Question

Answer as true or false with an explanation for the following: The wavelength of He-Ne laser light in water is less than its wavelength in the air. (The refractive index of water is \(1.33 .\)

Short answer

Explain your answer. Answer: The statement is True. The wavelength of He-Ne laser light in water is less than its wavelength in the air because the refractive index of water (1.33) is greater than that of air (1), causing the wavelength in water to decrease.

Step-by-step solution

Understand the relation between the speed of light, wavelength and refractive index

The relation between the speed of light in a vacuum 𝑣₀, the speed of light in a medium 𝑣, and the refractive index 𝑛 of that medium can be given as follows: \(n = \frac{v_0}{v}\) Also, the relation between speed, frequency (𝑓), and wavelength (𝜆) is given by \(v = f \lambda\). Considering these two equations, we have: \(n = \frac{v_0}{f\lambda}\).

Analyze the statement of the problem

We want to know if the wavelength of He-Ne laser light in water is less than its wavelength in the air. For this, let's compare the wavelengths in water and air. Let the wavelength in air be \(\lambda_1\) and the wavelength in water be \(\lambda_2\). Since the frequency remains constant for the same source of light: \(\frac{n_2}{n_1} = \frac{v_{01}}{v_{02}} = \frac{f\lambda_1}{f\lambda_2}\) where \(n_1=1\) (refractive index of air) and \(n_2=1.33\) (refractive index of water).

Calculate the ratio of wavelengths

From the equation above, \(\frac{1.33}{1} = \frac{\lambda_1}{\lambda_2}\) \(\implies 1.33\lambda_2 = \lambda_1\)

Determine if the statement is true or false

From the equation \(1.33\lambda_2 = \lambda_1\), we can see that the wavelength in air 𝜆₁ is greater than the wavelength in water 𝜆₂, so the statement "The wavelength of He-Ne laser light in water is less than its wavelength in the air" is True. This is because the refractive index of water is more than that of air, which causes the wavelength to decrease in water.

Key concepts

Wavelength

When we talk about the wavelength of light, we're referring to the distance between two consecutive peaks or troughs in a wave. It's a crucial characteristic that tells us about the color of the light.
In a vacuum, light travels at its maximum speed, and its wavelength is longest. However, when light enters a different medium, like water, its speed changes, affecting its wavelength but not its frequency.

• Wavelength in a vacuum or air is generally longer than in denser media like water.
• This change in wavelength is due to the refractive index of the new medium.

The wavelength in water decreases because water has a higher refractive index than air, specifically 1.33, meaning light moves slower in water. Therefore, the wavelength of light in water is less than its wavelength in the air.

Speed of Light

The speed of light is a fundamental constant of nature, approximately 299,792 kilometers per second (or about 186,282 miles per second) in a vacuum. It's the ultimate speed limit in the universe.
However, this speed can decrease when light passes through different media like glass or water. That's why when light enters a medium with a higher refractive index than air, its speed reduces.
The relationship between speed, frequency, and wavelength is given by the equation: \[ v = f \lambda \]

• Here, \( v \) is the speed of light in the medium.
• \( f \) is the frequency.
• \( \lambda \) is the wavelength.

By knowing the refractive index (in the case of water, 1.33), we know that the speed of light is slower in water than in air, directly affecting the light's wavelength but leaving the frequency unchanged.

Frequency

Frequency refers to how many wave cycles pass a point in one second, typically measured in hertz (Hz). For a given light source, such as a laser, its frequency remains constant regardless of the medium through which it travels.
Understanding that frequency stays the same is critical when comparing how light behaves in different environments because while the speed and wavelength may change, the frequency does not.
This means:

• Frequency is independent of the medium, and hence a constant for a given source of light.
• If wavelength decreases as light enters a denser medium, like water, the frequency remains the same.

Consequently, the constancy of frequency allows us to infer changes in other properties like wavelength and speed, using the relationships in the fundamental equation \( v = f \lambda \).