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

Knight Calc 5th Edition

Ch 34: Ray Optics

Problem 47a

Chapter 34, Problem 47a

A light ray in air is incident on a transparent material whose index of refraction is n. Find an expression for the (non-zero) angle of incidence whose angle of refraction is half the angle of incidence.

Verified step by step guidance

Start by applying Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media:

n

₁⁢sin⁢θ₁=n₂⁢sin⁢θ₂. Here,

n

₁ is the index of refraction of air (approximately 1),

n

₂ is the index of refraction of the material (

n

θ

₁ is the angle of incidence, and

θ

₂ is the angle of refraction.

Substitute the given condition that the angle of refraction is half the angle of incidence:

θ

₂=θ₁2. This means

\sin θ

₂=sin⁢θ₁2.

Using the trigonometric identity for the sine of a half-angle, rewrite

\sin θ

₁2 as

\sqrt{\frac{1}{2}} \sqrt{(1 - \cos θ}

₁).

Substitute this expression for

\sin θ

₂ into Snell's Law:

\sin θ

₁=n⁢12⁢(1-cos⁢θ₁).

Square both sides of the equation to eliminate the square root, and solve for

\cos θ

₁. This will give you an expression for the angle of incidence

θ

₁ in terms of the index of refraction

n

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Key Concepts

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

Snell's Law

Snell's Law describes the relationship between the angles of incidence and refraction when a light ray passes between two different media. It is mathematically expressed as n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the indices of refraction of the two media, and θ1 and θ2 are the angles of incidence and refraction, respectively.

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how much light slows down when it enters a material compared to its speed in a vacuum. It is defined as n = c/v, where c is the speed of light in a vacuum and v is the speed of light in the material. A higher index indicates that light travels slower in that medium.

Angle Relationships

In the context of the problem, the relationship between the angle of incidence (θ1) and the angle of refraction (θ2) is crucial. The question specifies that θ2 is half of θ1, which can be expressed as θ2 = θ1/2. This relationship will be used in conjunction with Snell's Law to derive the expression for the angle of incidence.

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