Problem 8 - Physics

Question

A flat sheet of ice has a thickness of \(2.0 \mathrm{~cm} .\) It is on top of a flat sheet of crystalline quartz that has a thickness of \(1.1 \mathrm{~cm} .\) Light strikes the ice perpendicularly and travels through it and then through the quartz. In the time it takes the light to travel through the two sheets, how far (in centimeters) would it have traveled in a vacuum?

Step-by-Step Solution

Answer

The light would travel 4.23 cm in a vacuum.

Step 1: Identify the Known Variables

We know the thickness of the ice is \[ d_\text{ice} = 2.0 \text{ cm} \]and the thickness of the quartz is \[ d_\text{quartz} = 1.1 \text{ cm} \].The refractive index of ice is approximately \( n_\text{ice} = 1.31 \)and that of quartz is \( n_\text{quartz} = 1.46 \).

Step 2: Calculate the Speed of Light in Each Material

The speed of light in a material is given by \[ v = \frac{c}{n} \].First, calculate the speed of light in ice:\[ v_\text{ice} = \frac{c}{n_\text{ice}} = \frac{3.0 \times 10^8 \text{ m/s}}{1.31} \approx 2.29 \times 10^8 \text{ m/s} \].Now calculate the speed of light in quartz:\[ v_\text{quartz} = \frac{c}{n_\text{quartz}} = \frac{3.0 \times 10^8 \text{ m/s}}{1.46} \approx 2.05 \times 10^8 \text{ m/s} \].

Step 3: Calculate Time Taken by Light to Travel Through Each Material

To find the time, use the formula \[ t = \frac{d}{v} \].For ice, \[ t_\text{ice} = \frac{2.0 \times 10^{-2} \text{ m}}{2.29 \times 10^8 \text{ m/s}} \approx 8.73 \times 10^{-11} \text{ s} \].For quartz, \[ t_\text{quartz} = \frac{1.1 \times 10^{-2} \text{ m}}{2.05 \times 10^8 \text{ m/s}} \approx 5.37 \times 10^{-11} \text{ s} \].

Step 4: Calculate Total Time Taken by Light

The total time taken by light to travel through both materials is the sum of the individual times:\[ t_\text{total} = t_\text{ice} + t_\text{quartz} \].Substituting the values,\[ t_\text{total} = 8.73 \times 10^{-11} \text{ s} + 5.37 \times 10^{-11} \text{ s} = 1.41 \times 10^{-10} \text{ s} \].

Step 5: Calculate Distance Traveled in a Vacuum

In a vacuum, light travels at speed \( c = 3.0 \times 10^8 \text{ m/s} \).Using \[ d = c \times t_\text{total} \],we get:\[ d = 3.0 \times 10^8 \text{ m/s} \times 1.41 \times 10^{-10} \text{ s} \approx 4.23 \times 10^{-2} \text{ m} \].To convert this distance to centimeters, multiply by 100:\[ 4.23 \times 10^{-2} \text{ m} \times 100 = 4.23 \text{ cm} \].

Key Concepts

Refractive Index

The refractive index is a crucial concept in optics, determining how much light bends or changes speed when moving from one medium to another. When light enters a new material, it doesn't travel at the same speed as it does in a vacuum. Instead, it slows down, and the factor by which it slows down is what we call the refractive index. Mathematically, the refractive index (\( n \)) is defined as\[n = \frac{c}{v}\]where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
For instance, ice has a refractive index of \( n_{\text{ice}} = 1.31 \), meaning light travels 1.31 times slower in ice than in a vacuum. Crystalline quartz, on the other hand, has a refractive index of \( n_{\text{quartz}} = 1.46 \).
Some key points to remember about refractive index include:

• The higher the refractive index, the slower light travels in the medium.
• Materials with a high refractive index are often thought of as "optically denser."
• Refractive indices are always greater than or equal to 1, with 1 being the refractive index of a vacuum.

Speed of Light

The speed of light is one of the fundamental constants of nature and plays a critical role in understanding optics. In a vacuum, light travels at its maximum speed of approximately \( c = 3.0 \times 10^8 \text{ m/s} \). This speed is often denoted as \( c \)in scientific equations and discussions.
However, when light enters a medium, it slows down. The speed of light in a medium can be calculated using the formula:\[v = \frac{c}{n}\]where \( c \)is the speed of light in a vacuum, and \( n \)is the refractive index of the medium.
In the case of ice, the speed of light is \( v_{\text{ice}} \approx 2.29 \times 10^8 \text{ m/s} \) while for quartz, it is \( v_{\text{quartz}} \approx 2.05 \times 10^8 \text{ m/s} \).

• Light travels slower in denser mediums, which is indicated by higher refractive indices.
• Understanding the speed of light in different materials is essential for applications like fiber optics and lenses.

Light in a Medium

When light enters a medium like ice or quartz, it undergoes changes in speed and direction. This is due to the refractive index of the medium, which determines how much light bends as it crosses the boundary between two different materials.
As light travels from vacuum into a medium with refractive index \( n \), its speed changes, but its frequency remains constant. This changing speed causes the light to bend, a phenomenon known as refraction.
Here are some important aspects of light in a medium:

• Speed change: While the speed of light in a vacuum is constant (\( c \)), it slows down in materials as per their refractive index.
• Bending or refraction: Light bends at the interface of two mediums, with the amount of bending dependent on the relative refractive indices of the two materials.
• Propagation through materials: Light travels slower through denser mediums such as quartz compared to ice, impacting the path length and time taken for light to traverse these materials.

These optical principles help explain everyday phenomena, like how a straw looks bent in a glass of water.