Question

The speed of light in topaz is \(1.85 \times 10^{8} \mathrm{m} / \mathrm{s} .\) What is the index of refraction of topaz?

Short answer

The index of refraction of topaz is approximately 1.62.

Step-by-step solution

Understand the Index of Refraction

The index of refraction, denoted as \( n \), is a measure of how much light slows down when it enters a medium from a vacuum. It is defined as the ratio of the speed of light in a vacuum \( c \) to the speed of light in the medium \( v \). Mathematically, it is expressed as \( n = \frac{c}{v} \).

Identify Known Values

In the problem, we are given the speed of light in topaz, which is \( v = 1.85 \times 10^8 \ \text{m/s} \). The speed of light in a vacuum \( c \) is a constant \( 3.00 \times 10^8 \ \text{m/s} \).

Calculate the Index of Refraction

Substitute the known values into the formula for the index of refraction:\[ n = \frac{c}{v} = \frac{3.00 \times 10^8}{1.85 \times 10^8} \]Calculate the value:\[ n \approx 1.62 \]

Conclusion

The calculated index of refraction shows how much light is slowed in topaz compared to a vacuum. Conclude that the index of refraction of topaz is approximately 1.62.

Key concepts

Speed of Light

The speed of light is one of the most fundamental constants in physics. It represents the fastest speed at which energy, information, or matter can travel in the universe. In a vacuum, light travels incredibly fast at approximately 3.00 x 10^8 meters per second (m/s). This speed is denoted as \( c \) and is the ultimate speed limit according to our current understanding of physics.When light enters different materials, like glass or water, it slows down due to interactions with the atoms in the medium. This slowing down is described by the index of refraction. The difference in speed between light in a vacuum and in a medium tells us a lot about how light and materials interact, which is central to the study of optics.

Optics

Optics is the branch of physics that studies light, its properties, and how it behaves when it interacts with different materials or objects. Understanding optics allows us to design lenses, glasses, microscopes, and cameras, among many other things. When light travels through different materials, we say it refracts or bends. This bending is fundamental to optics and is precisely described by the index of refraction. Refraction occurs because light changes speed when it enters a new medium. The index of refraction quantifies this change, helping scientists predict how much light will bend. A higher index means light slows down more and bends more sharply. For instance, light moves slower in diamond than in water, so diamond has a higher index. These principles are why lenses are curved—to focus light effectively based on refraction.

Medium

A medium is any material through which light can pass, whether transparent like air and glass, or translucent, like frosted glass. Each medium affects light differently, altering its speed and direction.The medium's effect on light is captured by its index of refraction, noted as \( n \), which is calculated by dividing the speed of light in a vacuum \( c \) by its speed in that medium \( v \): \[ n = \frac{c}{v} \]This ratio lets us understand how optically "dense" a medium is. For example, topaz, with an index of 1.62, slows light considerably, indicating it refracts light significantly—or bends it more than substances with lower indices.Hence, knowing a medium's refractive index helps us understand how it will influence light passing through especially important in designing optical devices.