The Human Eye and the Colourful World — Notes

The eye is a nearly spherical organ about 2.3 cm across that works like a camera.

• Cornea: transparent front bulge; most refraction happens at its outer surface.
• Iris: dark muscular diaphragm that controls the size of the pupil.
• Pupil: opening that regulates the light entering — small in bright light, large in dim light (the change takes time, which is why a dark hall looks black at first).
• Eye lens: crystalline, flexible; gives fine adjustment of focal length and forms a real, inverted image on the retina.
• Ciliary muscles: change the curvature of the lens.
• Retina: light-sensitive cells produce electrical signals that the optic nerve carries to the brain, which interprets them.

Power of accommodation: the ability of the lens to change its focal length. Distant object → ciliary muscles relax → lens thin → focal length longer. Near object → muscles contract → lens thick → focal length shorter. The focal length cannot be reduced below a minimum, so very near objects look blurred.

Near point (least distance of distinct vision): about 25 cm for a normal young eye. Far point: infinity.

Cataract: the lens becomes milky and cloudy in some older people, causing partial or total loss of vision; treated by surgery.

Myopia (near-sightedness): near objects clear, distant objects blurred; the far point is nearer than infinity. The image of a distant object forms in front of the retina. Causes: excessive curvature of the eye lens; elongation of the eyeball. Correction: concave lens with $f = -$(far point distance).

Hypermetropia (far-sightedness): distant objects clear, near objects blurred; the near point is farther than 25 cm. Light from a near object would focus behind the retina. Causes: focal length of the lens too long; eyeball too small. Correction: convex lens that forms a virtual image of an object at 25 cm at the person’s near point.

Presbyopia: with age the near point recedes because the ciliary muscles weaken and the lens loses flexibility. Correction: convex lens. A person with both myopia and hypermetropia uses bifocal lenses: upper part concave (distance), lower part convex (reading). Contact lenses and surgery are also used.

Method for powers: (1) decide the defect; (2) find $u$ and $v$ with signs (distances in front of the lens are negative); (3) use the lens formula; (4) convert $f$ to metres and find $P = 1/f$ with its sign.

A prism has two triangular bases and three rectangular lateral faces; the angle between the two refracting faces is the angle of the prism $A$.

At the first face light (air → glass) bends towards the normal; at the second face (glass → air) it bends away from the normal. Because the faces are inclined, the two bends add up and the emergent ray is turned towards the base.

The angle of deviation $\delta$ is the angle between the incident ray produced forward and the emergent ray produced backward. Unlike a glass slab, the emergent ray is not parallel to the incident ray and $e \neq i$ in general.

Drawing tip: draw normals at both faces first; mark $i$, $r$, $e$ and $\delta$.

Dispersion: splitting of white light into its component colours. The band of colours is a spectrum: VIBGYOR. Different colours bend through different angles: violet the most, red the least.

Newton’s experiments: (i) a second, identical prism placed inverted recombines the colours into white light, showing that white light is a mixture; (ii) a single colour from the spectrum passed through another prism is not split further. Any light that gives a spectrum like sunlight is called white light.

Rainbow: a natural spectrum seen after a shower. Water droplets act like small prisms: sunlight is refracted and dispersed on entering, internally reflected, and refracted again on leaving. A rainbow is always seen opposite the Sun — the Sun must be behind the observer. Red is on the outer edge, violet on the inner. It can also be seen in the spray of a waterfall or fountain.

The refractive index of air depends on its temperature and density: hot air is lighter and optically rarer; cooler air is optically denser. Light passing through air of changing refractive index bends: atmospheric refraction.

• Wavering objects seen through hot air above a fire or a road: the moving hot and cool layers keep changing the refraction.
• Apparent star position: starlight bends towards the normal as it enters denser lower air, so a star appears slightly higher than it is (most noticeable near the horizon).
• Twinkling of stars: the atmosphere keeps changing, so the apparent position and the amount of starlight entering the eye fluctuate. Stars are so far away that they are point sources.
• Planets do not twinkle: they are much closer and act as extended sources (many point sources); the fluctuations average out.
• Advance sunrise and delayed sunset: the Sun is seen about 2 minutes before it actually rises and about 2 minutes after it actually sets, lengthening the apparent day by about 4 minutes.

Scattering: redirection of light in many directions by particles in its path.

Tyndall effect: scattering by colloidal particles makes the path of a beam visible — a sunbeam in a smoky or dusty room, sunlight through the canopy of a misty forest, a torch beam in milk-water (but not in salt solution).

Colour depends on particle size: very fine particles scatter mainly blue (short wavelengths); larger particles scatter longer wavelengths too; particles large enough (cloud droplets) scatter all colours and look white.

Blue sky: air molecules and fine particles are smaller than the wavelength of visible light and scatter blue much more than red (red light has a wavelength about 1.8 times that of blue). Scattered blue light reaches the eye from all over the sky. Without an atmosphere the sky would look dark; it does look much darker to passengers at very high altitudes, where scattering is not prominent.