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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 weld the detached portions 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

01

Calculate Energy Delivered per Pulse

To find the energy delivered with each pulse, we use the formula for energy in terms of power and time:\[ E = P \times t \]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 duration of the pulse.Substituting the given values, we get:\[ E = 0.250 \times 1.50 \times 10^{-3} = 3.75 \times 10^{-4} \text{ J} \]Therefore, the energy delivered per pulse is \(3.75 \times 10^{-4} \text{ J}\).

02

Calculate the Average Pressure Exerted by the Laser Beam

The average pressure exerted by the laser beam can be calculated using the formula:\[ p = \frac{I}{c} \]where \( p \) is the pressure, \( I \) is the intensity, and \( c \) is the speed of light (~ \(3 \times 10^8 \text{ m/s}\)). The intensity is given by:\[ I = \frac{P}{A} \]where \( A = \pi \left(\frac{d}{2}\right)^2 \) is the area of the circular spot with diameter \( d = 510 \times 10^{-6} \text{ m} \).Calculating the area,\[ A = \pi \left(\frac{510 \times 10^{-6}}{2}\right)^2 = \pi \times (255 \times 10^{-6})^2 = 2.043 \times 10^{-7} \text{ m}^2 \]Now, substituting into the intensity equation:\[ I = \frac{0.250}{2.043 \times 10^{-7}} = 1.223 \times 10^6 \text{ W/m}^2 \]Substituting into the pressure equation:\[ p = \frac{1.223 \times 10^6}{3 \times 10^8} \approx 4.076 \times 10^{-3} \text{ N/m}^2 \]Thus, the average pressure is approximately \(4.076 \times 10^{-3} \text{ N/m}^2\).

03

Calculate the Wavelength Inside the Eye

The wavelength of light when it enters a medium is given by:\[ \lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{n} \]where \( \lambda_{\text{vacuum}} = 810 \text{ nm} = 810 \times 10^{-9} \text{ m} \) and \( n = 1.34 \).Substituting the values:\[ \lambda_{\text{medium}} = \frac{810 \times 10^{-9}}{1.34} = 604.48 \times 10^{-9} \text{ m} = 604.48 \text{ nm} \]Thus, the wavelength inside the eye is approximately \(604.48 \text{ nm}\).

04

Calculate the Frequency of the Laser Light

The frequency of light remains unchanged when it enters a new medium. Therefore, we first determine its frequency in a vacuum:\[ f = \frac{c}{\lambda_{\text{vacuum}}} \]Substituting the values:\[ f = \frac{3 \times 10^8}{810 \times 10^{-9}} = 3.704 \times 10^{14} \text{ Hz} \]The frequency of the laser light inside the vitreous humor is still \(3.704 \times 10^{14} \text{ Hz} \).

05

Calculate the Maximum Electric Field

The maximum value of the electric field can be calculated from the intensity using the relation:\[ I = \frac{1}{2} c \varepsilon_0 E_0^2 \]where \( \varepsilon_0 \approx 8.85 \times 10^{-12} \text{ F/m} \) is the permittivity of free space.Rearranging for \( E_0 \):\[ E_0 = \sqrt{\frac{2I}{c \varepsilon_0}} \]Substitute the intensity \( I = 1.223 \times 10^6 \text{ W/m}^2 \):\[ E_0 = \sqrt{\frac{2 \times 1.223 \times 10^6}{3 \times 10^8 \times 8.85 \times 10^{-12}}} \approx 2.436 \times 10^5 \text{ V/m} \]Thus, the maximum electric field is approximately \(2.436 \times 10^5 \text{ V/m}\).

06

Calculate the Maximum Magnetic Field

The maximum magnetic field \( B_0 \) is related to the electric field \( E_0 \) by:\[ B_0 = \frac{E_0}{c} \]Substituting \( E_0 = 2.436 \times 10^5 \text{ V/m} \):\[ B_0 = \frac{2.436 \times 10^5}{3 \times 10^8} = 8.12 \times 10^{-4} \text{ T} \]Thus, the maximum magnetic field is approximately \(8.12 \times 10^{-4} \text{ T}\).

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Key Concepts

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

Energy Calculation

When we talk about energy calculation in the context of laser physics, we're interested in determining how much energy is delivered by a laser pulse. For a single laser pulse, energy is dependent on both the power of the laser and the duration of the pulse.
The formula for calculating energy is simple:

• Energy (E) is given by \( E = P \times t \),
• where \( P \) is the power and \( t \) is the time duration of the pulse.

In our problem, the laser power is 250 mW (or 0.250 W) and the pulse lasts for 1.50 ms (or 1.50 x 10^-3 seconds). By multiplying these values, we can determine that each pulse delivers about 3.75 x 10^-4 Joules of energy to the retina. This calculated energy is crucial for understanding how laser treatments functioning in medical procedures.

Intensity and Pressure

The intensity of the laser beam is another important factor in laser physics as it affects how much pressure is exerted by the beam. Intensity relates the power of a laser beam to the area over which it is spread. This is calculated using the formula:

• Intensity (I) = \( \frac{P}{A} \)
• where \( A \) is the area over which the beam is spread.

In our exercise, the beam's spot is a circle with a diameter of 510 μm which translates to an area of approximately 2.043 x 10^-7 m². Using this area and the given power, we calculate intensity as approximately 1.223 x 10⁶ W/m².
Pressure exerted by the laser beam is then calculated through:

• Pressure (p) = \( \frac{I}{c} \)
• where \( c \) is the speed of light (approximately 3 x 10⁸ m/s).

This results in a pressure of about 4.076 x 10^-3 N/m². Understanding the relationship between intensity and pressure can help predict how the laser will interact with different surfaces.

Wavelength and Frequency

Wavelength and frequency are fundamental properties of light that describe how it propagates through space. When light enters a medium like the vitreous humor of the eye, its wavelength changes while its frequency remains constant.
The new wavelength inside the medium can be found by dividing the original wavelength in a vacuum by the index of refraction of the medium:

• New wavelength = \( \frac{\lambda_{\text{vacuum}}}{n} \)

For our exercise, the original wavelength is 810 nm and the index of refraction is 1.34, resulting in a new wavelength of approximately 604.48 nm.
The frequency of the light remains unchanged and is calculated in a vacuum as:

• Frequency (f) = \( \frac{c}{\lambda_{\text{vacuum}}} \)

Using the given wavelength, the frequency is about 3.704 x 10¹⁴ Hz. Understanding how wavelength and frequency relate allows us to comprehend the behavior of laser light as it passes through different materials.

Electric and Magnetic Fields

The electric and magnetic fields of a laser beam describe its electromagnetic nature. These fields oscillate perpendicularly to each other and to the direction of the beam's travel. The maximum values of these fields are linked to the intensity of the beam.
For the electric field (E), the relationship with intensity is:

• \( I = \frac{1}{2} c \varepsilon_0 E^2 \)

where \( \varepsilon_0 \) is the permittivity of free space. Rearranging this equation allows us to solve for the maximum electric field:

• \( E = \sqrt{\frac{2I}{c \varepsilon_0}} \)

Using the provided data, this results in a maximum electric field of about 2.436 x 10⁵ V/m.
Similarly, the maximum magnetic field (B) can be found from the electric field:

• \( B = \frac{E}{c} \)

Which calculates to approximately 8.12 x 10^-4 T. These calculations enable us to describe the electromagnetic properties of the laser beam, which are crucial for understanding its interactions with materials.