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Chapter 33: Problem 2

BIO 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 \(380 \mathrm{nm}\) (violet) to \(750 \mathrm{nm}\) (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
(a) Wavelength within the vitreous humor ranges from \(283.6 \, nm\) (violet) to \(559.7 \, nm\) (red), (b) Frequency remains the same as in the air, ranging from \(7.89 \times 10^{14} Hz\) (violet) to \(4.00 \times 10^{14} Hz\) (red) and (c) The speed of light within the vitreous humor is \(2.24 \times 10^8 m/s\).

Step by step solution

01

Determine wavelength in vitreous humor

First, calculate the wavelength of light within the vitreous humor. You can use the equation: \(\lambda_{medium} = \frac{\lambda_{air}}{n}\) where \(\lambda_{medium}\) is the wavelength in the medium and \(\lambda_{air}\) is the wavelength in air, and \(n\) is the refractive index of the medium. For violet light, \(\lambda_{violet,medium} = \frac{380\text{ nm}}{1.34} = 283.6 \text{ nm}\). For red light, \(\lambda_{red,medium} = \frac{750\text{ nm}}{1.34} = 559.7 \text{ nm}\).
02

Determine frequency in vitreous humor

The frequency of light does not change when it moves from one medium to another. Therefore, the frequencies of the red and violet light in the vitreous humor are same as in air. We can calculate the frequency using the equation \(f = \frac{c}{\lambda_{air}}\) where \(c\) is the speed of light in vacuum. Using the given wavelengths in air, we find the frequencies in air and hence vitreous humor, to be \(f_{violet} = \frac{3 \times 10^8 m/s}{380 \times 10^{-9} m} = 7.89 \times 10^{14} Hz\) and \(f_{red} = \frac{3 \times 10^8 m/s}{750 \times 10^{-9} m} = 4.00 \times 10^{14} Hz\).
03

Determine speed in vitreous humor

Finally, calculate the speed of light within the vitreous humor using the index of refraction. The equation is \(v = \frac{c}{n}\), so the speed of light in the vitreous humor is \(v = \frac{3 \times 10^8 m/s}{1.34} = 2.24 \times 10^8 m/s\).

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Key Concepts

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

Index of Refraction
The index of refraction is a critical concept in understanding how light behaves as it travels through different media, such as the vitreous humor in the eye. It is a measure of how much light slows down in a medium compared to its speed in a vacuum. Mathematically, it is defined as: \[ n = \frac{c}{v} \]where
  • \(n\) is the index of refraction,
  • \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8\) m/s),
  • \(v\) is the speed of light in the medium.

In the case of vitreous humor, the index of refraction is about 1.34.
This means that light travels slower in the eye's vitreous humor than it does in a vacuum.
The more the light slows down, the larger the index of refraction of the medium.
Wavelength
Wavelength refers to the distance between consecutive peaks of a wave.
In the context of light, wavelength determines the color we perceive.
When light enters a different medium, like the vitreous humor, its wavelength changes according to the medium's index of refraction.the wavelength in the medium is calculated as: \[ \lambda_{medium} = \frac{\lambda_{air}}{n} \]Where
  • \(\lambda_{air}\) is the wavelength of light in air,
  • \(n\) is the refractive index.

For vitreous humor, with an index of 1.34, if the original wavelength of violet light is 380 nm, it changes to approximately 283.6 nm. Similarly, red light changes from 750 nm to roughly 559.7 nm.
Frequency
Frequency is a term that describes how many wave crests pass a certain point in a given amount of time.
It's typically measured in hertz (Hz), where 1 Hz equals one wave per second.Importantly, the frequency of light remains unchanged as it moves between different materials.
Using the formula \[f = \frac{c}{\lambda_{air}}\], we can find the frequency of light in air and it remains the same when the medium changes. For example:
  • Violet light with a wavelength of 380 nm in air has a frequency of roughly \(7.89 \times 10^{14} \text{ Hz}\).
  • Red light measuring 750 nm in air has a frequency of about \(4.00 \times 10^{14} \text{ Hz}\).
Speed of Light
The speed of light is an important constant in physics, denoted by \(c\), and is approximately \(3 \times 10^8\) meters per second in a vacuum. However, when it enters another medium, its speed
changes. The speed of light in any medium can be described using the equation: \[v = \frac{c}{n}\]Where
  • \(v\) is the speed of light in the medium,
  • \(n\) is the medium's index of refraction.

In the vitreous humor, with an index of 1.34, the speed of light becomes approximately \(2.24 \times 10^8\) m/s.
The decrease in speed is due to the interaction of light with the material's particles, which cause it to slow down slightly before moving onward.
Vitreous Humor
The vitreous humor is a clear, gel-like substance filling most of the eye's interior.
It helps to maintain the eye's shape and provides a pathway for light to reach the retina.
  • The vitreous humor is located between the lens and the retina.
  • It has an index of refraction of about 1.34, which affects how light bends as it travels through.
This component is crucial for clear vision as it must remain transparent to allow unhindered light passage.
The vitreous humor protects the retina by cushioning it against shocks and vibrations.
Any changes or damages to the vitreous humor can impact vision, highlighting its role in focusing light precisely on the retina.

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Most popular questions from this chapter

A layer of liquid sits on top of the horizontal surface of a transparent solid. For a ray traveling in the solid and incident on the interface of the two materials, the critical angle is \(38.7^{\circ}\). (a) For a ray traveling in the solid and reflecting at the interface with the liquid, for what incident angle with respect to the normal is the reflected ray \(100 \%\) polarized? (b) What is the polarizing angle if the ray is traveling in the liquid?

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?

BIO Heart Sonogram. Physicians use high-frequency \((f=1-5 \mathrm{MHz})\) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is \(1480 \mathrm{~m} / \mathrm{s}\) in muscle and \(344 \mathrm{~m} / \mathrm{s}\) in air. We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell's law then applies to the refraction of sound waves. (a) At what angle from the normal does an ultrasound beam enter the heart if it leaves the lungs at an angle of \(9.73^{\circ}\) from the normal to the heart wall? (Assume that the speed of sound in the lungs is \(344 \mathrm{~m} / \mathrm{s}\).) (b) What is the critical angle for sound waves in air incident on muscle?

A ray of light traveling in water is incident on an interface with a flat piece of glass. The wavelength of the light in the water is \(726 \mathrm{nm}\) and its wavelength in the glass is \(544 \mathrm{nm}\). If the ray in water makes an angle of \(56.0^{\circ}\) with respect to the normal to the interface, what angle does the refracted ray in the glass make with respect to the normal?

BIO Seeing Polarized Light. Some insect eyes have two types of cells that are sensitive to the plane of polarization of light. In a simple model, one cell type (type \(\mathrm{H}\) ) is sensitive to horizontally polarized light only, and the other cell type (type \(\mathrm{V}\) ) is sensitive to vertically polarized light only. To study the responses of these cells, researchers fix the insect in a normal, upright position so that one eye is illuminated by a light source. Then several experiments are carried out. First, light with a plane of polarization at \(45^{\circ}\) to the horizontal shines on the insect. Which statement is true about the two types of cells? (a) Both types detect this light. (b) Neither type detects this light. (c) Only type \(\mathrm{H}\) detects the light. (d) Only type \(\mathrm{V}\) detects the light.
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