Ray Optics: Spherical Mirrors Notes

Fundamentals of Spherical Mirrors

• Spherical Mirror Types:

  • Convex Mirror: A curved mirror with the reflecting surface on the outside of the curve.
  • Concave Mirror: A curved mirror with the reflecting surface on the inside of the curve.

• Key Terminology:

  • Center of Curvature ($C$): The center of the sphere of which the mirror is a part.
  • Radius of Curvature ($R$): The radius of the sphere of which the mirror is a part.
  • Pole ($P$): The center of the spherical mirror.
  • Principal Axis: The straight line passing through the center of curvature and the pole.
  • Principal Focus ($F$):
  • In a concave mirror, parallel rays meet at this point on the principal axis after reflection.
  • In a convex mirror, parallel rays diverge and appear to come from this point on the principal axis.
  • Focal Length ($f$): The distance from the pole to the principal focus.

• Focal Length Formula:

  • $f = \frac{R}{2}$

Light Ray Rules & Aberration

• Paraxial Rays & Spherical Aberration:

  • Paraxial Rays: Rays parallel to the principal axis making an angle of incidence close to zero ($< 10^\circ$).
  • Spherical Aberration: Distortion or blur in an image produced by a spherical concave mirror when marginal rays and paraxial rays fail to meet at the same point.
  • Minimization: Can be minimized by using a smaller mirror or an aspherical mirror.

• Rules for Drawing Ray Diagrams:

  1. A ray parallel to the principal axis passes through (or appears to diverge from) the focus after reflection.
  2. A ray passing through (or directed toward) the focus becomes parallel to the principal axis after reflection.
  3. A ray passing through (or directed toward) the center of curvature follows the same path back after reflection.
  4. A ray incident at the pole is reflected back making the same angle with the principal axis.

Image Formation Summary

• Concave Mirror Image Formation:

  • At Infinity: Image at $F$; real, inverted, and point-sized.
  • Between Infinity and $C$ ($> 2f$): Image between $F$ and $C$; real, inverted, and reduced.
  • At $C$ ($2f$): Image at $C$; real, inverted, and same size.
  • Between $C$ and $F$: Image between $C$ and infinity; real, inverted, and enlarged.
  • At $F$: Image at infinity; real, inverted, and infinitely large.
  • Between $P$ and $F$: Image behind the mirror; virtual, erect, and enlarged.

• Convex Mirror Image Formation:

  • Any Position (Distant or Nearby): Image behind the mirror between $F$ and $P$; virtual, erect, and reduced.

Uses of Spherical Mirrors

• Uses of Concave Mirrors:

  • Reflectors in searchlights, vehicle headlights, and torches.
  • Reflectors in solar cookers to focus solar radiation.
  • Shaving mirrors, makeup mirrors, and dentist mirrors.
  • Reflecting telescopes.

• Uses of Convex Mirrors:

  • Reflectors in street lamps.
  • Rear-view mirrors in motor vehicles due to their large field of view.
  • Shop security mirrors and traffic safety mirrors.

Cartesian Sign Convention

• Sign Rules for Mirrors:
  1. The principal axis is taken as the x-axis with the pole ($P$) as the origin. All axial distances are measured from the pole.
  2. The direction of incident light is taken as the positive direction of the x-axis.
  3. Upward vertical distances perpendicular to the principal axis are positive.
  4. Downward vertical distances perpendicular to the principal axis are negative.