Ray Optics – Quick Revision

Optics Overview

• Optics ⇒ study of light phenomena: reflection, refraction, diffraction, polarisation

• Branches: Ray optics, Wave optics, Quantum optics

Light – Key Facts

• Form of energy producing sight sensation

• Particle nature: photons; Wave nature: electromagnetic, transverse (dual nature)

• Speed in vacuum: $c \approx 3 \times 10^{8}\ \text{m/s}$

• Visible wavelength range: $400\ \text{nm} - 700\ \text{nm}$

• Propagates in straight lines (rays)

Rays & Beams

• Ray: idealised straight-line path of light

• Beam: bundle of rays
  • Converging – rays meet at a point (concave mirror / convex lens)
  • Diverging – rays spread from a point (convex mirror / concave lens)
  • Parallel – rays remain parallel (distant sources)

Types of Light

• Monochromatic: single $\lambda$ (e.g.
  laser) ⇒ only refraction

• Polychromatic: multiple $\lambda$ (white light) ⇒ refraction + dispersion

Reflection of Light

• Definition: bouncing back of light in same medium

• Regular (specular): smooth surface ⇒ incident parallel beam remains parallel

• Irregular (diffuse): rough surface ⇒ rays scatter

Laws of Reflection

1. Incident ray, reflected ray, and normal lie in same plane

2. Angles equal: $\angle i = \angle r$

Key terms: incident ray, reflected ray, normal, point of incidence, glancing angles (complementary to $\angle i, \angle r$)

Plane Mirror Essentials

• Image: virtual, erect, same size, laterally inverted

• Object distance = image distance; focal length $f$ and radius of curvature $R$ → $\infty$

• Minimum mirror height for full image = half object height

• Rotation: mirror rotated by $\theta$ ⇒ reflected ray turns $2\theta$

Spherical Mirrors

• Concave (converging) – reflecting inner surface

• Convex (diverging) – reflecting outer surface

Principal Elements

• Pole $P$

• Centre of Curvature $C$ (radius $R$)

• Principal Axis (line $PC$)

• Principal Focus $F$; Focal length $f = PF$
  • Concave: real focus in front
  • Convex: virtual focus behind

• Relation: $f = \dfrac{R}{2}$ (small aperture)

• Aperture: usable diameter of mirror

Standard Rays (concave shown, reverse for convex)

1. $\text{Ray }\parallel\,\text{axis} \rightarrow \text{passes through }F$

2. $\text{Ray through }F \rightarrow \parallel\,\text{axis}$

3. $\text{Ray through }C \rightarrow \text{retrace path}$

4. $\text{Ray to pole} \rightarrow \text{reflect obeying }\angle i = \angle r$

Image Formation Summary

Concave mirror (object → image):

• Between $P$ & $F$ → behind mirror, virtual, erect, enlarged

• F < u < C → beyond $C$, real, inverted, enlarged

• $u = C$ → at $C$, real, inverted, same size

• u > C → between $F$ & $C$, real, inverted, diminished

• $u = F$ → at $\infty$, real, inverted, highly enlarged

• $u = \infty$ → at $F$, real, inverted, highly diminished

Convex mirror:

• Any finite $u$ → image between $P$ & $F$, virtual, erect, diminished

• $u = \infty$ → at $F$, virtual, erect, highly diminished

Mirror Formula & Magnification

• Mirror equation: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ (sign convention)

• Linear magnification: $m = \frac{h<em>i}{h</em>o} = -\frac{v}{u}$
  • m>0 ⇒ virtual, erect; m<0 ⇒ real, inverted

Sign Convention (pole as origin)

• Distances measured from $P$ along principal axis
  • Along incident light (left) ⇒ negative
  • Opposite direction (right) ⇒ positive

• Heights: above axis positive; below negative

• Object distance $u$ always negative; $h_o$ always positive

• Concave: $f, v, h_i, R$ usually negative (except virtual case)

• Convex: $f, v, h_i, R$ positive

Inclined Plane Mirrors – Number of Images

• Two mirrors at angle $\alpha$, object between:
  $n = \frac{360^{\circ}}{\alpha} - 1$
  • $\alpha = 180^{\circ}$ ⇒ $n = 1$
  • $\alpha = 90^{\circ}$ ⇒ $n = 3$
  • $\alpha \rightarrow 0^{\circ}$ (parallel) ⇒ infinite images

Practical Uses

• Concave: headlights, torches, solar furnace, shaving & dentist mirrors

• Convex: vehicle rear-view, CCTV, street mirrors

• Plane: household mirrors, optical instruments