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\(\bullet\) Light inside the eye. The vitreous humor, a transparent, gelatinous fluid that fills most of the eyeball, has an index of refraction of 1.34 . Visible light ranges in wavelength from 400 \(\mathrm{nm}\) (violet) to \(700 \mathrm{nm}(\mathrm{red}),\) as measured in air. This light travels through the vitreous humor and strikes the rods and cones at the surface of the retina. What are the ranges of (a) the wavelength, (b) the frequency, and (c) the speed of the light just as it approaches the retina within the vitreous humor?

Short Answer

Expert verified
Wavelength in vitreous humor: 298.51-522.39 nm. Frequency: 4.29-7.50 x 10^{14} Hz. Speed: 2.24 x 10^8 m/s.

Step by step solution

01

Understanding Refractive Index

The refractive index of the vitreous humor is given as \(n = 1.34\). The refractive index relates the speed of light in a medium to its speed in a vacuum according to the formula \(c = n \cdot v\), where \(c\) is the speed of light in a vacuum (approximately \(3.00 \times 10^8\) m/s), and \(v\) is the speed of light in the medium.
02

Finding the Speed of Light in the Vitreous Humor

To find the speed of light in the vitreous humor, we use the formula: \(v = \frac{c}{n}\). Substituting the values: \[v = \frac{3.00 \times 10^8 \text{ m/s}}{1.34} \approx 2.24 \times 10^8 \text{ m/s}\]. Therefore, the speed of light in the vitreous humor is approximately \(2.24 \times 10^8\) m/s.
03

Calculating the Wavelength in Vitreous Humor

The wavelength of light in a medium is given by \(\lambda' = \frac{\lambda}{n}\), where \(\lambda\) is the wavelength in air. For violet light (400 nm), \(\lambda'_\text{violet} = \frac{400 \text{ nm}}{1.34} \approx 298.51 \text{ nm}\). For red light (700 nm), \(\lambda'_\text{red} = \frac{700 \text{ nm}}{1.34} \approx 522.39 \text{ nm}\). Thus, the wavelength range in the vitreous humor is approximately 298.51 nm to 522.39 nm.
04

Calculating the Frequency of Light in Vitreous Humor

The frequency, \(f\), of light remains constant when transitioning between mediums. It is given by the formula \(f = \frac{c}{\lambda}\). For violet light, \(f_\text{violet} = \frac{3.00 \times 10^8 \text{ m/s}}{400 \times 10^{-9} \text{ m}} = 7.50 \times 10^{14} \text{ Hz}\). For red light, \(f_\text{red} = \frac{3.00 \times 10^8 \text{ m/s}}{700 \times 10^{-9} \text{ m}} \approx 4.29 \times 10^{14} \text{ Hz}\). Thus, the frequency range remains \(4.29 \times 10^{14}\) Hz to \(7.50 \times 10^{14}\) Hz.

