Problem 79 - College Physics

Question

\(\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?

Step-by-Step Solution

Answer

The refractive index of the jelly is approximately 1.23.

Step 1: 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.

Step 2: 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.

Step 3: 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: \[ L = c \times t_{air} = 3 \times 10^8 \times 8.72 \times 10^{-9} = 2.616 \times 10^{-9} \text{ meters} \]

Step 4: Calculate Time Taken by Light in Jelly

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

Step 5: Calculate Speed of Light in Jelly

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

Step 6: Compute the Refractive Index

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

Step 7: Conclusion

The refractive index of the jelly is approximately 1.23.

Key Concepts

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’s 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’s 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’s 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.