Precise light: derivations, corrective lenses and the shape of a rainbow

This layer pushes further than a school syllabus usually goes. It asks why the formulas you have been using actually work, follows the history of measuring light's speed all the way to a precision method, and puts real numbers on total internal reflection, the rainbow and the colour of the sky.

None of this is harder to read than Investigate. It is harder in the sense that it asks for one more step of reasoning each time — not just what happens, but why it has to happen that way.

Rømer's astronomical estimate in 1676 and Fizeau's toothed wheel in 1849 both got light's speed roughly right using completely different apparatus. Neither was precise enough to be the last word. In 1879, the American physicist Albert Michelson built a rotating-mirror apparatus at the US Naval Academy that measured the speed to within a whisker of the modern value — 299,910 km/s, only about 0.04% above 299,792.458 km/s. Michelson kept refining his methods for decades and won the 1907 Nobel Prize in Physics, the first American to win a science Nobel, substantially for this work.

Michelson's apparatus was, in spirit, a much faster and much more precise cousin of Fizeau's toothed wheel. Instead of a wheel with teeth, he used an octagonal rotating mirror: light bounced off one face of the spinning mirror, travelled about 35 km to a fixed mirror on a distant mountain and back, and was only caught by the eyepiece if the octagon had turned by exactly one-eighth of a full turn (or a whole number of eighths) in the time the light was away. Spinning the mirror faster or slower shifted the returning beam out of alignment with the eyepiece, and finding the exact matching speeds — Michelson used several, as a cross-check — gave the round-trip time with far less guesswork than judging by eye exactly when a wheel's light had vanished.

The relationship 1/v = 1/f − 1/u (with u, v and f all measured as positive distances for a real object and, respectively, a real image or a positive focal length) is not an arbitrary rule to memorise — it falls directly out of similar triangles, in almost the same way as the shadow-size rule from Understand.

Drawing a ray diagram, three rays are enough to fix an image exactly, and each obeys a simple, memorable rule:

The ray model explains what refraction does, but not why a boundary bends light in the first place. In 1678, the Dutch physicist Christiaan Huygens — the same Huygens who turned Rømer's observation into a speed — proposed a picture that answers exactly that: every point on an advancing wavefront of light can itself be treated as a source of tiny new wavelets, and the overall wavefront a moment later is simply the outer edge of all those wavelets added together.

A normally sighted eye can focus on anything from about 25 cm away (the near point) out to the far horizon (the far point, effectively infinity). Two very common departures from this are treated with an extra lens placed in front of the eye's own lens system — spectacles or contact lenses — precisely so that the combined power lands back on 25 cm and infinity.

An optical fibre keeps light bouncing down a hair-thin glass core by total internal reflection at the boundary with a surrounding layer of slightly lower refractive index (the cladding), at an angle always kept steeper than the critical angle. Almost the entire internet's long-distance traffic — video calls, this very page, financial trades between cities — travels this way today, not as electricity in a copper wire.

The 42° figure is not a modern discovery. In 1637, the French philosopher and mathematician René Descartes worked it out using nothing but Snell's law (which he had a hand in formulating) and a large glass sphere filled with water standing in for a giant raindrop, decades before Newton split light with a prism and nearly three centuries before anyone had a wave or particle theory of light to explain why Snell's law holds.

Investigate told you the primary bow sits at about 42° because deviation reaches a minimum near one particular entry angle, and rays bunch up around that minimum. Here is the actual shape of that minimum, worked out point by point rather than asserted.

Rayleigh's scattering law states that the intensity scattered by particles much smaller than a wavelength is proportional to 1 ÷ λ⁴, where λ is the wavelength. That fourth power is what makes the sky's blue so decisive rather than a weak trend: red light (700 nm) is scattered about 5.9 times less than blue (450 nm), and about 9.4 times less than violet (400 nm) — because (700 ÷ 450)⁴ ≈ 5.9 and (700 ÷ 400)⁴ ≈ 9.4.

Additive and subtractive colour mixing are opposite operations on the same underlying spectrum. Additive mixing (coloured lights) adds wavelengths together: shine red and green light on the same white screen and both sets of wavelengths reach your eye at once, which your brain reads as yellow — no yellow wavelength is actually present. Subtractive mixing (paints, inks, filters) works by removing wavelengths: a cyan pigment absorbs red light and reflects blue and green; a yellow pigment absorbs blue and reflects red and green; mix cyan and yellow paint and each removes what the other would have reflected, leaving mostly green.