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Chapter 23: Problem 78

The refractive index of a certain glass is \(1.66 .\) For what angle of incidence is light that is reflected from the surface of this glass completely polarized if the glass is immersed in (a) air or (b) water?

Short Answer

Expert verified
(a) Approximately 59.4掳 in air, (b) approximately 51.3掳 in water.

Step by step solution

01

Understanding Brewster's Law

Brewster's Law gives the condition for the angle of incidence at which light will be perfectly polarized upon reflection. This angle, called Brewster's angle, is given by the formula: \( \tan(\theta_B) = \frac{n_2}{n_1} \), where \( n_2 \) is the refractive index of the glass, and \( n_1 \) is the refractive index of the medium in which the glass is immersed.
02

Calculating Brewster's Angle in Air

For part (a), when the glass is in air, \( n_1 = 1.00 \). Substituting the given values into Brewster's Law: \( \tan(\theta_B) = \frac{1.66}{1.00} = 1.66 \). Solving for \( \theta_B \), we have \( \theta_B = \arctan(1.66) \), which can be computed using a calculator to get \( \theta_B \approx 59.4^\circ \).
03

Calculating Brewster's Angle in Water

For part (b), when the glass is immersed in water, \( n_1 = 1.33 \). Substituting the numbers into the formula: \( \tan(\theta_B) = \frac{1.66}{1.33} \approx 1.2481 \). Solving for \( \theta_B \), we find \( \theta_B = \arctan(1.2481) \), which computes to \( \theta_B \approx 51.3^\circ \).

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Key Concepts

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

Refractive Index
The refractive index is a fundamental concept in optics that describes how light propagates through a material. When light enters a medium from a vacuum or air, its speed decreases, causing it to bend or refract. The refractive index, denoted as \( n \), quantifies this bending. It is defined as the ratio of the speed of light in a vacuum to its speed in the medium: \[ n = \frac{c}{v} \] where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium. A higher refractive index means light slows down more and bends to a greater extent.

In this exercise, the glass has a refractive index of \( 1.66 \). This means light travels slower in this glass compared to air or water. Understanding how the refractive index affects light's behavior is crucial for analyzing scenarios involving light reflection and refraction.

Different materials have unique refractive indices:
  • Air typically has a refractive index of about 1.00.
  • Water's refractive index is around 1.33.
  • High-index glasses can reach values of 1.5 and above, like our glass in this problem.
Polarization
Polarization refers to the orientation of the oscillations of light waves. Normal light waves are usually unpolarized, meaning their waves oscillate in multiple planes. However, when light reflects off a surface at a particular angle, known as Brewster's angle, it can become fully polarized.

At Brewster's angle, the reflected light's electric field oscillates in a single plane parallel to the surface. This phenomenon occurs due to the interaction between the transmitted and reflected waves, resulting in the extinction of certain components of the electric field. The conditions for achieving polarization are determined by the properties of the media involved, primarily their refractive indices.

In practical terms:
  • Polarized sunglasses utilize this concept to reduce glare from surfaces like water or roads.
  • Cameras often use polarizing filters to enhance image quality by eliminating unwanted reflections.
By understanding polarization, we can manipulate light more effectively for various technological applications.
Angle of Incidence
The angle of incidence is the angle formed between the incoming light ray and the normal (perpendicular) to the surface it strikes. This angle plays a crucial role in determining how light behaves upon encountering a surface, influencing reflection, refraction, and potential polarization.

In this problem, we are particularly interested in the angle of incidence that causes complete polarization of the reflected light, known as Brewster's angle. Brewster's angle, \( \theta_B \), can be calculated using Brewster's Law:\[\tan(\theta_B) = \frac{n_2}{n_1}\] where \( n_2 \) is the refractive index of the refracting medium (glass, in this case), and \( n_1 \) is the refractive index of the initial medium (air or water).

For example, when the glass is in air (\( n_1 = 1.00 \)), Brewster's angle is approximately 59.4 degrees. When the glass is in water (\( n_1 = 1.33 \)), the angle is approximately 51.3 degrees. By calculating Brewster's angle, we can predict conditions under which light is polarized upon reflection, a valuable insight for numerous optical applications.

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Most popular questions from this chapter

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. To vary the angle as well as the intensity of polarized light, ordinary unpolarized light is passed through one polarizer with its transmission axis vertical, and then a second polarizer is placed between the first polarizer and the insect. When the light leaving the second polarizer has half the intensity of the original unpolarized light, which statement is true about the two types of cells? A. Only type H detects this light. B. Only type \(V\) detects this light. C. Both types detect this light, but type \(\mathrm{H}\) detects more light. D. Both types detect this light, but type \(\mathrm{V}\) detects more 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.

A sinusoidal electromagnetic wave emitted by a cellular phone has a wavelength of \(35.4 \mathrm{~cm}\) and an electric-field amplitude of \(5.40 \times 10^{-2} \mathrm{~V} / \mathrm{m}\) at a distance of \(250 \mathrm{~m}\) from the antenna. Calculate (a) the frequency of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.

A parallel-sided plate of glass having a refractive index of 1.60 is in contact with the surface of water in a tank. A ray coming from above makes an angle of incidence of \(32.0^{\circ}\) with the normal to the top surface of the glass. What angle does this ray make with the normal in the water?

NASA is doing research on the concept of solar sailing. A solar sailing craft uses a large, low-mass sail and the energy and momentum of sunlight for propulsion. (a) Should the sail be absorptive or reflective? Why? (b) The total power output of the sun is \(3.9 \times 10^{26} \mathrm{~W} .\) How large a sail is necessary to propel a \(10,000 \mathrm{~kg}\) spacecraft against the gravitational force of the sun? Express your result in square kilometers. (c) Explain why your answer to part (b) is independent of the distance from the sun.
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