Ch 33: The Nature and Propagation of Light

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 33: The Nature and Propagation of Light

Problem 22

Chapter 33, Problem 22

The indexes of refraction for violet light (λ = 400 nm) and red light (λ= 700 nm) in diamond are 2.46 and 2.41, respectively. A ray of light traveling through air strikes the diamond surface at an angle of 53.5° to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

Verified step by step guidance

Identify the given values: the angle of incidence θ₁ = 53.5°, the index of refraction for violet light n₁ = 2.46, and the index of refraction for red light n₂ = 2.41. The index of refraction for air is approximately n₀ = 1.00.

Use Snell's Law to find the angle of refraction for each color of light. Snell's Law is given by:

n

₀sin⁡(θ₁) = n₁sin⁡(θ_v) for violet light and

n

₀sin⁡(θ₁) = n₂sin⁡(θ_r) for red light.

Rearrange Snell's Law to solve for the angle of refraction for each color:

θ

_v = sin⁻1(n₀sin⁡(θ₁)n₁) for violet light and

θ

_r = sin⁻1(n₀sin⁡(θ₁)n₂) for red light.

Calculate the angle of refraction for both violet and red light using the rearranged Snell's Law equations. This involves substituting the known values into the equations and using a calculator to find the angles.

Determine the angular separation between the two colors of light by subtracting the angle of refraction for red light from the angle of refraction for violet light:

Δθ = θ

_v - θ_r.

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Key Concepts

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

Index of Refraction

The index of refraction, denoted as 'n', is a measure of how much light slows down as it passes through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. Different wavelengths of light have different indices of refraction, which causes dispersion.

Snell's Law

Snell's Law describes how light bends when it passes from one medium to another. It states that 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. This law is crucial for calculating the angle of refraction for different wavelengths.

Dispersion

Dispersion occurs when different wavelengths of light refract at different angles due to varying indices of refraction. This phenomenon causes the separation of light into its constituent colors, as seen in prisms. In this context, dispersion explains the angular separation between violet and red light as they pass through diamond.

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