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Chapter 33: Problem 38

A light beam is directed parallel to the axis of a hollow cylindrical tube. When the tube contains only air, it takes the light 8.72 ns to travel the length of the tube, but when the tube is filled with a transparent jelly, it takes the light 2.04 ns longer to travel its length. What is the refractive index of this jelly?

Short Answer

Expert verified
The refractive index of the jelly is approximately 0.81.

Step by step solution

01

Understand the Problem

We need to calculate the refractive index of the jelly in the tube. We know the time light takes to travel through air and through jelly, and we can use these to find the speeds of light in both mediums. Then, we can determine the refractive index, which is the ratio of the speed of light in air to the speed of light in jelly.
02

Write the Formula for Refractive Index

The refractive index \( n \) is given by the formula \( n = \frac{c}{v_j} \), where \( c \) is the speed of light in air and \( v_j \) is the speed of light in the jelly.
03

Calculate the Speed of Light in Air

The time taken for light to travel through the air is 8.72 ns. Assuming the length of the tube is \( L \), we have \( v_{air} = \frac{L}{8.72 \times 10^{-9}} \).
04

Calculate the Time Taken in Jelly

The total time taken when the jelly is in the tube is 8.72 ns + 2.04 ns = 10.76 ns.
05

Calculate the Speed of Light in Jelly

Using the same length \( L \), the speed of light in jelly is \( v_j = \frac{L}{10.76 \times 10^{-9}} \).
06

Substitute into the Refractive Index Formula

Now substitute \( v_{j} = \frac{L}{10.76 \times 10^{-9}} \) into the refractive index formula: \( n = \frac{8.72}{10.76} \).
07

Calculate the Refractive Index

Perform the division to get the refractive index: \( n = \frac{8.72}{10.76} \approx 0.81 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed of Light
The speed of light is a fundamental constant of nature, often symbolized by the letter "c" in physics. It is the speed at which light travels in a vacuum and has a value of approximately 299,792,458 meters per second.
When light travels through materials other than a vacuum, its speed is reduced.
  • In our exercise, we consider light traveling through air and through a jelly.
  • The speed in air is slightly less than in a vacuum because air is not completely devoid of material.
  • In the jelly, the speed reduces more significantly due to the jelly's density and composition.
This reduction in speed when passing through materials forms the basis for calculating the refractive index, which describes how much the medium slows the light.
Optics
Optics is the branch of physics concerned with the study of light. It deals with the behavior and properties of light, including reflection, refraction, and dispersion.
Understanding optics involves
  • how light interacts with different surfaces and media,
  • the principles that govern light's behavior, and
  • the design of devices that use light, called optical instruments.
In the context of our exercise, we use principles of optics to calculate how much the jelly bends and slows the light waves, essential for determining the refractive index. Optics provides the framework for how we can mathematically assess and predict light's behavior when transitioning between different media.
Light Propagation
Light propagation describes the way light moves through space and various media. In scientific terms, it's about how the wavefront of light changes over time and distance.
When light moves from one medium to another:
  • Its speed changes, often slowing down due to interaction with the medium's molecules.
  • Its direction may also change, a phenomenon known as refraction.
  • Depending on the medium's properties, light can also scatter or be absorbed.
In our example, light initially travels through air. It then enters a jelly, altering its speed. By calculating the time it takes to traverse each, we can measure the refractive index, which illustrates the extent to which light propagation is affected.
Medium
A medium is any material through which light can travel. Different media have unique optical properties that affect the speed and direction of light.
In the problem at hand:
  • We observe light traveling through two different media: air and jelly.
  • Air serves as a relatively fast medium for light, offering minimal resistance.
  • Jelly, on the other hand, presents a more significant obstacle, slowing light more significantly.
The difference in light speed between these mediums is quantified by the refractive index. Each medium's ability to slow or alter the path of light illustrates how crucial the medium's characteristics are to the behavior of light waves passing through it.

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Most popular questions from this chapter

A horizontal, parallel-sided plate of glass having a refractive index of 1.52 is in contact with the surface of water in a tank. A ray coming from above in air makes an angle of incidence of \(35.0^{\circ}\) with the normal to the top surface of the glass. (a) What angle does the ray refracted into the water make with the normal to the surface? (b) What is the dependence of this angle on the refractive index of the glass?

A ray of light in diamond (index of refraction 2.42 ) is incident on an interface with air. What is the largest angle the ray can make with the normal and not be totally reflected back into the diamond?

A parallel beam of light in air makes an angle of \(47.5^{\circ}\) with the surface of a glass plate having a refractive index of 1.66 . (a) What is the angle between the reflected part of the beam and the surface of the glass? (b) What is the angle between the refracted beam and the surface of the glass?

Three Polarizing filters. Three polarizing filters are stacked with the polarizing axes of the second and third at \(45.0^{\circ}\) and \(90.0^{\circ}\) , respectively, with that of the first. (a) If unpolarized light of intensity \(I_{0}\) is incident on the stack, find the intensity and state of polarization of light emerging from each filter \((b)\) If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

Light traveling in air is incident on the surface of a block of plastic at an angle of \(62.7^{\circ}\) to the normal and is bent so that it makes a \(48.1^{\circ}\) angle with the normal in the plastic. Find the speed of light in the plastic.
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