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: $$ n = \frac{\sin \left(\frac{A+\delta_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$$

References:

  • NCERT Physics, Class 11, Chapter 10: Ray Optics and Optical Instruments.
  • NCERT Physics, Class 12, Chapter 9: Ray Optics and Optical Instruments.