Light – Reflection & Refraction: Comprehensive Exam Notes

Nature of Light

• Light = form of energy enabling vision.
• FUN FACT: Light itself is invisible; we see only when it reaches our eyes after reflecting/scattering. ⇒ Explains why outer space appears dark.
• Rectilinear propagation: light travels in straight lines (foundation for ray diagrams).
• Speed of light (vacuum) $3\times10^{8}\,\text{m s}^{-1}$; speed changes with medium (basis for refraction).
• Ray = single straight-line path; beam = bundle of rays (parallel, convergent, divergent).

Reflection of Light

Key terminology

• Incident ray – incoming light.
• Reflected ray – bounced-back light.
• Normal – imaginary line ⟂ to surface at point of incidence.
• Angle of incidence $i$; angle of reflection $r$.

Laws of Reflection

1. $i=r$.
2. Incident ray, reflected ray, and normal lie in same plane.

Plane Mirror

• Characteristics of image:
  • Virtual, erect.
  • Laterally inverted (left–right reversal).
  • Same size as object.
  • Image distance $=\,$object distance (measured from mirror).
  • Focal length $f=\infty$.
• Lateral inversion application: word “AMBULANCE” painted reversed so drivers read correctly in rear-view mirrors.

Spherical Mirrors

Types

• Concave (converging, reflecting surface curved inward).
• Convex (diverging, reflecting surface curved outward).

Important geometrical terms

• Pole $P$: midpoint of mirror.
• Centre of curvature $C$: centre of the sphere of which mirror is a part.
• Radius of curvature $R$: $PC$; relation $R=2f$.
• Principal axis: straight line through $P$ and $C$.
• Principal focus $F$: point where paraxial rays converge (concave) or appear to diverge (convex).
• Focal length $f=PF$.
• Aperture: effective diameter of reflecting surface.

Ray rules for locating images (concave & convex)

1. Ray $\parallel$ principal axis ⇒ passes through $F$ (concave) / appears from $F$ (convex).
2. Ray through $F$ ⇒ emerges $\parallel$ principal axis.
3. Ray through $C$ ⇒ retraces path.
4. Ray striking at $P$ ⇒ reflects symmetrically making equal angles with axis.

Image formation by Concave Mirror

Image formation by Convex Mirror

Uses

• Concave: shaving/makeup mirrors; dentist’s mouth mirror (enlarged erect view); solar furnace (focus heats); vehicle headlights/torches/searchlights (bulb at $F$ gives parallel beam).
• Convex: rear-view mirrors (wide field, always virtual/diminished); security/surveillance mirrors.

Sign Convention for Mirrors (Cartesian)

• Object left of mirror ⇒ incident rays travel left→right.
• Distances measured from pole $P$.
  • Along incident-ray direction (+X) ⇒ positive.
  • Opposite (–X) ⇒ negative.
• Heights above axis (+Y) positive; below (–Y) negative.
• Consequences:
  • Object distance $u$ usually $-$ (since measured opposite to +X).
  • $f$ concave $<0$, convex $>0$.

Mirror Formula & Magnification

• Mirror formula: $\frac1f=\frac1v+\frac1u$.
• Linear magnification $m=\frac{h'}{h}=\frac{-v}{u}$.
• Sign & magnitude of $m$ tell image orientation & size.

Selected Numerical Illustrations

• Burning paper: place paper at focus ⇒ concentrated sunlight.
• Example: flask 15 cm tall forms 45 cm tall real image ⇒ $m=-3$ (option a).
• Object 5 cm at 10 cm from convex mirror $f=15\,\text{cm}$ ⇒ $v=+6\,\text{cm}, m=+0.6$ (image virtual, erect, height 3 cm).
• Object 4 cm at 15 cm in front of concave mirror $f=10\,\text{cm}$ ⇒ $v=-30\,\text{cm}, h'=-8\,\text{cm}$ (real, inverted, twice size).
• HW: if parallel ray meets axis 10 cm from pole ⇒ $f=10\,\text{cm}, R=2f=20\,\text{cm}$.

Refraction of Light

• Definition: bending of light at interface of two media owing to change in speed.
• Cause: speed decreases in denser medium ⇒ ray bends towards normal; increases in rarer ⇒ away.

Laws of Refraction (Snell’s law)

1. Incident ray, refracted ray & normal in same plane.
2. $\dfrac{\sin i}{\sin r}=n_{21}=\text{constant}$ (refractive index of second medium w.r.t. first).

