Light and Optics

Light is a form of energy that makes objects visible to us. In physics we say that light is an electromagnetic wave: it needs no medium to travel, so it reaches us from the Sun across empty space. Our eyes can detect only a narrow band of this family, roughly 400 nm (violet) to 700 nm (red), which we call visible light. In a uniform medium light travels in straight lines (rectilinear propagation), and that single fact explains shadows, eclipses and the pinhole camera.

Why did scientists need more than one theory? Because light behaves like a wave in some experiments (interference, diffraction, polarisation) and like a particle in others (photoelectric effect). Today we accept the dual nature of light. Interference is a direct result of the superposition principle of waves, and polarisation proves that light waves are transverse.

Speed of light. In vacuum light travels at about 3 × 10⁸ m/s (exactly 299,792,458 m/s), the highest speed possible in nature. Light from the Sun takes about 8 minutes 20 seconds to reach us, and moonlight about 1.3 seconds. A light year is the distance light covers in one year, about 9.46 × 10¹² km, and it is a unit of distance, not of time. Olaus Roemer made the first estimate of the speed of light in 1676 using the eclipses of Jupiter's moon Io; Fizeau (1849) and Foucault (1850) measured it on Earth.

Reflection. When light strikes a smooth surface and bounces back into the same medium, this is reflection. The two laws of reflection hold for every kind of surface: (1) the angle of incidence equals the angle of reflection (∠i = ∠r); (2) the incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane. The normal is the line perpendicular to the surface at that point (for a spherical mirror it is the radius through that point). Reflection from a smooth surface is regular; from a rough surface it is diffuse, but even then each tiny ray obeys the laws.

Plane mirror. The image in a plane mirror is virtual, erect, of the same size as the object, and formed as far behind the mirror as the object is in front. It is also laterally inverted: left appears as right. That is why AMBULANCE is written reversed on the front of the vehicle, so that a driver sees it correctly in the rear-view mirror. If a mirror is rotated by an angle θ, the reflected ray turns by 2θ. Two mirrors inclined at an angle θ form (360°/θ) − 1 images when 360°/θ is an even number; two parallel mirrors form an infinite number of images. A person needs a mirror at least half his height to see his full image. A kaleidoscope uses mirrors at 60° to make repeating patterns, and a periscope uses two plane mirrors to see over obstacles.

A spherical mirror is a part of a hollow sphere whose one side is polished. If the inner (hollow) surface reflects, it is a concave mirror; if the outer (bulging) surface reflects, it is a convex mirror. Think of the back and the front of a steel spoon: the inside of the bowl is concave, the outside is convex. Every numerical and every ray diagram uses a few fixed terms, so learn them first.

Why f = R/2? For a small-aperture mirror, a ray parallel to the axis meets the normal (the radius) at equal angles, so the reflected ray cuts the axis exactly halfway between P and C. Both P, F and C always lie on the principal axis, but the point of incidence does not.

Pattern behind the table: as the object moves from infinity towards F, the real image moves away from F towards infinity and grows. The moment the object crosses F and comes between F and P, the image jumps behind the mirror and becomes virtual, erect and magnified. A convex mirror always gives a virtual, erect, diminished image behind the mirror, wherever the object is, with the image lying between P and F.

Why these uses? A bulb placed at the focus of a concave mirror sends out a parallel beam, so headlights and torches throw light far. Parallel sunlight falling on a concave mirror is concentrated at its focus, which is the working of a solar furnace or solar cooker. A shaving mirror held close (inside F) shows an enlarged erect face. A convex mirror spreads the rays out, so it shows a wider field of view and an erect, smaller image, exactly what a driver needs in a rear-view mirror.

Sign convention and formulae (New Cartesian). Take the pole as origin. Distances measured in the direction of the incident light are positive; against it, negative. A real object is always in front, so u is negative. For a concave mirror f is negative; for a convex mirror f is positive. Then: mirror formula 1/v + 1/u = 1/f, and magnification m = h′/h = −v/u. A negative m means a real, inverted image; a positive m means a virtual, erect image; |m| > 1 means enlarged.

Worked example (as asked by SSC). A 1 cm object stands 24 cm in front of a concave mirror of focal length 16 cm. Here u = −24, f = −16. 1/v = 1/f − 1/u = −1/16 + 1/24 = −1/48, so v = −48 cm (real, in front). m = −v/u = −(−48)/(−24) = −2. Image height = −2 cm: inverted and 2 cm tall.

