Lenses: Bending Light on Purpose
Curve the glass and refraction becomes engineering: one number — the focal length — explains cameras, glasses, and why your arm gets longer every decade after forty.
Builds on: 5.1 Reflection & Refraction: Light Changes Lanes
A lens is organised refraction
A converging lens is shaped so that every parallel ray entering it is bent to pass through one point — the focus — a focal length f behind it. That single act of coordination is the whole invention; everything else is consequences. (Burning a hole in paper with sunlight is the focus made visible: the Sun’s parallel rays, concentrated to a dot.)
Where the image lands
Rays from any object point are bent to reconverge at a matching image point — a real image you can catch on a screen or a sensor, upside-down. The equation says where: distant objects (1/d₀ ≈ 0) image at the focal plane; as the object approaches, the image retreats. Focusing a camera is this equation — the lens motor changing dᵢ to keep the sensor on target.
Bring the object inside the focal length and the rays leave still diverging — no real image forms. But trace them backwards and they appear to come from a larger, upright virtual image behind the object: a magnifying glass. Same lens, two behaviours, and the focal length is the switch.
Your eye runs this equation
The eye is a converging lens (cornea plus a fine-tuning inner lens) imaging onto the retina at fixed dᵢ — so instead of moving the lens, muscles squeeze it rounder to shorten f for near objects. Myopia: eye too long, distant images land short — fixed by a diverging spectacle lens. Ageing stiffens the lens until near focus fails — hence reading glasses, and the lengthening arm. An optometrist’s prescription is in dioptres: 1/f in metres — a +2.0 D lens has f = 0.5 m, and dioptres add, which is why they are the trade’s unit.
The image on your retina is inverted, like every real image. Your brain simply wires “retinal down” to mean “up” — famously, test subjects wearing inverting goggles adapt within days, and the world flips “right way up” again. Optics delivers the data; interpretation is a software job.
⚗️ Lab — The Optical Bench
An object, a lens, a screen — slide things and watch the rays obey the equation.
- Put the object far away: the image sits at the focal plane, small and inverted.
- Slide the object toward the lens and chase the retreating image with the screen.
- Cross inside the focal length: the real image vanishes and a virtual magnifier appears.
Work it by hand — 0 / 2
No multiple choice here: compute the value and type it. Suffixes like 2.5k or 20m are understood; answers within ±2% count.