron volts. \[ E_{\text{photon}} = 3.05 \times 10^{-19} \times 6.242 \times 10^{18} \approx 1.91 \text{ eV} \].
06

Calculate Number of Photons in Each Pulse

Divide the total energy per pulse by the energy of one photon to find the number of photons: \[ \text{Number of Photons} = \frac{E}{E_{\text{photon}}} = \frac{0.012}{3.05 \times 10^{-19}} \approx 3.93 \times 10^{16} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laser Physics
The study of laser physics revolves around the fascinating process of producing light that is coherent, monochromatic, and highly focused. Lasers work on the principle of stimulating atoms or molecules to emit light of a particular wavelength. This emission then gets amplified to produce a concentrated beam. This is possible due to the nature of photons, which are particles of light that can be controlled in lasers to form precise and focused beams.
Lasers have diverse applications, and one critical use is in medical fields like ophthalmology. For instance, in the case of welding detached retinas, the laser emits light with a specific wavelength, which is utilized to perform delicate surgical procedures.
  • Laser light has a single wavelength, which means it is monochromatic.
  • It remains focused over long distances due to the collimated beam.
  • Laser beams are coherent, meaning the light waves are synchronized.
The precision of laser physics enables it to be used safely and effectively in surgeries without damaging surrounding tissues. Understanding this foundational concept is crucial when deciphering the energy involved in laser-based applications.
Energy Conversion
In the context of lasers, energy conversion refers to how we transform electric power into photon energy. This transformation is essential because it allows us to quantify the energy output of a laser pulse. For example, when you know that a laser emits light pulses with an average power of 0.600 W over 20.0 ms intervals, you can calculate the total energy produced in each pulse.
The relationship between power, energy, and time is expressed in the equation:
\( E = P \times t \).
Where:
  • \( E \) is energy (in joules).
  • \( P \) is power (in watts).
  • \( t \) is time (in seconds).
By using this formula and converting units appropriately, you can convert the energy in joules to electron volts (eV), which is another useful unit of energy in the context of photons and laser applications. One joule is equivalent to approximately \( 6.242 \times 10^{18} \) eV, allowing easy conversion between mechanical energy and electromagnetically stored energy.
Photon Quantity Calculation
Once you have calculated the energy per pulse from a laser, the next step is to determine how many photons that energy equates to. This can be achieved through understanding the energy of an individual photon.
Photons are tiny packets of light energy, each carrying an energy that can be determined with the equation:
\( E_{\text{photon}} = \frac{hc}{\lambda} \),where:
  • \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J} \cdot \text{s})\).
  • \( c \) is the speed of light \((3.00 \times 10^8 \text{ m/s})\).
  • \( \lambda \) is the wavelength in meters.
To find the number of photons, divide the total energy of the pulse by the energy of a single photon:
\[ \text{Number of Photons} = \frac{E}{E_{\text{photon}}} \].
This calculation yields the number of photons emitted in each pulse, shedding light on the quantum nature of light and providing insight into how lasers achieve their power and precision.

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Most popular questions from this chapter

A 2.50-W beam of light of wavelength 124 nm falls on a metal surface. You observe that the maximum kinetic energy of the ejected electrons is 4.16 eV. Assume that each photon in the beam ejects a photoelectron. (a) What is the work function (in electron volts) of this metal? (b) How many photoelectrons are ejected each second from this metal? (c) If the power of the light beam, but not its wavelength, were reduced by half, what would be the answer to part (b)? (d) If the wavelength of the beam, but not its power, were reduced by half, what would be the answer to part (b)?

If a photon of wavelength 0.04250 nm strikes a free electron and is scattered at an angle of 35.0\(^\circ\) from its original direction, find (a) the change in the wavelength of this photon; (b) the wavelength of the scattered light; (c) the change in energy of the photon (is it a loss or a gain?); (d) the energy gained by the electron.

A photon with wavelength 0.1100 nm collides with a free electron that is initially at rest. After the collision the wavelength is 0.1132 nm. (a) What is the kinetic energy of the electron after the collision? What is its speed? (b) If the electron is suddenly stopped (for example, in a solid target), all of its kinetic energy is used to create a photon. What is the wavelength of this photon?

Consider Compton scattering of a photon by a \(moving\) electron. Before the collision the photon has wavelength \(\lambda\) and is moving in the +\(x\)-direction, and the electron is moving in the -\(x\)-direction with total energy \(E\) (including its rest energy \(mc^2\)). The photon and electron collide head-on. After the collision, both are moving in the -\(x\)-direction (that is, the photon has been scattered by 180\(^\circ\)). (a) Derive an expression for the wavelength \(\lambda'\) of the scattered photon. Show that if \(E \gg mc^2\), where m is the rest mass of the electron, your result reduces to $$\lambda' = {hc \over E} (1 + {m^2c^4\lambda \over 4hcE}) $$ (b) A beam of infrared radiation from a CO\(_2\) laser (\(\lambda = 10.6 \mu{m}\)) collides head-on with a beam of electrons, each of total energy \(E\) = 10.0 GeV (1 GeV = 10\(^9\) eV). Calculate the wavelength \(\lambda'\) of the scattered photons, assuming a 180\(^\circ\) scattering angle. (c) What kind of scattered photons are these (infrared, microwave, ultraviolet, etc.)? Can you think of an application of this effect?

X rays are produced in a tube operating at 24.0 kV. After emerging from the tube, x rays with the minimum wavelength produced strike a target and undergo Compton scattering through an angle of 45.0\(^\circ\). (a) What is the original x-ray wavelength? (b) What is the wavelength of the scattered x rays? (c) What is the energy of the scattered x rays (in electron volts)?

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