Engineers are testing a new thin-film coating whose index of refraction is less than that of glass. They deposit a 560-nm-thick layer on glass, then shine lasers on it. A red laser with a wavelength of 640 nm has no reflection at all, but a violet laser with a wavelength of 400 nm has a maximum reflection. How the coating behaves at other wavelengths is unknown. What is the coating’s index of refraction?
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Index of Refraction
Problem 64
Textbook Question
Microwaves with a frequency of 10.5 GHz are aimed downward into a flat-bottomed beaker that contains sunflower oil. A microwave detector above the beaker finds that there are strong reflections when the oil depth is 2.76 cm and 3.68 cm but at no depths in between. What is the index of refraction of sunflower oil at microwave frequencies?

1
Step 1: Understand the problem. The strong reflections occur due to constructive interference of microwaves reflected from the top and bottom surfaces of the sunflower oil. This happens when the path difference between the reflected waves is an integer multiple of the wavelength in the oil.
Step 2: Write the condition for constructive interference. The path difference is twice the depth of the oil (since the wave travels down and back up). For constructive interference, this path difference must equal an integer multiple of the wavelength in the oil: , where is the depth, is an integer, and is the wavelength in the oil.
Step 3: Relate the wavelength in the oil to the wavelength in vacuum. The wavelength in the oil is given by , where is the wavelength in vacuum and is the index of refraction of the oil. The wavelength in vacuum can be calculated using the frequency of the microwaves: , where is the speed of light and is the frequency.
Step 4: Use the given depths to find the difference in constructive interference orders. The depths 2.76 cm and 3.68 cm correspond to consecutive constructive interference orders, so the difference in path lengths is equal to one wavelength in the oil: . Solve for .
Step 5: Calculate the index of refraction. Substitute and into the equation to solve for , the index of refraction of sunflower oil.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Microwave Frequency and Wavelength
Microwaves are a form of electromagnetic radiation with frequencies ranging from 300 MHz to 300 GHz. The frequency of 10.5 GHz corresponds to a wavelength of approximately 2.86 cm, calculated using the speed of light. Understanding the relationship between frequency and wavelength is crucial for analyzing how microwaves interact with materials, such as sunflower oil.
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Index of Refraction
The index of refraction (n) is a dimensionless number that describes how light propagates 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. For microwaves, the index of refraction can be determined by analyzing reflection and transmission at the boundaries of different media, which is essential for solving the given problem.
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Standing Waves and Resonance
Standing waves occur when two waves of the same frequency interfere with each other, creating nodes and antinodes. In the context of the beaker, strong reflections at specific depths indicate the formation of standing waves, which correspond to resonant conditions. These depths relate to the wavelength of the microwaves and are critical for determining the index of refraction of sunflower oil.
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