Describe the phenomenon of dispersion, and explain how it leads to the production of a spectrum.

Dispersion is a phenomenon in which the phase velocity of a wave depends on its frequency. In optics, it refers to the separation of light into different wavelengths or colors as it passes through a medium. This is because the refractive index of the medium varies with the wavelength of light, causing different wavelengths to travel at different speeds within the medium.

The refractive index of a substance is a measure of how much it slows down light as it passes through. The relationship between the refractive index (n) and wavelength (λ) is described by the equation: \[n(\lambda) = n_0 + \frac{C_1}{\lambda^2} + \frac{C_2}{\lambda^4} + \cdots\] where \(n_0\) is the refractive index at infinite wavelength, and \(C_1\) and \(C_2\) are the dispersion coefficients. As the wavelength changes, the refractive index also changes, which implies that different wavelengths of light travel at different speeds within the medium.

When parallel rays of light with different wavelengths pass through a medium with varying refractive index, their speeds differ, resulting in the change of their direction. This process separates the wavelengths of light, usually in the order of increasing wavelength from violet to red.

A spectrum is produced when the separated wavelengths of light are displayed in the order of their wavelengths. When a beam of white light passes through a dispersive medium such as a prism, the varying refractive index for different wavelengths of light causes the light to spread out into its component colors, creating a spectrum.

When white light enters a prism, it is refracted at the first surface and splits into its constituent colors. The shorter wavelengths (violet) are refracted more than the longer wavelengths (red). As the light continues through the prism and exits at the second surface, it is refracted again, further separating the colors. This results in the formation of a spectrum with colors ranging from violet to red. In conclusion, the dispersion phenomenon occurs when light passes through a medium with a refractive index that varies with wavelength, causing different wavelengths to travel at different speeds and change their direction. This separation of wavelengths in the medium ultimately leads to the production of a spectrum.