Rectangular Glass Slab

• Two refractions give emergent ray parallel to incident; lateral displacement depends on slab thickness, incidence angle & refractive index.

Refractive Index

• Absolute: $n=\dfrac{c}{v}$ (speed in vacuum / speed in medium).
• Relative: $n<em>{21}=\dfrac{v</em>1}{v<em>2}=\dfrac{n</em>2}{n_1}$.
• Optical density ≠ mass density; governed by interaction with light (slower speed ⇒ optically denser).

Sample Problems

1. Water $n=1.33$ ⇒ speed $v=\dfrac{3\times10^{8}}{1.33}=2.26\times10^{8}\,\text{m s}^{-1}$.
2. Given $n<em>{\text{water}}=4/3, n</em>{\text{glass}}=3/2, v<em>{\text{glass}}=2\times10^{8}\,\text{m s}^{-1}$: • Speed in water $v=\dfrac{c}{n}=\dfrac{3\times10^{8}}{4/3}=2.25\times10^{8}\,\text{m s}^{-1}$. • $n</em>{g\,w}=\dfrac{n<em>g}{n</em>w}=\dfrac{3/2}{4/3}=9/8=1.125$.
3. Optically denser medium between water (1.33) & alcohol (1.36) is alcohol (greater $n$ means denser).

Spherical Lenses

Classification

• Convex (converging), Concave (diverging).

Essential terminology

• Optical centre $O$ (undeviated ray).
• Principal axis, centres of curvature $C<em>1,C</em>2$ (two), radii $R<em>1,R</em>2$.
• Principal foci $F<em>1,F</em>2$ (two); focal length $f=OF<em>1=OF</em>2$; $R=2f$.
• Aperture = effective diameter.

Ray rules

1. Ray $\parallel$ axis ⇒ through $F<em>2$ (convex) / appears from $F</em>1$ (concave).
2. Ray through $F_1$ ⇒ emerges $\parallel$ axis.
3. Ray through $O$ ⇒ undeviated.

Image formation by Convex Lens

Image formation by Concave Lens

Sign Convention for Lenses

• Origin at $O$.
• Incident light travels left→right (for usual object on left).
• Distances along incident direction (+) : $v$ for real image >0, $u=-$ (object).
• Opposite direction (–).
• Heights above axis (+), below (–).
• $f<em>{\text{convex}}>0$, f{\text{concave}}<0.

Lens Formula & Magnification

• $\dfrac1f=\dfrac1v-\dfrac1u$.
• $m=\dfrac{h'}{h}=\dfrac{v}{u}$.
• Sign conventions similar meaning as mirrors.

Power of a Lens

• $P=\dfrac{1}{f(\text{in metres})}$; SI unit dioptre (D).
• Convex: +ve; Concave: –ve.
• Combination (thin lenses in contact): $P<em>{\text{total}}=P</em>1+P_2+\dots$.

Everyday / Instrumentation Uses

• Convex: magnifying glass, hypermetropia spectacles, objective/eyepiece of microscopes & telescopes.
• Concave: myopia correction, peepholes, beam expanders, laser devices.

Representative Numericals

• Identify virtual image in lens cases: only when u<f for convex.
• Greatest power among lenses $f=5\,\text{cm}$ ⇒ $P=+20\,\text{D}$ (lens R).
• Convex lens $f=10\,\text{cm}, u=-15$ ⇒ $v=+30\,\text{cm}, m=-2$ ⇒ real, inverted, double size.
• Concave lens $f=-10\,\text{cm}, u=-15$ ⇒ $v=-6\,\text{cm}, m=+0.4$ ⇒ virtual, erect, diminished.
• Screen 20 cm behind convex lens, object 30 cm front ⇒ $f=12\,\text{cm}$; image real (on screen).
• Palmistry: convex (magnifier) lens used because it gives enlarged image; for real magnified image object between $F$ & $2F$; given $f=10\, u=-5$ ⇒ $v=+10$ cm, $m=+2$ (erect virtual).

Ethical / Practical Connections

• Understanding reflection & refraction underpins optical instrument design, from vision correction (public health impact) to renewable-energy concentrators (solar furnace).
• Safety: proper placement of mirrors/lenses (e.g., rear-view mirrors, headlights) prevents accidents.

Motivational Connection

• Quote: “Reflect positivity, refract negativity, and always focus on your goals!” – Light
  • Metaphorically ties optical behaviours to personal growth: stay focused (focal point), reflect on experiences, bend (adapt) without breaking.