Notes on Refraction of Light - Ray Optics and Optical Instruments

1. Fundamental Concepts of Refraction

• Snell’s Law:
  • Relates the angles of incidence and refraction to the refractive indices of the two media.
  • Mathematically expressed as: $$ n_1 \sin\theta_1 = n_2 \sin\theta_2 $$
• Refractive index (n):
  • Ratio of the speed of light in vacuum to the speed of light in the medium.
  • $n = \frac{c}{v}$
• Relation between refractive index (n), speed of light (v), and wavelength (λ):
  • $$ v = \frac{c}{n} \text{ and } \lambda = \frac{\lambda_0}{n}$$

2. Refraction through Different Surfaces

• Plane surface:

  • Deviation produced is independent of the angle of incidence.
  • Refracted ray remains parallel to the incident ray.

• Spherical surface:

  • Refraction occurs at every point on the surface.
  • Focal length determined by the radius of curvature and the refractive indices.

• Total Internal Reflection (TIR):

  • Occurs when light strikes a boundary between two media with a higher refractive index to a lower refractive index at an angle greater than the critical angle.
  • Critical angle: $$ \theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right)$$

3. Lenses and their Characteristics

• Types of lenses:
  • Convex lens: Converging lens, positive focal length.
  • Concave lens: Diverging lens, negative focal length.
• Focal length (f):
  • Distance between the optical center and the principal focus of the lens.
  • Determined using the lens maker’s equation: $$ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $$
• Lens maker’s equation:
  • Relates the focal length of the lens to the refractive index of the lens material and the radii of curvature of its surfaces.

4. Image Formation and Ray Diagrams

• Ray diagrams:
  • Essential tool for understanding image formation.
  • Illustrate the path of light rays through the lens and help determine image location and characteristics.
• Image formation by lenses:
  • Real images: Formed by the convergence of light rays after passing through the lens.
  • Virtual images: Formed by the divergence of light rays that appear to originate from the lens.
• Image distance and magnification:
  • Image distance (v): Distance between the lens and the image.
  • Magnification (m): Ratio of the image height to the object height. $$m = \frac{h_i}{h_0} = -\frac{v}{u} $$
• Lateral magnification:
  • Ratio of the image width to the object width. Describes the change in the size of the image in the horizontal direction.

5. Aberrations in Lenses

• Spherical aberration:
  • Occurs due to the variation of focal length with the distance from the optical axis.
  • Results in images with blurred edges.
• Chromatic aberration:
  • Occurs due to the variation of refractive index with wavelength.
  • Results in images with colored fringes.
• Methods to minimize aberrations:
  • Using a diaphragm to reduce the effective aperture.
  • Using lenses made of special optical glasses that have low dispersion.

6. Optical Instruments

• Simple microscope:
  • Consists of a single convex lens.
  • Magnifying power: $$ M = 1 + \frac{D}{f} $$
• Compound microscope:
  • Consists of an objective lens and an eyepiece lens.
  • Magnifying power: $$ M = m_0 \times m_e = \frac{L}{(f_o + d)} \times \frac{D}{f_e} $$
• Reflecting telescope:
  • Uses mirrors to gather and focus light.
  • Magnifying power: $$ M = \frac{f_0}{f_e} $$
• Refracting telescope:
  • Uses lenses to gather and focus light.
  • Magnifying power: $$ M = \frac{f_0}{f_e} $$
• Prisms and their uses:
  • Prisms can disperse white light into a spectrum.
  • Used in prism spectrometers to measure wavelengths of light.

7. Wave Optics vs. Ray Optics

• Limitations of ray optics:
  • Ray optics assumes light travels in straight lines, which is not entirely true.
  • Fails to explain interference, diffraction, and polarization of light.
• Introduction to wave optics:
  • Considers the wave nature of light.
  • Explains interference, diffraction, and polarization of light.

Note:

• The NCERT books for class 11th and 12th cover all the topics and concepts mentioned above.
• For a comprehensive preparation for the JEE exam, it is recommended to thoroughly understand the concepts, solve related numerical problems and practice drawing accurate ray diagrams.