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Chapter 17: Problem 64

A beam of light traveling in air enters a pool of water. If the angle of refraction is 41掳 and the index of refraction of water is 1.33, what is the angle of incidence?

Short Answer

Expert verified
The angle of incidence is approximately 60.7 degrees.

Step by step solution

01

Understand Snell's Law

Snell's Law relates the angle of incidence and the angle of refraction when light passes from one medium into another. It is expressed as \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \), where \(n_1\) and \(n_2\) are the indices of refraction of the two media, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively.
02

Identify Known Values

In this problem, the index of refraction of air \(n_1\) is approximately 1.00, the index of refraction of water \(n_2\) is given as 1.33, and the angle of refraction \(\theta_2\) is 41 degrees. We need to find the angle of incidence \(\theta_1\).
03

Rearrange Snell's Law to Solve for \(\theta_1\)

Rearrange Snell's Law to solve for the sine of the angle of incidence: \( \sin(\theta_1) = \frac{n_2}{n_1} \sin(\theta_2) \). Substitute in the known values to find \(\sin(\theta_1)\).
04

Calculate \(\sin(\theta_1)\)

Substitute the known values into the rearranged formula: \(\sin(\theta_1) = \frac{1.33}{1.00} \sin(41^\circ)\). Calculate \(\sin(41^\circ)\) using a calculator which equals approximately 0.6561. Then, \(\sin(\theta_1) = 1.33 \times 0.6561 = 0.8726\).
05

Calculate the Angle of Incidence \(\theta_1\)

Use the inverse sine function to find \(\theta_1\): \(\theta_1 = \sin^{-1}(0.8726)\). Calculate \(\theta_1\) using a calculator which gives \(\theta_1 \approx 60.7^\circ\).

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Key Concepts

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

Index of Refraction
The index of refraction is a crucial concept in optics that helps to describe how light travels through different materials. It is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. This value is highly important when studying the behavior of light as it transitions between different substances:
  • In a vacuum, the index of refraction is exactly 1.00, as this is the medium in which light travels fastest.
  • For air, the index is approximately 1.00 as well, given that air's effect is nearly negligible.
  • For water, the index is higher, typically around 1.33, indicating that light slows down significantly as it enters the water.
Understanding these indexes helps predict how light will bend or change direction when crossing the boundary between two media, which is the basis for Snell's Law.
Angle of Incidence
The angle of incidence is the angle between an incoming ray of light and the normal (a line perpendicular) to the surface at the point of entry. This angle is one of the foundational elements of Snell's Law, which predicts how light behaves as it moves from one medium to another. Here is why this concept is significant:
  • The angle of incidence directly influences the degree to which light is refracted at the boundary.
  • By knowing the properties of the two media (like air and water in our exercise), we can calculate the angle of incidence if we know the other parameters like the angle of refraction.
  • This aspect allows us to further explore optical phenomena, including reflection and dispersion.
For the exercise, Snell's Law was rearranged to solve for the angle of incidence, demonstrating its crucial role in analyzing light behavior.
Angle of Refraction
The angle of refraction describes the angle at which light travels after passing through the boundary from one medium to another. It is measured from the normal to the refracted ray. In contexts such as the given exercise, understanding the angle of refraction is essential:
  • The angle of refraction reveals how much the light path is bent upon entering a new medium with different optical density.
  • By utilizing Snell's Law, we are able to derive other related optical properties, including the angle of incidence or the indices of refraction of media involved.
  • The refractive angle is integral to explaining why objects appear bent or displaced when viewed across substances like water and glass.
The exercise provided the angle of refraction (41掳) which, along with the indices of refraction, was used to find the angle of incidence, illustrating the interconnectedness of these angles.

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Most popular questions from this chapter

A physics professor shipwrecked on Hooligan's Island decides to build a telescope from his eyeglasses and some coconut shells. Fortunately, the professor's eyes require different prescriptions, with the left lens having a focal length of \(21 \mathrm{~cm}\) and the right lens having a focal length of \(55 \mathrm{~cm}\). (a) Which lens should he use as the objective? (b) What is the magnification of the professor's telescope?

An intracorneal ring is a small plastic device implanted in a person's cornea to change its curvature. By changing the shape of the cornea, the intracorneal ring can correct a person's vision. (a) If a person is nearsighted, should the ring increase or decrease the cornea's curvature? (b) Choose the best explanation from among the following: A. The intracorneal ring should increase the curvature of the cornea so that it bends light more. This will allow it to focus light coming from far away. B. The intracorneal ring should decrease the curvature of the cornea so that it is flatter and bends light less. This will allow parallel rays from far away to be focused.

One of the many works published by the Greek astronomer Ptolemy (A.D. circa 90-170) was Optics. In this book Ptolemy reported the results of refraction experiments he conducted by observing light passing from air into water. Two of his results are as follows: (1) angle of incidence \(=10.0^{\circ}\), angle of refraction \(=8.00^{\circ} ;(2)\) angle of incidence \(=20.0^{\circ}\), angle of refraction \(=15.5^{\circ}\). Find the percent error for each of Ptolemy's measurements, assuming that the index of refraction of water is \(1.33\).

ptolemy鈥檚 Optics One of the many works published by the Greek astronomer Ptolemy (a.d. circa 90鈥�170) was Optics. In this book Ptolemy reported the results of refraction experiments he conducted by observing light passing from air into water. Two of his results of refraction = 8.00; (2) angle of incidence = 20.0, angle of refraction = 15.5. Find the percent error for each of Ptolemy鈥檚 measurements, assuming that the index of refraction of water is 1.33. are as follows: (1) angle of incidence = 10.0, angle

Being Invisible A common science fiction plot device is for a character to become invisible. Suppose a character becomes invisible by altering his index of refraction to match that of air. If someone could actually do this, would the person be able to see? Explain.
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