Refract at one curved surface and light already forms an image; put two back to back and you have a lens. The thin-lens law 1/v − 1/u = 1/f and the magnification m = v/u place every image — real or virtual, upright or flipped. Drag the object across the focus and watch the image leap from behind the lens to in front of it, then invert. Build the glass yourself with the lens-maker's equation, and stack lenses into instruments.
One curved boundary between air and glass already bends a diverging pencil into an image. A ray heading for the centre of curvature C strikes the surface head-on and goes straight; a ray parallel to the axis refracts through the focus F. Where they cross is the image. Drag the object and read it against the surface law.
The focus sits at f₂ = n₂R / (n₂ − n₁) inside the glass. Two of these surfaces, a whisker apart, add their powers — and that sum is the thin lens.
Two rays fix the image: one parallel to the axis bends through F, one straight through the optical centre. Drag the object from far away inward. Beyond 2F the image is real, inverted, shrunk; at 2F it's life-size; between F and 2F it's real, inverted, magnified; at F it races to infinity; inside F it flips to a virtual, erect, magnified image on the near side — the magnifying glass.
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The focal length is baked in by the two surface radii and the glass index. Bend the surfaces and the parallel beam tightens or spreads — and the shape itself morphs from biconvex through meniscus to biconcave.
Optical power P = 1/f (in dioptres, f in metres) just adds — that's why opticians talk in D. In contact the powers sum outright; pulled apart by a gap d, the combination loses a little. Slide the two lenses together and apart and watch the joint focus march in.
In contact (d = 0): 1/F = 1/f₁ + 1/f₂, so P = P₁ + P₂. Two +5 D lenses touching make +10 D. Separate them and the power drops.
Silver the back of a lens and light gets processed three times: refracted in, reflected, refracted out. The whole sandwich behaves as one equivalent mirror whose power is P = P_L + P_M + P_L — the lens counts twice. A JEE-Advanced favourite, tamed.
Every silvered lens here acts as a concave mirror — even the plano-convex with a flat silvered back (the flat mirror adds nothing, but the lens still counts twice). Classic check: equiconvex, n = 1.5 → f_eq = R/4.
Fix a candle and a screen a distance D apart and slide the lens between them. If D > 4f there are exactly two lens positions that throw a sharp image — one magnified, one shrunk. Measure their separation x and the focal length falls out without ever hunting for the image: the lab-practical way to measure f.
The two positions are conjugates: swap object and image distances and the lens equation is equally happy. So m₁·m₂ = 1, and the object size is the geometric mean of the two image sizes, O = √(I₁I₂). Shrink D below 4f and watch sharpness become impossible anywhere.
The index is bigger for violet than for red, and 1/f = (n − 1)K — so a lens has a shorter focal length for violet. White light doesn't focus at a point; it smears along the axis. The gap fr − fv = ω f is the longitudinal chromatic aberration. Cement a weak diverging flint lens to a converging crown one so their dispersions cancel but a net convergence survives — an achromatic doublet, one focus for every colour.
A telescope objective and a good camera lens are achromats. Perfect cancellation is only possible for two chosen wavelengths — a faint secondary spectrum always survives.
A warm-up, an open bench where you drag a real lens into each imaging condition and it checks you, then a JEE-Advanced / Olympiad exam bank. Collapse any rung you're done with.
Drag the object along the axis and slide the focal length until the image matches each task, then hit check.
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