Geometric optics
Optics studies how light forms images. In geometric optics, we treat light as rays travelling in straight lines and use diagrams and equations to predict where images form.
Key terms
| Term | Meaning |
|---|---|
| Principal axis | Line through the centre of a mirror or lens |
| Focal point (F) | Where parallel rays converge (or appear to diverge from) |
| Focal length (f) | Distance from the mirror/lens to F |
| Centre of curvature (C) | For mirrors, the centre of the sphere; C = 2f |
| Real image | Formed where rays actually meet; inverted; can be projected |
| Virtual image | Formed where rays appear to come from; upright; can’t be projected |
Curved mirrors
Concave (converging) mirrors
| Object position | Image |
|---|---|
| Beyond C | Real, inverted, smaller, between C and F |
| At C | Real, inverted, same size, at C |
| Between C and F | Real, inverted, larger, beyond C |
| At F | No image (rays parallel) |
| Inside F | Virtual, upright, larger, behind mirror |
Uses: shaving and makeup mirrors (object inside F), car headlights and flashlights (bulb at F makes a parallel beam), reflecting telescopes, solar cookers.
Convex (diverging) mirrors
Always form virtual, upright, smaller images. They give a wide field of view, used in car side mirrors and store security mirrors.
Refraction and Snell’s law
The index of refraction of a material:
n = c ÷ v
| Material | n |
|---|---|
| Vacuum | 1.00 |
| Air | 1.0003 |
| Water | 1.33 |
| Crown glass | 1.52 |
| Diamond | 2.42 |
Snell’s law: n₁ sin θ₁ = n₂ sin θ₂
Example: Light passes from air (n = 1.00) into water (n = 1.33) at 40°.
sin θ₂ = (1.00 × sin 40°) ÷ 1.33 = 0.643 ÷ 1.33 = 0.483 → θ₂ ≈ 29°
Total internal reflection
When light travels from a higher-n to a lower-n medium, it bends away from the normal. At the critical angle (θ_c), the refracted ray skims the surface. Beyond it, all light reflects: total internal reflection.
sin θ_c = n₂ ÷ n₁
For water to air: sin θ_c = 1.00 ÷ 1.33 → θ_c ≈ 48.8°.
Applications:
- Fibre optics: light bounces along glass fibres for internet and medical endoscopes.
- Diamonds sparkle because their high n gives a small critical angle (about 24°), trapping light.
- Binoculars use prisms instead of mirrors.
Dispersion
The index of refraction varies slightly with wavelength. Violet bends more than red, so prisms and raindrops split white light into colours. Lenses can suffer chromatic aberration (coloured fringes) for the same reason.
Thin lenses
| Lens | Shape | f sign | Images |
|---|---|---|---|
| Converging (convex) | Thicker centre | Positive | Real or virtual (like concave mirror) |
| Diverging (concave) | Thinner centre | Negative | Always virtual, upright, smaller |
The mirror and thin lens equations
Both mirrors and lenses obey:
1/f = 1/d_o + 1/d_i
Magnification: M = h_i ÷ h_o = −d_i ÷ d_o
Sign conventions
| Quantity | Positive | Negative |
|---|---|---|
| f | Converging | Diverging |
| d_i | Real image | Virtual image |
| M | Upright | Inverted |
Example
An object is 30 cm in front of a converging lens with f = 10 cm.
- 1/d_i = 1/10 − 1/30 = 3/30 − 1/30 = 2/30 → d_i = 15 cm (real)
- M = −15 ÷ 30 = −0.5 (inverted, half size)
Optical instruments
| Instrument | How it works |
|---|---|
| Human eye | Cornea and lens focus a real, inverted image on the retina; the brain flips it |
| Camera | Converging lens forms a real image on a sensor |
| Magnifying glass | Object inside F gives a virtual, upright, enlarged image |
| Compound microscope | Objective lens forms a magnified real image; eyepiece magnifies it further |
| Refracting telescope | Large objective lens gathers light; eyepiece magnifies |
| Reflecting telescope | Concave mirror gathers light (the James Webb Space Telescope’s mirror is 6.5 m across) |
Vision correction
| Condition | Problem | Correction |
|---|---|---|
| Myopia (nearsightedness) | Image forms in front of retina | Diverging lens |
| Hyperopia (farsightedness) | Image forms behind retina | Converging lens |
Common mistakes
- Mixing sign conventions. Virtual images have negative d_i.
- Using degrees and radians inconsistently in Snell’s law.
- Thinking total internal reflection happens going into a denser medium. It only happens going to a lower n.
Try it yourself
- An object 20 cm from a concave mirror (f = 15 cm). Find d_i and M. Describe the image.
- Light goes from glass (n = 1.52) to air. Find the critical angle.
- A diverging lens has f = −12 cm. An object is 24 cm away. Find d_i and M.
(Answers: d_i = 60 cm, M = −3, real, inverted, enlarged; θ_c ≈ 41°; d_i = −8 cm, M = +0.33, virtual, upright, smaller)
Quick recap
- Real images can be projected; virtual images can’t.
- Concave mirrors and convex lenses converge light; convex mirrors and concave lenses diverge it.
- Snell’s law: n₁ sin θ₁ = n₂ sin θ₂.
- Total internal reflection occurs beyond the critical angle going to lower n.
- 1/f = 1/d_o + 1/d_i and M = −d_i/d_o apply to mirrors and lenses.