Problem 69 - College Physics

Question

\(\bullet$$\bullet\) Laser surgery. Very short pulses of high-intensity laser beams are used to repair detached portions of the retina of the eye. The brief pulses of energy absorbed by the retina welds the detached portion back into place. In one such procedure, a laser beam has a wavelength of 810 \(\mathrm{nm}\) and delivers 250 \(\mathrm{mW}\) of power spread over a circular spot 510\(\mu \mathrm{m}\) in diameter. The vitreous humor (the transparent fluid that fills most of the eye) has an index of refraction of 1.34 . (a) If the laser pulses are each 1.50 \(\mathrm{ms}\) long, how much energy is delivered to the retina with each pulse? (b) What average pressure does the pulse of the laser beam exert on the retina as it is fully absorbed by the circular spot? (c) What are the wavelength and frequency of the laser light inside the vitreous humor of the eye? (d) What are the maximum values of the electric and magnetic fields in the laser beam?

Step-by-Step Solution

Answer

(a) 0.375 J; (b) 6.12 Pa; (c) 604 nm, 3.70 x 10^14 Hz; (d) 136 kV/m, 0.453 mT

Step 1: Calculate Energy Delivered per Pulse

The energy delivered per pulse can be calculated using the formula: \[ E = P imes t \]where \( E \) is the energy, \( P = 250 \text{ mW} = 0.250 \text{ W} \) is the power, and \( t = 1.50 \text{ ms} = 1.50 \times 10^{-3} \text{ s} \) is the pulse duration. Plugging in the values, we get:\[ E = 0.250 \times 1.50 \times 10^{-3} = 0.375 \text{ J} \] Thus, each pulse delivers 0.375 joules of energy.

Step 2: Calculate Average Pressure on Retina

The average pressure \( P \) exerted by the laser can be calculated with:\[ P = \frac{F}{A} \]where \( F \) is the force and \( A \) is the area. The force exerted is equal to the change in momentum per unit time, which is given by the energy absorbed divided by the speed of light \( c \).\[ F = \frac{E}{c} = \frac{0.375}{3 \times 10^8} \text{ N} \]The area \( A \) of the circular spot is:\[ A = \pi \left( \frac{d}{2} \right)^2 = \pi \left( \frac{510 \times 10^{-6}}{2} \right)^2 = 2.043 \times 10^{-7} \text{ m}^2 \]Therefore, the average pressure is:\[ P = \frac{0.375}{3 \times 10^8 \times 2.043 \times 10^{-7}} = 6.12 \times 10^{-3} \text{ N/m}^2 \]

Step 3: Compute Wavelength and Frequency in Vitreous Humor

The wavelength inside the vitreous humor, \( \lambda_{in} \), can be calculated by:\[ \lambda_{in} = \frac{\lambda_{0}}{n} \]where \( \lambda_{0} = 810 \text{ nm} = 810 \times 10^{-9} \text{ m} \) is the original wavelength and \( n = 1.34 \) is the index of refraction:\[ \lambda_{in} = \frac{810 \times 10^{-9}}{1.34} = 6.04 \times 10^{-7} \text{ m} \]The frequency \( f \) of the light does not change and is given by:\[ f = \frac{c}{\lambda_{0}} = \frac{3 \times 10^8}{810 \times 10^{-9}} = 3.70 \times 10^{14} \text{ Hz} \]

Step 4: Calculate Maximum Electric and Magnetic Fields

The maximum electric field \( E_{0} \) can be found using the relation:\[ E_{0} = \sqrt{\frac{2P_{av}}{\epsilon_{0} c}} \]where \( P_{av} = \frac{250 \text{ mW}}{\pi (510 \times 10^{-6}/2)^2} \), \( \epsilon_{0} \approx 8.85 \times 10^{-12} \text{ C}^2/\text{N m}^2 \) is the permittivity of free space.Calculating \( P_{av} \):\[ P_{av} = \frac{0.250}{2.043 \times 10^{-7}} = 1.22 \times 10^6 \text{ W/m}^2 \]\[ E_{0} = \sqrt{\frac{2 \times 1.22 \times 10^6}{8.85 \times 10^{-12} \times 3 \times 10^8}} \approx 1.36 \times 10^5 \text{ V/m} \]The maximum magnetic field \( B_{0} \) can be found by:\[ B_{0} = \frac{E_{0}}{c} = \frac{1.36 \times 10^5}{3 \times 10^8} = 4.53 \times 10^{-4} \text{ T} \]

