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Chapter 21: Problem 10

Briefly describe the phenomenon of dispersion in a transparent medium.

Short Answer

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Answer: Dispersion is the process in which light is separated into its individual colors (or wavelengths) as it passes through a transparent medium. This phenomenon occurs because the velocity of light varies for different wavelengths in the medium, causing a slight delay in the propagation of some wavelengths relative to others. It is caused by the wavelength dependence of a material's refractive index, resulting in different wavelengths of light refracting by different angles when passing through the medium. This separation of wavelengths forms a spectrum, as seen in rainbows and light passing through a prism.

Step by step solution

01

Define Dispersion

Dispersion is the process in which light is separated into its individual colors (or wavelengths) as it passes through a transparent medium. This phenomenon occurs because the velocity of light varies for different wavelengths in the medium, causing a slight delay in the propagation of some wavelengths relative to others.
02

Refractive Index and Light Speed

Refractive index is the ratio of the speed of light in vacuum to its speed in a given medium. Light travels at different speeds in different mediums, and this change in speed affects its direction, causing refraction. The refractive index of a material is dependent on both the material and the wavelength of light. This dependence can be expressed as \(n(\lambda) = \frac{c}{v(\lambda)}\) where \(n(\lambda)\) is the refractive index as a function of wavelength, \(c\) is the speed of light in vacuum, and \(v(\lambda)\) is the speed of light in the material for wavelength \(\lambda\).
03

Wavelength Dependence of Refractive Index

As mentioned in Step 2, the refractive index varies with wavelength. This is due to the interaction of light with the atoms and molecules that make up the medium. For most materials, the refractive index decreases with increasing wavelength (i.e., with decreasing frequency). This means that longer wavelength (red) light travels faster through a medium than shorter wavelength (blue) light. Consequently, the different wavelengths of light will refract by different angles and spread apart, forming a spectrum.
04

Snell's Law and Dispersion

Snell's Law describes the behavior of light as it travels from one medium to another. The law states that the ratio of the sines of the angles of incidence and refraction is equal to the inverse ratio of the refractive indices of the two media. Mathematically, it can be written as \(\frac{\sin\theta_1}{\sin\theta_2} = \frac{n_2}{n_1}\). Due to the wavelength-dependent refractive index, the angle of refraction (\(\theta_2\)) will also be dependent on the wavelength. Therefore, each wavelength will be refracted by different angles, resulting in dispersion.
05

The Effects of Dispersion in Transparent Media

When a beam of white light enters a transparent medium, such as a glass prism, the different wavelengths of light are refracted by different angles due to the varying refractive indices. The result is that the light is split into its constituent colors, with shorter wavelengths (violet) refracting more and longer wavelengths (red) refracting less. Dispersion is responsible for the formation of rainbows, as water droplets in the atmosphere act like tiny prisms, separating sunlight into its spectral colors.

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