Refraction is the bending of light when it passes obliquely from one transparent medium into another. The cause is a change in speed: light slows down in a denser medium and speeds up in a rarer one. The ray bends towards the normal when it enters a denser medium (air to water) and away from the normal when it enters a rarer medium (water to air). A ray falling along the normal does not bend at all, though its speed still changes.

Laws of refraction. (1) The incident ray, the refracted ray and the normal at the point of incidence lie in the same plane. (2) Snell's law: for light of a given colour and a given pair of media, sin i / sin r = constant, and this constant is the refractive index of the second medium with respect to the first.

Refractive index. The absolute refractive index of a medium is n = c / v, the ratio of the speed of light in vacuum to its speed in that medium. It is a pure ratio, so it has no unit, and it is always greater than 1 for any material. A medium with a larger n is called optically denser. Optical density is not the same as mass density: turpentine and kerosene have a higher refractive index than water but a lower mass density.

Everyday effects of refraction. A pencil in a glass of water looks bent; a swimming pool looks shallower than it is, because rays from the bottom bend away from the normal as they leave the water (real depth / apparent depth = n); a coin at the bottom of a cup appears raised. When light crosses a rectangular glass slab, the emergent ray is parallel to the incident ray but shifted sideways (lateral displacement). A prism has two refracting faces inclined at an angle called the prism angle.

Total internal reflection (TIR). When light goes from a denser to a rarer medium, it bends away from the normal. As the angle of incidence grows, the refracted ray skims along the surface; the incidence angle at which this happens is the critical angle C, and sin C = 1 / n. If the angle of incidence becomes greater than C, there is no refraction at all and the whole light is reflected back into the denser medium. Two conditions are essential: (1) light must travel from a denser to a rarer medium; (2) the angle of incidence must be greater than the critical angle.

A lens is a piece of transparent material (usually glass) bounded by two curved surfaces. A convex lens is thicker at the middle than at the edges and converges parallel rays; a concave lens is thinner at the middle and diverges them. The mirror was about reflection; a lens works by refraction, and light passes through it. Like a mirror, it has an optical centre, a principal axis and foci, but a lens has two principal foci (one on each side).

The convex-lens table follows the same logic as the concave mirror, with one difference: real images form on the opposite side of the lens, while virtual images form on the same side as the object. A concave lens gives, for every position of the object, a virtual, erect, diminished image between F and O on the same side as the object. So a single convex lens can be a camera lens (real image) or a magnifying glass (virtual image), but a concave lens never gives a real image.

Lens formula and magnification. 1/v − 1/u = 1/f and m = v/u = h′/h. The sign convention is the same Cartesian one: u is negative for a real object; f is positive for a convex lens and negative for a concave lens. Note the difference from the mirror formula: in the lens formula the sign between 1/v and 1/u is minus.

Power of a lens. The power measures how strongly a lens bends light: P = 1/f, with f in metres. The SI unit is the dioptre (D). A convex lens has positive power and a concave lens negative power. A lens of focal length +50 cm (0.5 m) has power +2 D; a concave lens of power −2.5 D has f = −40 cm. For lenses placed in contact, the powers simply add: P = P₁ + P₂ (and 1/f = 1/f₁ + 1/f₂). Opticians prescribe spectacles in dioptres.

Worked example. An object is 30 cm from a convex lens of f = 20 cm. u = −30, f = +20. 1/v = 1/f + 1/u = 1/20 − 1/30 = 1/60, so v = +60 cm. m = v/u = 60/(−30) = −2: the image is real, inverted and twice the size.

The human eye works like a camera. Light enters through the transparent cornea, passes through the pupil, is focused by the crystalline lens, and forms a real, inverted image on the retina. The retina's light-sensitive cells send signals through the optic nerve to the brain, which turns the image upright and gives us the feeling of sight.

Important numbers. The eye lens can change its focal length to see objects at different distances, an ability called accommodation. A normal eye sees clearly from 25 cm (near point, also called the least distance of distinct vision) to infinity (far point). The image on the retina stays for about 1/16 second after the object is removed (persistence of vision), which is why a series of pictures looks like a moving film. Two eyes give binocular vision, which helps in judging depth. The cornea can be donated after death, but not the whole eye.

Optical instruments. A simple microscope is one convex lens of short focal length (magnifying glass). A compound microscope has two convex lenses: a small-focal-length objective near the object makes a real, enlarged image, and the eyepiece magnifies it further into a virtual, enlarged final image. A refracting telescope has a large objective of long focal length and a small eyepiece; a reflecting telescope replaces the objective lens by a large concave mirror (Newton built the first one) and avoids colour defects. A radio telescope collects radio waves from space; the cosmic microwave background, lying in the microwave–radio region, is studied with radio-type telescopes. India's MACE at Hanle (Ladakh) detects very high-energy gamma rays indirectly through the faint Cherenkov light they produce in the atmosphere. A camera works like the eye: lens = eye lens, sensor/film = retina, aperture = pupil.

