Problem 4 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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: \( 20.0 \text{ ms} = 0.020 \text{ s} \).

Step 2: 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 \( E \). \[ E = 0.600 \times 0.020 = 0.012 \text{ J} \].

Step 3: Convert Joules to Electron Volts

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

Step 4: 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). \[ E_{\text{photon}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{652 \times 10^{-9}} \approx 3.05 \times 10^{-19} \text{ J} \].

Step 5: 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. \[ E_{\text{photon}} = 3.05 \times 10^{-19} \times 6.242 \times 10^{18} \approx 1.91 \text{ eV} \].

Step 6: 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: \[ \text{Number of Photons} = \frac{E}{E_{\text{photon}}} = \frac{0.012}{3.05 \times 10^{-19}} \approx 3.93 \times 10^{16} \].

Key Concepts

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.