Question

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

The refractive index of the jelly is approximately 1.208.

Step-by-step solution

Understand the Problem

We have a light beam traveling through a cylindrical tube. Initially, the tube is filled with air, taking the light 8.72 ns to travel through it. When the tube is filled with jelly, the light takes an additional 1.82 ns, totaling 10.54 ns. We need to determine the refractive index of the jelly.

Define the Relationship

The refractive index, \( n \), of a medium is defined as the ratio of the speed of light in a vacuum, \( c \), to the speed of light in the medium, \( v \). It can also be determined by the ratio of the time taken by light in the medium to the time taken in a vacuum (or air when close to vacuum), \( n = \frac{t_{ ext{jelly}}}{t_{ ext{air}}} \).

Calculate the Time Ratios

First, note that \( t_{\text{air}} = 8.72 \) ns and \( t_{\text{jelly}} = 8.72 + 1.82 = 10.54 \) ns. We use these values to find the refractive index: \[ n = \frac{10.54}{8.72} \].

Perform the Calculation

Perform the division to calculate:\[ n = \frac{10.54}{8.72} = 1.208 \].

Interpret the Result

The refractive index of the jelly is 1.208, indicating that light travels slower in the jelly compared to air.

Key concepts

Speed of Light

The speed of light is a fundamental constant of nature and is symbolized by the letter \( c \). Light travels incredibly fast, reaching speeds of approximately \( 3 \times 10^8 \) meters per second (m/s) in a vacuum. This speed is crucial as it sets the upper limit for how fast information can travel in our universe. When light travels through other mediums, however, its speed decreases due to interactions with the particles in the medium.
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

A transparent medium is a material that allows light to pass through it with minimal scattering, letting objects be seen clearly through it. Glass, clear plastics, and water are common examples. Different transparent mediums may change the speed at which light travels through them.
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

A light beam is a narrow stream of light, typically emitted from a source like a laser or flashlight. Beams enable us to analyze how light behaves as it travels through different environments. This understanding applies in the exercise where a light beam travels through a cylindrical tube.
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

Time of travel refers to the duration it takes for light to traverse a particular distance in a given medium. This measure is crucial in determining the optical properties of materials, such as the refractive index.
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.