Question

A helium-neon laser has a wavelength of \(632.8 \mathrm{nm}\). a) What is the wavelength of this light as it passes through Lucite with an index of refraction \(n=1.500 ?\) b) What is the speed of light in the Lucite?

Short answer

Answer: The wavelength of the helium-neon laser light in Lucite is 421.9 nm, and the speed of light in Lucite is 2.0 x 10^8 m/s.

Step-by-step solution

Analyze the given information and formulas

We are given the wavelength of the helium-neon laser, \(\lambda = 632.8 \, nm\), and the index of refraction for Lucite, \(n = 1.500\). The relationship between the index of refraction, wavelength, and speed of light is given by the equation: \(n = \frac{c}{v}\) where \(n\) is the index of refraction, \(c\) is the speed of light in a vacuum, and \(v\) is the speed of light in the medium. We will use this equation to find both the wavelength and the speed of light in the Lucite.

Find the wavelength of this light as it passes through Lucite

To find the wavelength in the Lucite, we must first find the speed of light in the Lucite using the equation \(n = \frac{c}{v}\). Rearrange the equation to find \(v\): \(v = \frac{c}{n}\) Now, substitute the given values into the equation: \(v_L = \frac{3.0 \times 10^8 \, m/s}{1.500} = 2.0 \times 10^8 \, m/s\) Now we can find the wavelength of the light as it passes through Lucite. The speed of light and the wavelength have an inverse relationship: \(\lambda_L = \frac{\lambda}{n}\) Substitute the given values into the equation: \(\lambda_L = \frac{632.8 \, nm}{1.500} = 421.9 \, nm\) The wavelength of this light as it passes through Lucite is \(421.9 \, nm\).

Calculate the speed of light in Lucite

We have already calculated the speed of light in Lucite in Step 2. The speed of light in Lucite is: \(v_L = 2.0 \times 10^8 \, m/s\)

Summary

The wavelength of the helium-neon laser as it passes through Lucite with an index of refraction of \(1.500\) is \(\boxed{421.9 \, nm}\), and the speed of light in Lucite is \(\boxed{2.0 \times 10^8 \,m/s}\).

Key concepts

Wavelength of Light

Light, as it travels through various mediums, is subject to a change in its wavelength. This property is particularly significant when studying optical phenomena. The wavelength of light is the distance between two consecutive peaks (or troughs) in a light wave. In a vacuum, all light travels at a universal speed, known as the speed of light, which is approximately \(3.0 \times 10^8 \text{m/s}\). However, when light enters a medium such as water, glass, or Lucite, its speed reduces and, as a result, its wavelength changes too.

This alteration is dictated by the medium’s index of refraction, often denoted as \(n\). The index of refraction is a dimensionless number that describes how much the light will slow down and bend when entering the medium. To find the new wavelength \( \lambda_{\text{medium}}\) of light within a medium when you know its wavelength \( \lambda_{\text{vacuum}}\) in a vacuum, you use the simple equation:\[\lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{n}\]Considering a wavelength of \(632.8 \text{nm}\) of a helium-neon laser light, and Lucite with an index of refraction of \(1.500\), the new wavelength in the Lucite is calculated to be \(421.9 \text{nm}\).

Understanding how wavelength alters in different mediums is pivotal for applications such as fiber optic communications, microscopy, and laser technology.

Speed of Light in Medium

The speed of light is a fundamental concept in physics, symbolized by \(c\) when referring to its unyielding value in a vacuum. However, in any other medium, the speed of light \(v\) will differ, generally slowing down relative to \(c\). The ratio of the speed of light in a vacuum to the speed in the medium is the medium's index of refraction \(n\), represented by the equation \(n = \frac{c}{v}\).

To find the speed of light in a medium like Lucite, you simply rearrange the formula to \(v = \frac{c}{n}\). For Lucite with an index of refraction of \(1.500\), the speed of light within it is calculated as \(2.0 \times 10^8 \text{m/s}\), slower than \(c\) in a vacuum.

Knowing the speed of light in various mediums is essential in fields such as astronomy, optical engineering, and telecommunications where precise measurements of light propagation are crucial. This information allows for the precise design and analysis of optical systems, such as lenses and telescopes, ensuring their optimal performance.

Helium-Neon Laser

A helium-neon laser, often abbreviated as He-Ne laser, is a type of gas laser that has become a staple in various scientific and commercial applications due to its stable output and relatively low cost. It operates by using a mixture of helium and neon gases to produce a coherent beam of red light, typically at a wavelength of \(632.8 \text{nm}\).

Helium-neon lasers are employed in a myriad of applications including but not limited to barcode scanning, alignment, holography, and as a standard in interferometry. The reason for its widespread use lies in its coherency, good monochromacity, and a beam that can be accurately controlled and modulated for precision tasks.

When used in educational contexts, such as in the given exercise, a helium-neon laser helps students understand optical principles like refraction, reflection, and wavelength shift in different mediums, thus providing a practical context for theoretical physics concepts.