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Chapter 34: Problem 20

It is common knowledge that the visible light spectrum extends approximately from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\). Roughly, \(400 \mathrm{nm}\) to \(500 \mathrm{nm}\) corresponds to blue light, \(500 \mathrm{nm}\) to \(550 \mathrm{nm}\) corresponds to green, \(550 \mathrm{nm}\) to \(600 \mathrm{nm}\) to yelloworange, and above \(600 \mathrm{nm}\) to red. In an experiment, red light with a wavelength of \(632.8 \mathrm{nm}\) from a HeNe laser is refracted into a fish tank filled with water with index of refraction 1.33. What is the wavelength of the same laser beam in water, and what color will the laser beam have in water?

Short Answer

Expert verified
Answer: The wavelength of the red laser beam in water is approximately 475.7 nm, and it would appear blue in water.

Step by step solution

01

Identify the given values

We are given the wavelength of the red laser beam in air (vacuum) as \(632.8 \mathrm{nm}\) and the index of refraction of water as \(1.33\).
02

Apply the relationship between wavelength and index of refraction

The relationship between the wavelength of light in two different media (in this case, air and water) through their indices of refraction (n) is given as: $$ \frac{\lambda_1}{\lambda_2} = \frac{n_2}{n_1} $$ Where \(\lambda_1\) and \(\lambda_2\) are the wavelengths in medium 1 (air) and medium 2 (water), and \(n_1\) and \(n_2\) are their respective indices of refraction.
03

Calculate the wavelength in water

We know that the index of refraction of air is approximately 1. We can use this value and the formula derived in step 2 to calculate \(\lambda_2\) (the wavelength of the red laser beam in water): $$ \frac{632.8 \ \text{nm}}{\lambda_2} = \frac{1.33}{1} $$ Solving for the wavelength of the red laser beam in water, we get: $$ \lambda_2 = \frac{632.8 \ \text{nm}}{1.33} \approx 475.7 \ \text{nm} $$
04

Identify the corresponding color in the visible spectrum

The approximately \(475.7 \mathrm{nm}\) wavelength in water is close to \(500 \mathrm{nm}\), which is categorized as blue light in the visible spectrum. Thus, the laser beam will appear blue in water. The final answer is that the wavelength of the red laser beam in water is approximately \(475.7 \mathrm{nm}\), and its color in water would be blue.

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