A glass prism is a transparent solid with two triangular bases and three rectangular faces. When a ray passes through it, it bends twice, once at each refracting face, and bends towards the base of the prism. The angle between the two refracting faces is the prism angle (angle of the prism), and the angle between the incident ray and the emergent ray is the angle of deviation. Unlike a rectangular slab, a prism does not give a parallel emergent ray.

Dispersion. White light is a mixture of seven colours. When it passes through a prism, each colour bends by a different amount, so the white light splits into a band of colours called a spectrum. This splitting is dispersion. The reason: in glass the different colours travel at different speeds, so glass has a slightly different refractive index for each colour. Violet is slowed the most (highest refractive index) and bends the most; red is slowed the least and bends the least. In vacuum all colours travel at the same speed, so there is no dispersion there. Isaac Newton (1666) was the first to obtain the spectrum of sunlight with a glass prism; with a second inverted prism he recombined the colours into white light.

Colour of objects. An opaque object shows the colour it reflects and absorbs the rest. A red rose reflects red; in green light it looks black because there is no red to reflect. A white object reflects all colours; a black object absorbs all. A transparent coloured glass transmits its own colour and absorbs the others.

Rainbow. A rainbow is dispersion of sunlight by millions of tiny water droplets. Inside each drop three things happen in order: refraction and dispersion as light enters, total internal reflection at the back of the drop, and refraction again as the light leaves. You see a rainbow only when the Sun is behind you and the rain is in front, so it appears in the sky opposite to the Sun, usually in the morning or evening when the Sun is low. In a primary rainbow red is on the outer (upper) edge and violet on the inner edge. A fainter secondary rainbow forms above it after two internal reflections, and its colour order is reversed: violet outside, red inside.

Scattering is the spreading of light in all directions when it strikes tiny particles such as air molecules, dust, smoke or water droplets. The amount of scattering depends on the size of the particle and the wavelength of light. For very small particles (air molecules, smaller than the wavelength) scattering is far stronger for short wavelengths: blue and violet scatter much more than red (Rayleigh scattering, roughly proportional to 1/λ⁴). Large particles (water droplets, dust) scatter all colours almost equally, so they look white.

Tyndall effect. When a beam of light passes through a colloid or a suspension, the particles scatter the light and make the path of the beam visible. John Tyndall studied this. It is seen when sunlight enters a dense forest through the canopy of leaves and mist, when a torch beam passes through smoke or a dusty room, or when a beam of light comes through a small hole into a dark room. The colour of the scattered light depends on the size of the scattering particles: very fine particles scatter mainly blue light, larger particles give whitish light.

Sea water looks blue partly because it reflects the blue sky and partly because water absorbs red light more than blue. Raman effect: C. V. Raman discovered (1928) that when light is scattered by molecules, a small part of it changes wavelength. He won the Nobel Prize in Physics in 1930, and India celebrates National Science Day on 28 February for this discovery.

Atmospheric refraction. The air is not uniform: its density, and so its refractive index, changes with height. Light from a star passes through layers of different density and bends continually towards the normal, so the apparent position of a star is slightly higher than its real position. Because the layers keep moving, the apparent position shifts and the amount of starlight reaching the eye flickers: stars twinkle. Planets do not twinkle because they are much closer and act as an extended source, a collection of many point sources, so the flickering of different points averages out. The same bending makes the Sun appear to rise about 2 minutes before it actually crosses the horizon and set about 2 minutes later, and makes the Sun's disc look flattened at sunrise and sunset.

Mirage. On a hot summer day the road looks wet in the distance, or desert travellers see 'water' ahead. The ground heats the air just above it, so the layers near the ground are hotter and rarer (lower refractive index) while higher layers are cooler and denser. Light from the sky or from a tree travelling downward goes from denser to rarer air, bends away from the normal and finally undergoes total internal reflection; to the eye it appears to come from below the ground, like a reflection in water. In cold regions the opposite effect gives a superior mirage (looming), in which distant ships seem to float in the air.

Auroras (Aurora Borealis in the north, Aurora Australis in the south) are glowing lights near the poles. Charged particles of the solar wind are guided by Earth's magnetic field towards the poles, where they collide with gases of the upper atmosphere (oxygen gives green and red, nitrogen gives blue and purple) and make them glow. So auroras come from the interaction of solar wind and the magnetic field, not from lightning, volcanoes or earthquakes.