### 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.