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Key Concepts

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

Visible Light Wavelengths
Visible light is the part of the electromagnetic spectrum that can be detected by the human eye. It encompasses a range of wavelengths that include all the colors we perceive in a rainbow. In air, these wavelengths range from approximately 400 nanometers (nm) for violet light to 700 nm for red light. This spectrum is crucial as it defines the colors we see in our daily environment.
When light passes from air into another medium such as water or glass, its wavelength changes but its frequency remains constant. This concept is important to understand how light behaves in different surroundings. In the case of the vitreous humor found in the eye, this change in wavelength plays a significant role in how we detect and process visual information. Think of it as light adjusting its stride when stepping from one terrain to another.
Speed of Light in Mediums
The speed of light is not a constant 3.00 x 10^8 meters per second (m/s) when it's traveling through different materials; rather, it changes based on the medium. The speed of light in a vacuum is indeed about 3.00 x 10^8 m/s, but this speed is reduced in denser mediums.
In any given medium, the speed of light can be calculated using the formula: \[ v = \frac{c}{n} \]
Here, \( v \) is the speed of light in the medium, \( c \) is the speed of light in vacuum, and \( n \) is the refractive index of that medium. This index provides a measure of how much light slows down in a medium. For example, inside the vitreous humor of the eye, light travels at approximately 2.24 x 10^8 m/s. This slowdown is a critical factor in understanding how light is bent or refracted when it moves from one medium to another.
Vitreous Humor
The vitreous humor is the clear, gel-like substance filling the space between the lens and the retina in the eyeball. It maintains the eye's shape and allows light to pass through so it can reach the retina. The refractive index of the vitreous humor, which is about 1.34, indicates how much it slows down light passing through it compared to in a vacuum.
Understanding the vitreous humor's role is essential in comprehending how visual processing occurs. It affects the path and speed of light, ensuring it hits the retina correctly for proper image formation. Essentially, the vitreous humor plays a silent but essential role in helping us see the world clearly.
Wave Frequency
Frequency is a measure of how many waves pass a fixed point in a second. It is denoted in hertz (Hz) and can be thought of as the number of crests or troughs passing a given point per second. For visible light, the frequency defines the 'color' of light.
Interestingly, when light transitions between different mediums, its frequency remains unchanged. This means that regardless of the medium, the frequency of violet light remains about 7.50 x 10^14 Hz, and that of red light is about 4.29 x 10^14 Hz. Maintaining its frequency ensures that light maintains its color, regardless of how its path or speed may be altered.
Wave Speed
When discussing wave speed, it is crucial to note that it depends on both the medium and the wavelength. Wave speed can be calculated with the formula:\[ v = f \times \lambda \]
where \( v \) is the wave speed, \( f \) is the frequency, and \( \lambda \) is the wavelength.
In a vacuum, the speed of light is consistent, but in a medium like the vitreous humor, this speed changes due to the medium's refractive index. This altered wave speed affects how images are formed inside the eye, confirming the intertwined relationship between speed, wavelength, and medium properties. Wave speed is pivotal in understanding how light travels and bends, impacting our visual perception.

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Most popular questions from this chapter

\(\bullet$$\bullet\) A thin layer of ice \((n=1.309)\) floats on the surface of water \((n=1.333)\) in a bucket. A ray of light from the bottom of the bucket travels upward through the water. (a) What is the largest angle with respect to the normal that the ray can make at the ice-water interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

\(\bullet$$\bullet\) You want to support a sheet of fireproof paper horizontally, using only a vertical upward beam of light spread uniformly over the sheet. There is no other light on this paper. The sheet measures 22.0 \(\mathrm{cm}\) by 28.0 \(\mathrm{cm}\) and has a mass of 1.50 \(\mathrm{g}\) . (a) If the paper is black and hence absorbs all the light that hits it, what must be the intensity of the light beam? (b) For the light in part (a), what are the maximum values of its electric and magnetic fields? (c) If the paper is white and hence reflects all the light that hits it, what intensity of light beam is needed to support it? (d) To see if it is physically reasonable to expect to support a sheet of paper this way, calculate the intensity in a typical 0.500 \(\mathrm{mW}\) laser beam that is 1.00 \(\mathrm{mm}\) in diameter and compare this value with your answer in part (a).

\(\bullet$$\bullet\) A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is \(\theta\) , the intensity of the emerging beam is \(I\) . If you instead want the intensity to be \(I / 2,\) what should be the angle (in terms of \(\theta )\) between the polarizing angle of the filter and the original direction of polarization of the light?

\(\bullet\) The critical angle for total internal reflection at a liquid-air interface is \(42.5^{\circ} .\) (a) If a ray of light traveling in the liquid has an angle of incidence of \(35.0^{\circ}\) at the interface, what angle does the refracted ray in the air make with the normal? (b) If a ray of light traveling in air has an angle of incidence of \(35.0^{\circ}\) at the interface, what angle does the refracted ray in the liquid make with the normal?

\(\bullet$$\bullet\) A parallel-sided plate of glass having a refractive index of 1.60 is in contact with the surface of water in a tank. A ray coming from above makes an angle of incidence of \(32.0^{\circ}\) with the top surface of the glass. What angle does this ray make with the normal in the water?

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