Geometric Optics – Spherical Mirrors

Spherical Mirrors: Fundamental Concepts

• Spherical mirror = polished section of a sphere; reflection may occur on

  • inner surface → concave mirror

  • outer surface → convex mirror

• Everyday applications

  • Concave: shaving, cosmetic, dentist mirrors – give magnified, upright image at close range

  • Convex: vehicle rear-view mirrors, shop security, pilot’s visor – give reduced, upright image & wide field of view

Geometry & Terminology

• Pole (P): geometric centre of the mirror surface

• Centre of curvature (C): centre of the parent sphere

• Principal axis: straight line through C and P

• Radius of curvature: $R = CP$

• Focal point (F)

  • Concave: point where incident rays parallel to principal axis converge after reflection (in front of mirror)

  • Convex: point from which reflected rays appear to diverge (behind mirror)

• Focal length: $f = PF = \frac{R}{2}$

  • f>0 for concave

  • f<0 for convex

Ray-Tracing Rules (any 2 locate the image; 3rd for check)

1. Parallel ray → reflects through $F$ (concave) or appears from $F$ (convex)

2. Ray through $F$ → reflects parallel to principal axis

3. Ray through $C$ (normal incidence) → retraces its path

Image Formation with a Concave Mirror

Image Formation with a Convex Mirror

• For any object position → image is virtual, upright, diminished, behind mirror (between $F$ and $P$)

Mirror Equation & Magnification

• Mirror formula (spherical approximation): $\boxed{\frac{1}{f}=\frac{1}{u}+\frac{1}{v}}$ where

  • $u$: object distance (from P)

  • $v$: image distance (from P)

• Linear magnification: $\boxed{m=-\frac{v}{u}=\frac{h<em>i}{h</em>o}}$

  • $m>0$: image upright

  • $m<0$: image inverted

Sign Convention (Cartesian)

1. Principal axis chosen positive toward incoming light (left → right)

2. $f$ : concave $+$, convex $-$

3. $u$ : real object (+) in front of mirror, virtual object (–) behind mirror

4. $v$ : real image (+) in front, virtual image (–) behind

5. $h_i$ positive (upright), negative (inverted)

Worked Examples

• Example A (Concave – virtual, magnified)

• Given: $f=10\,\text{cm}$, $u=+6\,\text{cm}$, $h_o=1.2\,\text{m}=120\,\text{cm}$

• $\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{1}{10}-\frac{1}{6}=\frac{-1}{15} \Rightarrow v=-15\,\text{cm}$ (virtual)

• $m=-\frac{v}{u}=2.5 \Rightarrow h_i=2.5(120)=300\,\text{cm}$ (upright, 2.5× larger)

• Example B (Convex – reduced)

• Radius $R=60\,\text{cm}\Rightarrow f=-30\,\text{cm}$

• $u=+15\,\text{cm}$

• $\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{-1}{30}-\frac{1}{15}=-\frac{1}{10} \Rightarrow v=-10\,\text{cm}$ (virtual)

• $m=-\frac{v}{u}=0.67$ (upright, two-third size)

• Example C (Concave – long distance)

• Screen 3.00 m ($v=+300\,\text{cm}$) forms image of object at $u=+10\,\text{cm}$

• $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}=\frac{1}{10}+\frac{1}{300}=\frac{31}{300} \Rightarrow f=9.68\,\text{cm}$

• $R=2f=19.4\,\text{cm}$

• $m=-\frac{v}{u}=-30 \Rightarrow h_i=-30(0.5)=-15\,\text{mm}$ (real, inverted, 30×)

• Past-year quick-answer (Convex, given: $R=30\,\text{cm}$, image half-size):

• $f=-15\,\text{cm}$, $m=+0.5$ (upright)

• $m=-\frac{v}{u} \Rightarrow v=-0.5u$

• Substitute into mirror formula: $\frac{1}{-15}=\frac{1}{u}+\frac{1}{-0.5u}$ → $u=15\,\text{cm}$

Concise Summary Table (Concave)

Typical Exercises (selected)

1. Concave ($f=5\,\text{cm}$, $h_o=2\,\text{cm}$): draw/image for $u=3,5,7,C,12\,$cm & state characteristics

2. Convex ($f=-5\,\text{cm}$): drawings for $u=4,7\,$cm

3. Predict concave-mirror image when object at $C$ (real, inverted, same size)

4. Describe mirror that gives $m=0.1$ virtual image at $u=90\,$cm → convex with $f≈-10\,$cm (radius $≈-20\,$cm)

Practical & Conceptual Points

• Concave mirrors concentrate energy (solar furnaces); focal length determines intensity

• Convex mirrors increase safety by enlarging field of view; diminished images avoid overestimation of distance (“Objects in mirror are closer than they appear”)

• Sign convention consistency prevents algebraic errors in design/analysis

• Ethical / safety implication: correct mirror choice in vehicles, medical tools

Motivational Quote

“Be not afraid of going slowly; be afraid only of standing still.”

End of Chapter 1 notes – Geometric Optics: Spherical Mirrors