Ray Optics and Optical Instruments

• Reflection of Light

  • Laws of Reflection
    • Angle of incidence = Angle of reflection ($\angle i = \angle r$)
    • Incident ray, reflected ray, and normal lie in the same plane
  • Spherical Mirrors
    • Concave Mirror: Converging
    • Convex Mirror: Diverging
    • Mirror Formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
    • Linear Magnification ($m$): $m = -\frac{v}{u} = \frac{h_i}{h_o}$

• Refraction of Light

  • Laws of Refraction (Snell's Law)
    • $\frac{\sin i}{\sin r} = \text{constant} = n_{21}$
  • Refractive Index ($n$): $n = \frac{c}{v}$
  • Total Internal Reflection (TIR)
    • Condition 1: Ray travels from denser to rarer medium
    • Condition 2: Angle of incidence > Critical Angle ($i > i_c$)
    • Applications: Optical fibres, Mirage, Diamond sparkling
  • Refraction at Spherical Surfaces
    • Formula: $\frac{n_2}{v} - \n_1{u} = \frac{n_2 - n_1}{R}$
  • Lens Maker's Formula
    • $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
  • Thin Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$
  • Power of a Lens ($P$): $P = \frac{1}{f \text{ (in meters)}}$ (Unit: Dioptre)
  • Combination of Thin Lenses in Contact
    • Equivalent focal length: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} + \dots$
    • Equivalent power: $P = P_1 + P_2 + \dots$

• Refraction through a Prism

  • Angle of Deviation ($\delta$)
  • Prism Formula: $n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
  • Dispersion of Light
    • Splitting of white light into constituent colors
    • Angular Dispersion ($\delta_v - \delta_r$)
    • Dispersive Power ($\omega = \frac{n_v - n_r}{n - 1}$)

• Natural Optical Phenomena

  • Rainbow Formation (Dispersion + TIR + Refraction)
  • Scattering of Light
    • Rayleigh Scattering ($I \propto \frac{1}{\lambda^4}$)
    • Phenomena: Blue sky, Red sunset

• Optical Instruments

  • The Human Eye
    • Accommodation and defects (Myopia, Hypermetropia, Presbyopia, Astigmatism)
  • Simple Microscope (Magnifying Glass)
    • Magnifying Power ($m$)
    • Image at Least Distance of Distinct Vision ($D$): $m = 1 + \frac{D}{f}$
    • Image at Infinity: $m = \frac{D}{f}$
  • Compound Microscope
    • Two lenses: Objective and Eyepiece
    • Magnifying Power: $m = m_o \times m_e \approx -\frac{L}{f_o} \left(1 + \frac{D}{f_e}\right)$
  • Telescopes
    • Astronomical Telescope (Refracting)
      • Magnifying Power: $m = -\frac{f_o}{f_e}$
    • Reflecting Telescopes
      • Advantage: Free from chromatic and spherical aberration (Cassegrain design)