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Chapter 23: Problem 79

\(\bullet$$\bullet\) 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 1.23.

Step by step solution

01

Understanding the Problem

We are given the time light takes to travel through a tube filled with air and the time in a tube filled with jelly. We need to calculate the refractive index of the jelly, using the difference in time.
02

Define the Known Quantities

Time for light to travel through air-filled tube: 8.72 ns. Additional time with jelly: 2.04 ns. Speed of light in vacuum (and approximately in air) is approximately \(3 \times 10^8\) m/s.
03

Calculate the Length of the Tube

Convert 8.72 ns into seconds: \(8.72 \times 10^{-9}\) s. Calculate length of the tube using speed \(c = 3 \times 10^8\) m/s: \[ L = c \times t_{air} = 3 \times 10^8 \times 8.72 \times 10^{-9} = 2.616 \times 10^{-9} \text{ meters} \]
04

Calculate Time Taken by Light in Jelly

Total time for light to travel through jelly-filled tube: \[ t_{jelly} = t_{air} + 2.04 \times 10^{-9} = 8.72 \times 10^{-9} + 2.04 \times 10^{-9} = 10.76 \times 10^{-9} \text{ seconds} \]
05

Calculate Speed of Light in Jelly

Using the formula for speed: \[ v_{jelly} = \frac{L}{t_{jelly}} = \frac{2.616 \times 10^{-9}}{10.76 \times 10^{-9}} \approx 2.43123 \times 10^{8} \text{ m/s} \]
06

Compute the Refractive Index

Refractive index \( n \) is given by: \[ n = \frac{c}{v_{jelly}} = \frac{3 \times 10^8}{2.43123 \times 10^8} \approx 1.23 \]
07

Conclusion

The refractive index of the jelly is approximately 1.23.

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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 that is essential in the realm of physics and optics. Designated by the symbol \( c \), it represents the speed at which light waves propagate through a vacuum. This speed is approximately \(3 \times 10^8\) meters per second (m/s). Understanding the speed of light helps us make sense of phenomena involving time and distance in the universe. When light travels through different mediums, its speed changes depending on the medium鈥檚 properties. For instance, when light travels through materials like air or jelly, it does so at different speeds, which affects the overall travel time.
Time of Flight
Time of flight refers to the period it takes for a wave, such as light, to travel from one point to another. This concept is crucial for calculating distances and understanding light propagation through various materials. In the described experiment, the time taken for light to travel through the tube is measured twice: once with air and once with jelly filling the tube.
  • For the air-filled tube, the flight time is 8.72 nanoseconds (ns).
  • With the jelly, the flight time increases by 2.04 ns, totaling 10.76 ns.
This time difference provides insights into the optical property known as refractive index, which indicates how much the medium affects light speed.
Light Propagation in Materials
When light enters a different medium, its speed decreases in a way that is characteristic of that medium. This change is determined by the material鈥檚 refractive index. Refractive index \( n \) is defined as the ratio of the speed of light in a vacuum \( c \) to the speed of light in the material \( v \). Mathematically, it is expressed as:\[ n = \frac{c}{v} \]In our experiment, the speed of light in the jelly-filled tube is approximately \(2.43123 \times 10^8\) m/s. Calculating the refractive index with this value gives us \( n = 1.23 \), meaning that light travels slower in jelly than in air. This slowing down of light is why the flight time is extended when the tube is filled with the jelly.
Hollow Cylindrical Tube Experiment
The hollow cylindrical tube experiment involves observing how light travels through a tube filled with different media. Firstly, the experiment measures how long light takes to traverse a tube containing only air, which provides a baseline for comparison. The time recorded here aligns with the expected speed of light in air. In the second phase, the tube is filled with jelly, significantly altering the medium's optical characteristics. Due to the jelly鈥檚 refractive index, light interacts with its molecules more frequently, which results in a slower propagation speed compared to air. Thus, measuring the time of flight in both scenarios lets us determine the refractive index of the jelly, aiding our understanding of light behavior in different environments.

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Most popular questions from this chapter

\(\bullet$$\bullet\) A parallel-sided plate of glass having a refractive index of 1.60 is in contact with the surface of water in a tank. A ray coming from above makes an angle of incidence of \(32.0^{\circ}\) with the top surface of the glass. What angle does this ray make with the normal in the water?

\(\bullet$$\bullet\) A ray of light traveling in a block of glass \((n=1.52)\) is incident on the top surface at an angle of \(57.2^{\circ}\) with respect to the normal in the glass. If a layer of oil is placed on the top surface of the glass, the ray is totally reflected. What is the maximum possible index of refraction of the oil?

\(\bullet\) Light of original intensity \(I_{0}\) passes through two ideal polarizing filters having their polarizing axes oriented as shown in Figure \(23.62 .\) You want to adjust the angle \(\phi\) so that the intensity at point \(P\) is equal to \(I_{0} / 10 .\) (a) If the original light is unpolarized, what should \(\phi\) be? (b) If the original light is linearly polarized in the same direction as the polarizing axis of the first polarizer the light reaches, what should \(\phi\) be?

\(\bullet\) A beam of light strikes a sheet of glass at an angle of \(57.0^{\circ}\) with the normal in air. You observe that red light makes an angle of \(38.1^{\circ}\) with the normal in the glass, while violet light makes a \(36.7^{\circ}\) angle. (a) What are the indexes of refraction of this glass for these colors of light? (b) What are the speeds of red and violet light in the glass?

\(\bullet\) The indices of refraction for violet light \((\lambda=400 \mathrm{nm})\) and red light \((\lambda=700 \mathrm{nm})\) in diamond are 2.46 and \(2.41,\) respectively. A ray of light traveling through air strikes the diamond surface at an angle of \(53.5^{\circ}\) to the normal. Calculate the angular separation between these two colors of light in the refracted ray.
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