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Ch 33: The Nature and Propagation of Light
Chapter 33, Problem 33

The indexes of refraction for violet light λ = 400 nm2 and red light λ= 700 nm2 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.

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1
Calculate the angle of refraction for each color using Snell's Law, which is given by \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \). Here, \( n_1 = 1 \) (for air), \( \theta_1 = 53.5^\circ \), and \( n_2 \) is 2.46 for violet light and 2.41 for red light.
Rearrange Snell's Law to solve for \( \theta_2 \) for each color: \( \theta_2 = \arcsin\left(\frac{n_1 \sin(\theta_1)}{n_2}\right) \). Plug in the values for violet light and calculate \( \theta_{2,\text{violet}} \).
Repeat the previous step for red light to find \( \theta_{2,\text{red}} \).
Calculate the angular separation between the two colors by finding the difference between the two angles of refraction: \( \Delta \theta = |\theta_{2,\text{violet}} - \theta_{2,\text{red}}| \).
The result from the previous step gives you the angular separation between the violet and red light as they pass through the diamond.

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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 how light bends when it passes from one medium to another. It states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two media. This principle is essential for calculating the angles at which light refracts when entering a new medium, such as diamond from air.
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Index of Refraction

The index of refraction is a dimensionless number that describes how fast light travels 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. Different wavelengths of light have different indices of refraction, which leads to phenomena such as dispersion, where colors separate when passing through a prism or other optical materials.
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Angular Separation

Angular separation refers to the difference in angles of refraction for different wavelengths of light as they pass through a medium. In this context, it is the angle difference between the refracted rays of violet and red light in diamond. This separation occurs due to the varying indices of refraction for different colors, leading to distinct paths for each wavelength after refraction.
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