SATHEE: Notes from Toppers

Refraction Through a Prism and Dispersion - Ray Optics and Optical Instruments

1. Refraction through a prism:

Refraction is the bending of light as it passes from one medium to another with a different refractive index.
Snell’s law describes quantitative behaviour of refraction: $$ n_1 \sin i_1 = n_2 \sin i_2 $$ where (n_1) and (n_2) are the refractive indices of the two media, and (i_1) and (i_2) are the angles of incidence and refraction, respectively.
Prism angle ((A)) is the angle between the two refracting surfaces of the prism.
Angle of deviation ((\delta)) is the angle between the incident ray and the emergent ray after passing through the prism.
(\delta) and ( A ) are related through the refractive index of the material ((n)) and angle of incidence ((i)):

$$ n = \frac{\sin (i + \delta ) / 2}{\sin i/2}$$

2. Dispersion

Dispersion is the spreading out of light into its component colors when passing through a prism due to the variation of refractive index with wavelength
Refractive index (n) : Measure of the bending of light as it passes from one medium to another and depends on the wavelength of light.
Relation between refractive index and wavelength: For most materials, the refractive index increases with decreasing wavelength.
Visible spectrum: ROYGBIV (Red, Orange, Yellow, Green, Blue, Indigo, Violet).
Different wavelengths (colors) of light refract at different angles causing dispersion (separation) of colors.

3. Types of spectra

Normal spectrum: The spectrum formed by white light passing through a prism, with the colors arranged in the order from Red (least refracted) to Violet (most refracted).
Abnormal spectrum: When light from a source other than a point source passes through a prism, the resulting spectrum is called an abnormal spectrum.

4. Applications:

Spectrometers: Prisms are used to separate light into different wavelengths, allowing for analysis of light sources, materials, and chemical compositions.
Rainbows: Dispersion of sunlight by raindrops leads to rainbow formation. Primary rainbow has an outer red band and inner violet band while the secondary rainbow has the order of colors reversed.
Optical instruments: Prisms are crucial components in various devices like telescopes, binoculars, periscopes, spectrometers, and polarimeters.

5. Prismatic dispersion formula:

Using geometry and Snell’s law, the angular dispersion (\delta_a) between two wavelengths (\lambda_1) and (\lambda_2) can be derived: $$ \delta_a = \left(\frac{d\delta }{d\lambda}\right) ( \lambda_2-\lambda_1)$$
where (\frac{d\delta}{d\lambda} = \frac{\Delta \delta}{\Delta \lambda }) is the rate of change of the angle of deviation with respect to the wavelength.
Prismatic dispersion depends on the refractive index variation with wavelength, known as the dispersive power ((\omega)) of the material: $$ \omega = \frac{n_{v} - n_{r}}{n_{y}-1} $$ where (n_v) and (n_r) are the refractive indices for violet and red light, respectively.

6. Chromatic Aberrations:

When light of different colors (wavelengths) is focused by a lens, the focal length varies for each color, causing chromatic aberration-colored fringes in images.
Achromat lens: Combines two different types of glass with different dispersive powers to correct for chromatic aberration, resulting in a lens that focuses different wavelengths at almost the same point.
Apochromatic lens: An advanced type of lens that uses three or more types of glass to minimize chromatic aberrations even further.

7. Prism shaped Lens

A prism can act as a lens due to its ability to refract light rays.
The focal length (f) of a prism-shaped lens can be calculated using the formula: $$ \frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2})$$
where (n) is the refractive index, (R_1) and (R_2) are the radii of curvature of the prism surfaces.

8. Rainbow

Refraction, dispersion, and total internal reflection of sunlight in spherical raindrops cause rainbows.
Raindrops act as tiny prisms that separate different wavelengths of light and reflect the red light at a higher angle than blue light.
Primary rainbows are formed when light undergoes one internal reflection, while secondary rainbows are created after two internal reflections within raindrops.

9. Experimental determination

Determination of the refractive index of a prism:
Use a spectrometer to measure the angle of deviation ((\delta)) of light for different incident angles((i)).
Plot a graph of (\sin i) versus (\sin(\delta+ i / 2)) and calculate (n) from the slope of the linear fit.
Determination of the angle of minimum deviation ((\delta_m)):
- Measure the angle of deviation ((\delta)) for different angles of incidence ((i)) and identify the angle at which the deviation is minimum ((\delta_m)).
- Calculate the refractive index using the formula: $$ n = \frac{\sin \left(\frac{A+\delta_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$$

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