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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 1.24.

Step by step solution

01

Understanding the Problem

We have a cylindrical tube and a laser light traveling through it. We are given the time taken for light to travel through the tube filled with air and filled with jelly. We need to find the refractive index of the jelly.
02

Identify Key Variables

Let the length of the tube be denoted by \( L \). The time taken by light through air is 8.72 ns, and through jelly is 8.72 ns + 2.04 ns = 10.76 ns. Speed of light in air (or vacuum) is \( c = 3 \times 10^8 \) m/s.
03

Formulate Speed Equations

The speed of light through air is \( v_{\text{air}} = \frac{L}{8.72} \). The speed of light through jelly is \( v_{\text{jelly}} = \frac{L}{10.76} \).
04

Calculate Length of the Tube

Using speed in air, \( L = v_{\text{air}} \times 8.72 = c \times 8.72 \times 10^{-9} = 2.616 \times 10^{-9} \) meters.
05

Calculate Speed of Light in Jelly

Using the length of the tube calculated previously, \( v_{\text{jelly}} = \frac{L}{10.76} = \frac{2.616 \times 10^{-9}}{10.76 \times 10^{-9}} = 2.43 \times 10^8 \) m/s.
06

Calculate the Refractive Index

The refractive index \( n \) is given by \( n = \frac{c}{v_{\text{jelly}}} = \frac{3 \times 10^8}{2.43 \times 10^8} \approx 1.24\).

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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, often symbolized as "c," and has an approximate value of \(3 \times 10^8\) meters per second (m/s) in a vacuum. This speed is the fastest known in the universe and plays a crucial role in many areas of physics, including Einstein's theory of relativity.

In different mediums, the speed of light can change. For instance, when light travels through air, its speed is slightly less than in a vacuum. Materials like water and glass slow it down even more, affecting how we perceive everything from rainbows to eyeglasses.
  • Speed in vacuum: \(c = 3 \times 10^8\) m/s
  • Speed in air: slightly less than \(c\)
  • Speed in other media: depends on the medium
Changes in speed cause phenomena such as bending of light rays, known as refraction, important for the concept we're discussing here.
Light Transmission
Light transmission is the process of light traveling through a medium. This process is lengthened if the medium has properties that slow down the light, such as a higher refractive index.

In our everyday experience, we witness light transmission in transparent materials like glass, water, or air. When light moves from one medium to another, its speed changes, and this can affect how light behaves within the medium. It can create interesting effects like distortion in water or magnification in lenses.
  • Media transmitting light: air, glass, water, etc.
  • Speed alteration: caused by the medium's properties
  • Refraction: bending of light based on speed change
Understanding these changes in speed during light transmission is key to technologies such as fiber optics and cameras.
Optics
Optics is the branch of physics that studies the behavior and properties of light. It includes understanding phenomena like reflection, refraction, and diffraction. Optics is crucial in technologies that use lenses and involve light manipulation.

In optics, concepts such as the refractive index are fundamental. They describe how much a medium can slow down light compared to a vacuum. This is essential in designing lenses for glasses, microscopes, and even within cameras.
  • Refractive index: measure of light speed reduction
  • Applications: eyeglasses, cameras, sensors
  • Phenomena: reflection (bouncing back), refraction (bending)
By mastering optics, we can enhance visibility, improve vision correction, and even design better-quality optical equipment.
Transparent Media
Transparent media are materials that allow light to pass through them with little to no obstruction, making them crucial for vision and various technological applications. Examples include air, glass, and certain plastics.

The transparency of a medium is directly related to its refractive index. A lower refractive index means that less light is slowed down or bent as it travels through the medium. This is why clear glass provides an almost perfect vision through it, while water slightly bends images.
  • Examples: air, glass, clear plastics
  • Refractive index role: determines degree of light bending
  • Importance: essential for lenses and improving sight
These materials are vital for crafting everything from simple windows to complex optical lenses used in scientific research.

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Most popular questions from this chapter

Light with a frequency of \(5.80 \times 10^{14} \mathrm{Hz}\) travels in a block of glass that has an index of refraction of \(1.52 .\) What is the wavelength of the light (a) in vacuum and (b) in the glass?

A ray of light traveling in water is incident on an interface with a flat piece of glass. The wavelength of the light in the water is 726 \(\mathrm{nm}\) and its wavelength in the glass is 544 \(\mathrm{nm} .\) If the ray in water makes an angle of \(42.0^{\circ}\) with respect to the normal to the interface, what angle does the refracted ray in the glass make withrespect to the normal?

A ray of light is incident on a plane surface separating two sheets of glass with refractive indexes 1.70 and 1.58 . The angle of incidence is \(62.0^{\circ},\) and the ray originates in the glass with \(n=1.70 .\) Compute the angle of refraction.

A thin layer of ice \((n=1.309)\) floats on the surface of water \((n=1.333)\) in a bucket. A ray of light from the bottom of the bucket travels upward through the water. (a) What is the largest angle with respect to the normal that the ray can make at the ice-water interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

(a) A tank containing methanol has walls 2.50 \(\mathrm{cm}\) thick made of glass of refractive index \(1.550 .\) Light from the outside air strikes the glass at a \(41.3^{\circ}\) angle with the normal to the glass. Find the angle the light makes with the normal in the methanol. (b) The tank is emptied and refilled with an unknown liquid. If light incident at the same angle as in part (a) enters the liquid in the tank at an angle of \(20.2^{\circ}\) from the normal, what is the refractive index of the unknown liquid?

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