Problem 54 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A laser used to weld detached retinas puts out 23 -mslong pulses of 640 -nm light which average 0.63 -W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain?

Step-by-Step Solution

Answer

The energy per pulse is 0.01449 J, and each pulse contains approximately \(4.67 \times 10^{16}\) photons.

Step 1: Convert Pulse Duration to Seconds

First, convert milliseconds to seconds because the standard unit of time in physics is seconds. The pulse duration is given as 23 ms. To convert this to seconds, divide by 1000: \[ 23 \text{ ms} = \frac{23}{1000} \text{ s} = 0.023 \text{ s} \]

Step 2: Calculate Energy Per Pulse

The energy per pulse can be calculated using the formula \( E = P \cdot t \), where \( E \) is the energy, \( P \) is the power, and \( t \) is the time duration of the pulse. Substituting the given values (\( P = 0.63 \text{ W} \) and \( t = 0.023 \text{ s} \)): \[ E = 0.63 \text{ W} \times 0.023 \text{ s} = 0.01449 \text{ J} \]

Step 3: Calculate the Energy of a Single Photon

Each photon has energy given by the formula \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \) is Planck's constant, \( c = 3 \times 10^{8} \text{ m/s} \) is the speed of light, and \( \lambda = 640 \times 10^{-9} \text{ m} \) is the wavelength. Calculate: \[ E_{\text{photon}} = \frac{(6.626 \times 10^{-34}) \times (3 \times 10^{8})}{640 \times 10^{-9}} \approx 3.1 \times 10^{-19} \text{ J} \]

Step 4: Calculate Number of Photons Per Pulse

Using the total energy per pulse and the energy of a single photon, calculate the number of photons per pulse: \[ n = \frac{E_{\text{pulse}}}{E_{\text{photon}}} = \frac{0.01449}{3.1 \times 10^{-19}} \approx 4.67 \times 10^{16} \]

Key Concepts

Wavelength and Frequency

Wavelength and frequency are fundamental concepts in understanding light and electromagnetic waves. Wavelength, often denoted by the symbol \( \lambda \), is the distance between successive peaks of a wave. It is typically measured in meters or nanometers. Frequency, on the other hand, represented by \( f \), is the number of complete wave cycles that pass a point in one second, measured in Hertz (Hz). These two properties of waves are inversely related. The formula that connects them is \( c = \lambda \times f \), where \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \).

• A shorter wavelength means a higher frequency, as more waves pass a given point each second.
• Conversely, a longer wavelength results in a lower frequency.

Understanding the relationship between wavelength and frequency is crucial in photon energy calculations, as both properties determine the energy carried by light.

Energy of a Photon

The energy of a photon is determined by its wavelength or frequency. This energy can be calculated using Planck's equation: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of the photon, \( h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \) is Planck's constant, \( c = 3 \times 10^{8} \text{ m/s} \) is the speed of light, and \( \lambda \) is the wavelength of light.

• The shorter the wavelength, the higher the energy of the photon.
• Inversely, a longer wavelength indicates a lower energy photon.

This concept is especially important in calculations involving lasers, where the energy output can be significant, even from a small number of high-energy photons. Being able to precisely determine the energy of a single photon allows scientists to calculate the number of photons present in a given light pulse, which is key in laser applications.

Laser Physics

Lasers are devices that emit light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. They are unique in their ability to produce coherent light, which means the light waves are in phase and have the same frequency and wavelength. This coherence allows lasers to focus intensely and deliver significant amounts of energy over a small area, as in the case of retinal welding lasers. Lasers are characterized by:

• Their wavelength (color of the laser beam), which determines the photon energy.
• Output power, which can be controlled and varies depending on the specific application.

Laser physics utilizes the distinctive properties of lasers to apply them in various fields such as medicine, telecommunications, and manufacturing. By understanding how laser light is generated and manipulated, we can harness its power for tasks that require precision and accuracy, such as retinal welding, where controlled energy is vital for delicate procedures.