Problem 2 - University Physics with Modern Physics

Question

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 nm (violet) to 750 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?

Step-by-Step Solution

Answer

(a) Wavelength: 283.58 nm to 559.70 nm. (b) Frequency: 4.00 to 7.89 x 10^14 Hz. (c) Speed: 2.24 x 10^8 m/s.

Step 1: Calculate Wavelength in Vitreous Humor

The wavelength of light in a medium is given by \( \lambda_{\text{medium}} = \frac{\lambda_{\text{air}}}{n} \), where \( \lambda_{\text{air}} \) is the wavelength in air and \( n \) is the index of refraction of the medium. For the vitreous humor, \( n = 1.34 \). - For 380 nm light in air: \[ \lambda_{\text{vitreous}} = \frac{380 \,\text{nm}}{1.34} \approx 283.58 \, \text{nm} \]- For 750 nm light in air: \[ \lambda_{\text{vitreous}} = \frac{750 \,\text{nm}}{1.34} \approx 559.70 \, \text{nm} \]Thus, the wavelength range in the vitreous humor is from approximately 283.58 nm to 559.70 nm.

Step 2: Calculate Frequency in Vitreous Humor

Frequency remains constant when light enters a new medium. The frequency of light is calculated using the equation \( f = \frac{c}{\lambda_{\text{air}}} \), where \( c \) is the speed of light in a vacuum (approximately \( 3 \times 10^{8} \,\text{m/s} \)). - For 380 nm light: \[ f = \frac{3 \times 10^{8} \, \text{m/s}}{380 \times 10^{-9} \, \text{m}} \approx 7.89 \times 10^{14} \, \text{Hz} \]- For 750 nm light: \[ f = \frac{3 \times 10^{8} \, \text{m/s}}{750 \times 10^{-9} \, \text{m}} \approx 4.00 \times 10^{14} \, \text{Hz} \]Thus, the frequency range of light in the vitreous humor remains \( 4.00 \times 10^{14} \, \text{Hz} \) to \( 7.89 \times 10^{14} \, \text{Hz} \).

Step 3: Calculate Speed of Light in Vitreous Humor

The speed of light in a medium is given by \( v = \frac{c}{n} \), where \( c \) is the speed of light in vacuum and \( n \) is the index of refraction. For the vitreous humor:\[ v = \frac{3 \times 10^{8} \text{ m/s}}{1.34} \approx 2.24 \times 10^{8} \, \text{m/s} \]Hence, the speed of light in the vitreous humor is approximately \( 2.24 \times 10^{8} \, \text{m/s} \).

Key Concepts

Index of Refraction

The index of refraction, represented by the symbol \( n \), plays a crucial role in optical physics. It is the ratio of the speed of light in a vacuum to the speed of light in a given medium. When light enters a medium other than a vacuum, its speed decreases, depending on the medium's index of refraction. For instance, in the exercise, the vitreous humor has an index of refraction of 1.34. This means that light travels 1.34 times slower in the vitreous humor than it does in a vacuum. The index of refraction is key as it tells us how much the path of light is bent, or refracted, when entering a new medium. It provides essential information for calculating other optical properties like wavelength and speed within that medium.

Wavelength Calculation

Wavelength is an important characteristic of waves like light. When light travels through a medium, its wavelength changes, but its frequency remains constant. In the given problem, we calculate the wavelength of light in the vitreous humor using the formula:\[ \lambda_{\text{medium}} = \frac{\lambda_{\text{air}}}{n} \]where \( \lambda_{\text{air}} \) is the wavelength in air, and \( n \) is the index of refraction of the medium. - For violet light (380 nm) entering the vitreous humor, the resulting wavelength is approximately 283.58 nm.- For red light (750 nm), the resulting wavelength in the vitreous humor is about 559.70 nm.These calculations show us how the medium impacts the wavelength, compressing it compared to when it travels through air.

Light Frequency

Light frequency, denoted by \( f \), is the number of wave cycles that pass a point per unit time. This is a fundamental property that does not change when light transitions between different media. Frequency can be calculated using the equation:\[ f = \frac{c}{\lambda_{\text{air}}} \]where \( c \) is the speed of light in a vacuum.- For violet light (380 nm in air), the frequency is roughly \( 7.89 \times 10^{14} \text{ Hz} \).- For red light (750 nm), the frequency is about \( 4.00 \times 10^{14} \text{ Hz} \).These frequencies remain constant even as light enters the vitreous humor. Understanding this constancy helps in analyzing how light behaves across different environments.

Speed of Light in Medium

Light's speed diminishes when it passes from one medium into another, due to the medium's index of refraction. To find the speed of light within a medium, we use the formula:\[ v = \frac{c}{n} \]where \( c \) is the speed of light in a vacuum.In the vitreous humor, for example, since \( n = 1.34 \), the speed of light becomes approximately \( 2.24 \times 10^{8} \text{ m/s} \). This decreased speed compared to its pace in a vacuum explains why lenses are effective – they can bend light to focus images. It highlights how different substances can tailor the paths and speed of light traveling through them.

Visible Light Spectrum

The visible light spectrum encompasses the range of wavelengths that the human eye can perceive, from about 380 nm to 750 nm. These include colors from violet to red. Each wavelength corresponds to a specific color, and this range of colors is what we experience as visible light. As light enters different mediums, such as the vitreous humor in the eye, although wavelengths vary, the spectrum itself remains unchanged. The eyes' structures are adapted to these specific wavelengths, allowing us to see. Understanding how visible light interacts with various media enhances comprehension of phenomena such as refraction and lenses in everyday life and advanced optical technologies.