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Chapter 33: Problem 3

A beam of light has a wavelength of 650 nm in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.477 (b) What is the wavelength of these waves in the liquid?

Short Answer

Expert verified
(a) Speed in the liquid: 2.03 x 10^8 m/s; (b) Wavelength in the liquid: 440.15 nm.

Step by step solution

01

Identifying Given Variables

We are given the wavelength in vacuum \(\lambda_0 = 650\,\text{nm}\) and the index of refraction \(n = 1.477\). We need to find the speed of light \(v\) in the liquid and the wavelength \(\lambda\) in the liquid.
02

Using the Refractive Index Formula

The speed of light in a medium is given by \(v = \frac{c}{n}\), where \(c = 3 \times 10^8\,\text{m/s}\) is the speed of light in vacuum. Substitute \(n = 1.477\) into the formula to find \(v\).
03

Calculating the Speed of Light in the Liquid

\[ v = \frac{3 \times 10^8\,\text{m/s}}{1.477} \approx 2.03 \times 10^8\,\text{m/s} \] So, the speed of light in the liquid is approximately \(2.03 \times 10^8\,\text{m/s}\).
04

Finding the Wavelength in the Liquid

Use the relationship \(\lambda = \frac{\lambda_0}{n}\) to find the wavelength of light in the liquid. Substitute \(\lambda_0 = 650\,\text{nm}\) and \(n = 1.477\) into this formula.
05

Calculating the Wavelength in the Liquid

\[ \lambda = \frac{650\,\text{nm}}{1.477} \approx 440.15\,\text{nm} \] Therefore, the wavelength of the light in the liquid is approximately \(440.15\,\text{nm}\).

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Key Concepts

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

Understanding the Index of Refraction
The index of refraction, often denoted as \( n \), is a fundamental concept in physics that helps us understand how light behaves as it travels through different materials. It's a measure that compares the speed of light in a vacuum to its speed in another medium. Every material has its own index of refraction, which essentially tells us how much slower light travels in that material compared to in a vacuum.
  • For example, the index of refraction of water is about 1.33, which means light travels 1.33 times slower in water than in a vacuum.
  • In our exercise, a liquid has an index of refraction of 1.477. This tells us that light travels 1.477 times slower in this liquid.
The higher the index of refraction, the more the light slows down. This slowing down of light when it enters a medium is what causes it to bend, a phenomenon known as refraction. This property is crucial for designing lenses and understanding optical phenomena.
Speed of Light in Different Media
The speed of light is a constant in a vacuum, approximately \(3 \times 10^8 \text{ m/s}\). However, this speed changes when light enters a medium, decreasing based on the medium's index of refraction. To calculate the speed of light in a different medium, we use the formula:\[ v = \frac{c}{n} \]where \( v \) is the speed of light in the medium, \( c \) is the speed of light in a vacuum, and \( n \) is the index of refraction.
  • In our problem, the medium is a liquid with an index of refraction 1.477.
  • Plugging in the values, the speed of light in the liquid becomes \( v = \frac{3 \times 10^8 \text{ m/s}}{1.477} \).
  • This gives us a speed of approximately \( 2.03 \times 10^8 \text{ m/s} \).
Understanding how light speed changes in different media is essential for designing glasses, cameras, and even scientific instruments. It affects how images are focused and how we perceive objects underwater or through glass.
Wavelength in Medium
The wavelength of light changes when it travels from one medium to another, and this concept is intertwined with the index of refraction. In a vacuum, light has its fundamental wavelength, but this wavelength decreases when light enters a medium. The formula to calculate the wavelength in a medium is:\[ \lambda = \frac{\lambda_0}{n} \]where \( \lambda \) is the wavelength in the medium, \( \lambda_0 \) is the wavelength in a vacuum, and \( n \) is the index of refraction.
  • For our problem, the initial wavelength in vacuum is 650 nm, and the index of refraction of the liquid is 1.477.
  • Thus, the new wavelength in the liquid is \( \lambda = \frac{650 \text{ nm}}{1.477} \).
  • This calculates to about 440.15 nm.
When light enters a medium and its wavelength changes, it affects various properties such as color and the way we see light patterns. It's essential for understanding optics and how lenses manipulate light.

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Most popular questions from this chapter

Old photographic plates were made of glass with a light-sensitive emulsion on the front surface. This emulsion was some what transparent. When a bright point source is focused on the front of the plate, the developed photograph will show a halo around the image of the spot. If the glass plate is 3.10 \(\mathrm{mm}\) thick and the halos have an inner radius of \(5.34 \mathrm{mm},\) what is the index of refraction of the glass? (Hint: Light from the spot on the front surface is scattered in all directions by the emulsion. Some of it is then totally reflected at the back surface of the plate and returns to the front surface.)

A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal larger than \(48.7^{\circ}\) , no light is refracted into the water. What is the refractive index of the glass?

A quarter-wave plate converts linearly polarized light to circularly polarized light. Prove that a quarter-wave plate also converts circularly polarized light to linearly polarized light.

In a physics lab, light with wavelength 490 \(\mathrm{nm}\) travels in air from a laser to a photocell in 17.0 \(\mathrm{ns}\) . When a slab of glass 0.840 \(\mathrm{m}\) thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light 21.2 \(\mathrm{ns}\) to travel from the laser to the photocell. What is the wavelength of the light in the glass?

(a) Light passes through three parallel slabs of different thicknesses and refractive indexes. The light is incident in the first slab and finally refracts into the third slab. Show that the middle slab has no effect on the final direction of the light. That is, show that the direction of the light in the third slab is the same as if the light had passed directly from the first slab into the third slab. Generalize this result to a stack of \(N\) slabs. What determines the final direction of the light in the last slab?
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