Question

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

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

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 \(1.33\).

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.

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: $$ \lambda_2 = \frac{632.8 \ \text{nm}}{1.33} \approx 475.7 \ \text{nm} $$

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.

Key concepts

Wavelength

The concept of wavelength is fundamental when understanding light and its properties. Wavelength is the distance between two consecutive peaks of a wave. It is usually measured in nanometers (nm) when discussing visible light. Visible light ranges from approximately 400 nm, which is violet, to 700 nm, which corresponds to red light.

Different colors of light have different wavelengths. For example:

• Blue light: 400 nm - 500 nm
• Green light: 500 nm - 550 nm
• Yellow-orange: 550 nm - 600 nm
• Red light: above 600 nm

Understanding the wavelength helps in identifying the color of the light, as well as in tracking its behavior when it moves through various media like air or water.

Index of Refraction

The index of refraction is a crucial concept in physics that describes how light travels through different materials. It measures how much light slows down in a medium compared to a vacuum.

The formula used is:\[n = \frac{\text{speed of light in vacuum}}{\text{speed of light in medium}}\]Where "n" stands for the index of refraction. For instance, air has an index of refraction close to 1, while water has an index of 1.33. This essentially tells us that light travels slower in water than in air.

When light enters a medium with a different index of refraction, its speed changes, as does its wavelength, but its frequency remains constant. This change is why a red laser might appear blue under water.

Color Perception

Color perception is a fascinating topic, deeply rooted in physics and human biology. We perceive different colors based on the wavelengths of light that enter our eyes. These wavelengths correspond to different colors in the visible light spectrum.

However, our perception can also change depending on the medium through which light travels. Light might shift in color due to changes in its wavelength, as happens when it passes from air to water. For example, a red laser beam (632.8 nm in air) might appear blue in water because the wavelength shortens to approximately 475.7 nm.

This transformation occurs due to the alteration in the index of refraction, affecting how the light is perceived. This phenomenon explains why objects underwater sometimes look different in color than they do above the water.

HeNe Laser

HeNe lasers, short for Helium-Neon lasers, emit a distinctive red light, usually at a wavelength of 632.8 nm. These lasers are widely used in scientific experiments and educational settings because of their stability and visibility.

The fascinating aspect of HeNe lasers lies in their predictability and coherence, making them ideal for studying light phenomena, including refraction and dispersion in different materials.

Refraction experiments often include HeNe lasers because their wavelength solution is easily calculated in new mediums using the index of refraction. For instance, the predictable red wavelength of 632.8 nm turns into a shorter wavelength of blue (about 475.7 nm) when it moves into water.

Such reliable behavior helps in practical applications and theoretical understanding of light properties, making HeNe lasers a staple in optics studies.