Problem 2 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

Wavelength: 284-560 nm; Frequency: 4.00-7.89 x 10^{14} Hz; Speed: 2.24 x 10^8 m/s

Step 1: Determine the Wavelength in the Vitreous Humor

The wavelength of light changes as it enters a medium other than air, and can be calculated using the formula:\[\lambda' = \frac{\lambda}{n}\]where \(\lambda\) is the wavelength in air, \(\lambda'\) is the wavelength in the medium, and \(n\) is the index of refraction. For the vitreous humor, \(n = 1.34\). Therefore, the wavelength range within the vitreous humor is:\[\lambda'_{\text{violet}} = \frac{380\, \text{nm}}{1.34} \approx 283.58\, \text{nm}\]\[\lambda'_{\text{red}} = \frac{750\, \text{nm}}{1.34} \approx 559.70\, \text{nm}\]Thus, the wavelength range in the vitreous humor is approximately 284 nm to 560 nm.

Step 2: Determine the Frequency of Light in the Vitreous Humor

The frequency of light does not change as it enters a new medium. Therefore, the frequency remains the same as in the air. Frequency can be calculated using the formula:\[f = \frac{c}{\lambda}\]where \(c = 3 \times 10^8\, \text{m/s}\) is the speed of light in vacuum, and \(\lambda\) is the wavelength in air.\[f_{\text{violet}} = \frac{3 \times 10^8\, \text{m/s}}{380 \times 10^{-9}\, \text{m}} \approx 7.89 \times 10^{14}\, \text{Hz}\]\[f_{\text{red}} = \frac{3 \times 10^8\, \text{m/s}}{750 \times 10^{-9}\, \text{m}} \approx 4.00 \times 10^{14}\, \text{Hz}\]Thus, the frequency range is approximately 4.00 to 7.89 x 10^{14} Hz.

Step 3: Determine the Speed of Light in the Vitreous Humor

The speed of light in a medium can be calculated using the formula:\[v = \frac{c}{n}\]For the vitreous humor with \(n = 1.34\):\[v = \frac{3 \times 10^8\, \text{m/s}}{1.34} \approx 2.24 \times 10^8\, \text{m/s}\]Thus, the speed of light in the vitreous humor is approximately 2.24 x 10^8 m/s.

Key Concepts

Wavelength

The concept of wavelength is crucial for understanding how light behaves when moving through different media. Wavelength refers to the distance between successive peaks of a wave, which is crucial in identifying different types of light. In air, visible light has wavelengths ranging from 380 nanometers (nm) for violet light to 750 nm for red light.

However, when light enters a different medium, such as the vitreous humor in the eye, its speed and wavelength change due to the medium's index of refraction. The wavelength inside a medium like the vitreous humor can be calculated using the formula:

• \[ \lambda' = \frac{\lambda}{n} \]

Here, \( \lambda \) represents the original wavelength in air, while \( \lambda' \) is the wavelength within the medium, and \( n \) is the medium's index of refraction.

For the vitreous humor with an index of refraction of 1.34, the wavelength of violet light decreases from 380 nm to approximately 284 nm, and red light decreases from 750 nm to approximately 560 nm when it enters the medium. This shift is what helps the eye to refocus light onto the retina.

Frequency

Frequency is a measure of how many times a wave passes a point in one second. It is measured in Hertz (Hz) and remains unchanged as light moves between different media. Unlike wavelength, frequency does not depend on the medium, since it is defined by the light source itself.

To determine the frequency of light, the formula used is:

• \[ f = \frac{c}{\lambda} \]

where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength in air. For violet light at 380 nm, the frequency is approximately \( 7.89 \times 10^{14} \text{ Hz} \), and for red light at 750 nm, it is approximately \( 4.00 \times 10^{14} \text{ Hz} \).

These values remain constant even as light passes through different media such as the vitreous humor, signifying the consistent nature of frequency in light propagation.

Index of Refraction

The index of refraction, denoted as \( n \), is a dimensionless number that describes how light propagates through a medium. It is the ratio of the speed of light in vacuum \( c \) to its speed in a medium \( v \), given by the equation:

• \[ n = \frac{c}{v} \]

A higher index indicates that light travels slower in the medium compared to vacuum. In the context of the eye, the vitreous humor has an index of refraction of 1.34.

This value indicates that light slows down when it enters the vitreous humor, leading to phenomena such as focusing within the eye. The speed of light drops from \( 3 \times 10^8 \text{ m/s} \) in vacuum to approximately \( 2.24 \times 10^8 \text{ m/s} \) in the vitreous humor. This slowing down of light helps in bending the paths of the light rays precisely to converge onto the retina, thereby forming clear images.