Chapter 33: Problem 36
A light beam is directed parallel to the axis of a hollow cylindrical tube. When the tube contains only air, the light takes 8.72 ns to travel the length of the tube, but when the tube is filled with a transparent jelly, the light takes 1.82 ns longer to travel its length. What is the refractive index of this jelly?
Short Answer
Step by step solution
Understand the Problem
Define the Relationship
Calculate the Time Ratios
Perform the Calculation
Interpret the Result
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Speed of Light
When a light beam moves through different materials, its speed changes. For example, air slightly slows down light compared to a vacuum. This slowing effect becomes more pronounced in denser materials like water, glass, or in this case, jelly. Understanding how the speed of light varies in different substances is key to solving problems about refractive indices and travel times.
To grasp how the speed of light influences calculations like those needed for the refractive index, it's helpful to remember that a medium's refractive index indicates how much the medium slows light down compared to its speed in a vacuum.
Transparent Medium
The transparency of a material doesn't mean light travels at the same speed as in a vacuum. Instead, even transparent media can slow down light, which impacts the refractive index—a measure of how much the speed of light is reduced. Each medium has its own refractive index, calculated by comparing light's speed in the medium to its speed in a vacuum (or air, as a close approximation).
In our exercise, the jelly filling the cylindrical tube represents the transparent medium. We noted the difference in time it took for the light beam to pass through when the jelly was present compared to when air was present. By identifying the extra time taken by the light through the jelly, we can determine the jelly’s refractive index.
Light Beam
When light beams pass through different materials, their speed and direction can change due to the material's optical properties. This behavior is central to concepts like refraction—the bending of light as it passes from one medium to another. Here, we explore how the light beam's speed changes depending on whether the tube contains air or jelly.
The additional time the light beam takes to pass through the jelly, as compared to air, allows us to calculate the medium's refractive index. Light beams provide a means to visually and mathematically investigate properties like speed and refractive index, critical in optics studies.
Time of Travel
In the given exercise, light takes 8.72 nanoseconds (ns) to travel through air in the tube. When the same tube is filled with jelly, this time increases by 1.82 ns, totaling 10.54 ns. Understanding this increase in travel time is fundamental in calculating the jelly's refractive index.
Another way to view the increased time is to recognize that the speed of light is effectively reduced in the jelly. By comparing the time of travel in different mediums, we can quantify how much slower light travels in jelly than in air, and thus, find the refractive index: \[ n = \frac{t_{\text{jelly}}}{t_{\text{air}}} = \frac{10.54 \text{ ns}}{8.72 \text{ ns}} \approx 1.208 \]. The refractive index provides insight into the material’s characteristics and its optical density.