Key Concepts

Retina Repair Using Lasers

Laser surgery has revolutionized the way doctors can repair delicate parts of the human body, such as the retina of the eye. The retina can become detached, which can lead to vision problems. In laser surgery, precise beams with high energy are utilized to weld the detached retina back in place.

This process involves targeting a very small area with a short pulse of laser light. The energy from the laser is absorbed by the retina, creating a small amount of heat that essentially "welds" the retina back into its normal location.

Though the idea might sound complex, it is quite straightforward thanks to the focused nature of laser beams. The key aspects that allow for such precision include the ability to control the laser's energy pulse and the size of the beam. By using specific wavelengths, only targeted tissues are affected, which is essential in tissues as sensitive as those within the eye.

This technique is highly effective, offering a minimally invasive solution to conditions like retinal detachment, leading to better outcomes for patient vision.

Laser Beam Characteristics

One of the defining features of laser beams used in surgical procedures is their precision. This precision is a result of several key characteristics that are unique to laser light.

First, laser beams are highly collimated, meaning the light waves travel parallel to each other over long distances without spreading out. This is critical for focusing on small targets like the retina without affecting surrounding tissues.

Secondly, laser beams are coherent, which means that the light waves are in phase with each other. This coherence allows the laser light to be extremely intense and focused, contributing to its effectiveness in medical applications.

Additionally, lasers have monochromatic light, indicating that they emit light of a single wavelength. This property is important because different tissues in the body absorb different wavelengths to varying degrees. Therefore, having control over the wavelength of the laser beam allows for precise targeting, minimizing damage to non-targeted areas.

These characteristics distinguish lasers from other light sources, making them an indispensable tool in modern surgery.

Wavelength in Mediums

When considering the passage of light through different media, it's important to understand how the wavelength is affected. Laser surgery in the eye involves the beam passing through the vitreous humor, which affects its wavelength due to the medium's refractive index.

The wavelength of light changes when it enters a different medium due to this refractive index, which measures how much the speed of light is reduced inside the medium. In the case of the vitreous humor, with an index of refraction of 1.34, the wavelength of the laser light is shortened.

Mathematically, the wavelength in the medium, \( \lambda_{in} \), is given by:\[ \lambda_{in} = \frac{\lambda_{0}}{n} \],
where \( \lambda_{0} \) is the original wavelength in a vacuum, and \( n \) is the refractive index.

Despite this change in wavelength, the frequency of the laser light remains constant across media, which is crucial for maintaining the desired energy and focus of the laser as it moves through different structures within the eye.

Electric and Magnetic Fields in Lasers

Lasers not only entail light waves but also include electric and magnetic fields that play a role in their characteristics and effects. The electromagnetic nature of lasers means they have properties of both electric and magnetic fields, interacting with charged particles.

The electric field in a laser beam, represented as \( E_0 \), indicates the strength of the electric component at any point in space, explaining how powerful the laser can be. The magnetic field component, denoted as \( B_0 \), is typically weaker. However, it complements the electric field by influencing the direction and propagation of the light wave.

These fields in a laser are maximized and calculated through their power and area, and their interaction with materials is crucial in processes like retinal repair. One can calculate \( E_0 \) and \( B_0 \) using: \[ E_{0} = \sqrt{\frac{2P_{av}}{\epsilon_{0} c}} \] \[ B_{0} = \frac{E_{0}}{c} \]
Understanding these fields helps in determining how the laser light transfers energy to the targeted tissues, ensuring safety and efficacy in medical treatments like laser surgery on the eye.