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Ch 34: Ray Optics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 34: Ray OpticsProblem 58b
Chapter 34, Problem 58b

There's one angle of incidence β onto a prism for which the light inside an isosceles prism travels parallel to the base and emerges at angle β. A laboratory measurement finds that β=52.2° for a prism shaped like an equilateral triangle. What is the prism's index of refraction?

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Step 1: Identify the geometry of the prism. The prism is an equilateral triangle, meaning all angles inside the prism are 60°. The angle of incidence β is given as 52.2°, and the light travels parallel to the base inside the prism.
Step 2: Use Snell's Law at the first interface where the light enters the prism. Snell's Law is expressed as: n1⁢sin⁢β=n2⁢sin⁢θ, where n₁ is the refractive index of air (approximately 1), n₂ is the refractive index of the prism, β is the angle of incidence, and θ is the angle of refraction.
Step 3: Recognize that for the light to travel parallel to the base, the angle of refraction θ inside the prism must equal the prism angle α (60°). Substitute θ = 60° into Snell's Law: sin⁢β=n⁢sin⁢α, where n is the refractive index of the prism.
Step 4: Rearrange Snell's Law to solve for the refractive index n of the prism: n=sin⁢βsin⁢α. Substitute β = 52.2° and α = 60° into the equation.
Step 5: Perform the calculation using the sine values of the angles. This will yield the refractive index n of the prism. Ensure the sine values are calculated accurately using a scientific calculator or trigonometric table.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Refraction

Refraction is the bending of light as it passes from one medium to another with a different index of refraction. This phenomenon occurs due to the change in the speed of light in different materials, which is described by Snell's Law. The law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant and equal to the ratio of the indices of refraction of the two media.
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Index of Refraction

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how much light slows down in a medium compared to its speed in a vacuum. It is defined as the ratio of the speed of light in vacuum to the speed of light in the medium. For example, an index of refraction of 1.5 indicates that light travels 1.5 times slower in that medium than in a vacuum, which is crucial for understanding how light behaves in prisms.
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Prism Geometry

The geometry of a prism, particularly an isosceles or equilateral prism, plays a significant role in determining how light refracts within it. In the case of an equilateral prism, the angles are all 60 degrees, which influences the relationship between the angles of incidence and refraction. Understanding the angles involved helps in applying Snell's Law to find the index of refraction when light travels parallel to the base of the prism.
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