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Chapter 37: Problem 1

The speed of light in water is (3/4)c. What is the effect, on the frequency and wavelength of light, of passing from vacuum (or air, to good approximation) into water? Compute the refractive index of water. The same number of wave peaks leave the air each second as enter into the water. Hence, the frequency is the same in the two materials. But because Wavelength \(=(\) Speed \() /(\) Frequency \()\), the wavelength in water is three fourths that in air. The (absolute) refractive index of water is $$ n=\frac{\text { Speed in vacuum }}{\text { Speed in water }}=\frac{c}{(3 / 4) \mathrm{c}}=\frac{4}{3}=1.33 $$

Short Answer

Expert verified
The frequency remains the same, the wavelength decreases by 3/4, and the refractive index of water is 1.33.

Step by step solution

01

Understanding the Problem

The speed of light changes when it passes from one medium to another. In this problem, the speed of light in water is given as \((3/4)c\), where \(c\) is the speed of light in vacuum. We need to find how this change in speed affects the frequency and wavelength and calculate the refractive index of water.
02

Frequency of Light

The frequency of light remains constant when it moves from one medium to another. Therefore, the frequency of light in water is the same as in vacuum or air. This is because the number of wave peaks entering the water per second is the same as the number of wave peaks in air.
03

Wavelength of Light

We use the relationship \(\text{Wavelength} = \text{Speed} / \text{Frequency}\) to determine the effect on wavelength. Since the speed of light in water is \((3/4)\) of its speed in vacuum, the wavelength in water is also \((3/4)\) of its wavelength in air.
04

Calculating Refractive Index

The refractive index \(n\) is calculated as the ratio of the speed of light in vacuum to its speed in water. Using the formula: \[ n = \frac{c}{(3/4)c} = \frac{4}{3} = 1.33 \] This indicates how much the light slows down when entering water.

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Key Concepts

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

Speed of Light
The speed of light is a fundamental concept in physics, representing how fast light travels in a vacuum. This speed, denoted by the symbol \(c\), is approximately 299,792,458 meters per second. This speed is specifically for a vacuum, the emptiest environment possible, free from any atoms or molecules that might interact with the light.

However, when light enters a different medium, such as water, its speed decreases. This occurs because the light waves interact with the particles in the medium, which slows them down. In water, the light's speed is \((3/4)c\), meaning it travels slower than in a vacuum.
  • This reduction in speed leads to the phenomenon of refraction.
  • Despite this change in speed, the light's frequency remains constant.
Understanding the change in speed helps comprehend how light behaves in various materials.
Frequency
Frequency is the number of wave peaks that pass a point in one second, measured in hertz (Hz). It's an important property of light and has a direct impact on what we see as color.

When light travels between media with different densities, such as from air to water, the frequency does not change. This constancy arises because the source of light continually emits the same number of wave peaks per second, regardless of the medium.
  • Since frequency is unchanged, the color of the light also remains unchanged when light passes between different media.
  • The frequency is connected to energy; higher frequencies mean higher energy.
Thus, understanding frequency helps clarify why certain optical properties are preserved even when light slows down.
Wavelength in Different Media
Wavelength is the physical length of one complete oscillation of a wave. It is related to the speed and frequency of the light according to the formula: \( \text{Wavelength} = \text{Speed} / \text{Frequency} \).

Because the speed of light changes when it enters a different medium, so does the wavelength. In our example, light entering water from air, the speed reduces to \((3/4)c\), and so the wavelength in water is also reduced to three-fourths of its original value in air.
  • Shorter wavelengths in a denser medium are why underwater objects often appear closer than they are.
  • This proportional reduction maintains the equality in the equation \( \text{Speed} = \text{Frequency} \times \text{Wavelength} \).
Understanding how wavelengths change helps in designing lenses and understanding various optical effects seen in everyday life.

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Most popular questions from this chapter

A glass plate is \(0.60 \mathrm{~cm}\) thick and has a refractive index of \(1.55 .\) How long does it take for a pulse of light incident normally to pass through the plate? $$ t=\frac{x}{v}=\frac{0.0060 \mathrm{~m}}{\left(2.998 \times 10^{8} / 1.55\right) \mathrm{m} / \mathrm{s}}=3.1 \times 10^{-11} \mathrm{~s} $$

A laser beam is incident in air on the surface of a thick flat sheet of glass having an index of refraction of \(1.500\). The beam within the glass travels at an angle of \(35.0^{\circ}\) from the normal. Determine the angle of incidence at the air-glass interface. [Hint: Recall Snell's Law. Here \(\theta_{t}=35.0^{\circ}\), and we need to find \(\theta_{i}\), which should be greater than that.]

What is the critical angle when light passes from glass \((n=1.50)\) into air?

A thick layer of olive oil, having an index of refraction of 1.47, is floating on a quantity of pure water. A narrow beam of light in the oil arrives at the oil-water interface at an angle of \(50.0^{\circ}\) with respect to the normal. At what angle measured from the normal does the beam progress into the water? [Hint: Here \(\theta_{i}=50.0^{\circ}\), and we need to find \(\theta_{t}\), which should be greater than that. Since the indices don't differ by much, the two angles should be close.]

A thick layer of olive oil, having an index of refraction of 1.47, is floating on a quantity of pure water. A narrow beam of light in the water arrives at the water-oil interface at an angle of \(50.0^{\circ}\) with respect to the normal. At what angle measured from the normal does the beam progress into the oil? [Hint: Here \(\theta_{i}=\) \(50.0^{\circ}\), and we need to find \(\theta_{t}\), which should be less than that. Since the indices don't differ by much, the two angles should be close.]
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