Problem 5 - University Physics with Modern Physics

Question

A light beam travels at \(1.94 \times 10^8\) m/s in quartz. The wavelength of the light in quartz is 355 nm. (a) What is the index of refraction of quartz at this wavelength? (b) If this same light travels through air, what is its wavelength there?

Step-by-Step Solution

Answer

The index of refraction of quartz is 1.546, and the wavelength of light in air is 548.83 nm.

Step 1: Understand the Problem

We want to find the index of refraction for quartz and the wavelength of light in air given the speed and wavelength of light in quartz.

Step 2: Use the Formula for Index of Refraction

The index of refraction, \( n \), is given by the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in vacuum \((3.00 \times 10^8\ m/s)\), and \( v \) is the speed of light in the medium (quartz). Use these values to calculate \( n \).

Step 3: Calculate Index of Refraction

Substitute the given values into the formula: \( n = \frac{3.00 \times 10^8}{1.94 \times 10^8} \). This gives \( n = 1.546 \).

Step 4: Use the Formula for Wavelength in Different Medium

The wavelength of light changes when it moves between two mediums. In air, the wavelength \( \lambda_{air} \) is given by \( \lambda_{air} = \frac{\lambda_{quartz} \times n}{n_{air}} \). The index of refraction of air \( n_{air} \) is approximately 1.

Step 5: Calculate Wavelength in Air

Substitute the values into the formula: \( \lambda_{air} = \frac{355 \times 1.546}{1} \). This simplifies to \( \lambda_{air} = 548.83 \ nm \).

Step 6: Verification

Verify the units and calculations to ensure accuracy: 355 nm and the calculated refractive index together yield a wavelength in air that is longer than in quartz, which makes sense because light travels faster in air.

Key Concepts

Speed of Light in Different Media

The speed of light is famously approximately \(3 imes 10^8\, ext{m/s}\) in a vacuum, which is often rounded to 300,000 kilometers per second. However, when light travels through any other optical medium, like glass, water, or quartz, its speed decreases due to interactions with the atoms and molecules in that medium.

This reduction in speed is quantified by the medium's index of refraction, denoted as \(n\). The index of refraction is a dimensionless number defined by \(n = \frac{c}{v}\), where \(c\) is the speed of light in vacuum, and \(v\) is the speed of light in the medium.

For example, when light travels through quartz, with a calculated speed of \(1.94 \times 10^8\, ext{m/s}\), it moves slower than it would in a vacuum. This decrease in speed is why we calculated an index of refraction for quartz of 1.546.

Wavelength and Its Change in Different Media

Light's wavelength is another critical aspect of its behavior in different media. Wavelength refers to the distance between two consecutive peaks of a wave, which we generally measure in nanometers.

When light travels from one medium to another — for example, from quartz into air — its wavelength changes. This is because the speed of light alters due to the different optical properties of the media.

Using the formula \(\lambda_{air} = \frac{\lambda_{quartz} \times n}{n_{air}}\), where \(\lambda_{quartz}\) is the initial wavelength in quartz, and \(n\) and \(n_{air}\) are the refractive indices of quartz and air respectively, we deduced that the wavelength of light in air is 548.83 nm, longer than in quartz, where it was 355 nm.

Optical Medium and Its Influence on Light

An optical medium is any material through which light can propagate. Common examples include air, water, glass, and quartz. Each optical medium affects the speed and behavior of light differently.

Two fundamental properties influenced by the optical medium are the speed of light and its wavelength. In general, the denser the medium, the slower the speed of light within it, and the shorter the wavelength becomes.

For instance, in our exercise, we observed that light traveling in quartz moved slower compared to air. As a direct consequence, its wavelength in air increased. This behavior is due to quartz having a higher index of refraction compared to air, indicating that it is denser and reduces the speed of light more significantly.