The critical angle C is given by sin C = 1/n, where n is the refractive index of the denser medium (glass). So sin C = 1/1.5 = 0.667, giving C = sin inverse (0.667) = 41.8 degrees (approximately 42 degrees). For angles of incidence greater than this, total internal reflection occurs.

sin C = 1/n

The lens maker's formula is 1/f = (n - 1)(1/R1 - 1/R2), where f is the focal length of the lens, n is the refractive index of the lens material with respect to the surrounding medium, and R1 and R2 are the radii of curvature of the two surfaces of the lens (taken with proper sign convention). It shows that the focal length depends on the material and the shape (curvature) of the lens.

1/f = (n - 1)(1/R1 - 1/R2)

Using the sign convention, u = -30 cm and f = +20 cm. From the lens formula 1/v - 1/u = 1/f, we get 1/v = 1/f + 1/u = 1/20 + 1/(-30) = 1/20 - 1/30 = (3 - 2)/60 = 1/60, so v = +60 cm. The magnification m = v/u = 60/(-30) = -2. The image is real (v positive for a lens), inverted (m negative) and magnified twice, formed 60 cm on the other side of the lens.

1/v - 1/u = 1/f

The power of a lens is a measure of its ability to converge or diverge light and is defined as the reciprocal of its focal length in metres: P = 1/f (f in metres). Its SI unit is the dioptre (D). A converging (convex) lens has positive power and a diverging (concave) lens has negative power. For two thin lenses in contact, the total power is the sum: P = P1 + P2.

P = 1/f; P = P1 + P2

When a ray passes through a prism, it is deviated; the deviation is least at the angle of minimum deviation Dm, when the ray passes symmetrically through the prism. The refractive index of the prism material is given by n = sin((A + Dm)/2)/sin(A/2), where A is the refracting angle of the prism and Dm is the angle of minimum deviation. This formula is used to determine the refractive index of a transparent material.

n = sin((A + Dm)/2)/sin(A/2)

An astronomical telescope has an objective lens of large focal length fo and an eyepiece of small focal length fe. In normal adjustment (final image at infinity), the magnifying power is M = fo/fe, and the length of the telescope is fo + fe. For high magnification, the objective should have a large focal length (and large aperture) and the eyepiece a small focal length. The final image is inverted.

M = fo/fe

Huygens' principle states that: (1) every point on a given wavefront acts as a source of new disturbance called secondary wavelets, which spread out in all directions with the speed of the wave; and (2) the new wavefront at a later instant is the forward envelope (tangential surface) of all these secondary wavelets. This principle is used to explain the laws of reflection and refraction and the propagation of light as a wave.

In Young's double slit experiment, two coherent sources produce alternate bright and dark fringes on a screen. The fringe width (the distance between two consecutive bright or dark fringes) is beta = lambda D/d, where lambda is the wavelength of light used, D is the distance between the slits and the screen, and d is the separation between the two slits. All fringes are of equal width, and the width increases with wavelength and D but decreases as d increases.

beta = lambda D/d

Given lambda = 600 nm = 600 x 10^-9 m, d = 1 mm = 1 x 10^-3 m, D = 1 m. Fringe width beta = lambda D/d = (600 x 10^-9 x 1)/(1 x 10^-3) = 600 x 10^-6 = 6 x 10^-4 m = 0.6 mm.

beta = lambda D/d

For constructive interference (bright fringe), the path difference between the two interfering waves must be an integral multiple of the wavelength: path difference = n lambda, where n = 0, 1, 2, ... For destructive interference (dark fringe), the path difference must be an odd multiple of half the wavelength: path difference = (2n - 1) lambda/2. Equivalently, the phase difference is 2n pi for constructive and (2n - 1) pi for destructive interference.

constructive: n lambda; destructive: (2n-1) lambda/2

Coherent sources are two sources of light that emit waves of the same frequency (or wavelength) and have a constant phase difference between them. Conditions for sustained (steady) interference: (1) the two sources must be coherent; (2) they must have the same frequency and nearly equal amplitudes; and (3) they must be narrow and close together, and the light should preferably be monochromatic. In practice, coherent sources are obtained from a single source (for example, using two slits).

Diffraction is the bending of light around the edges of an obstacle or aperture and its spreading into the geometrical shadow region. In diffraction at a single slit of width a, a central bright maximum is flanked by alternate dark and bright fringes. The condition for the dark fringes (minima) is a sin theta = n lambda, where n = 1, 2, 3, ... and theta is the angle of diffraction. The central maximum is the brightest and widest.

a sin theta = n lambda

(1) Interference is due to the superposition of waves from two (or more) different coherent sources, whereas diffraction is due to the superposition of secondary wavelets coming from different parts of the same wavefront. (2) In interference all bright fringes are of equal intensity and equal width, whereas in diffraction the central maximum is the brightest and the intensity of the secondary maxima decreases rapidly on either side.

Polarisation is the phenomenon of restricting the vibrations of the electric field of a light wave to a single plane perpendicular to the direction of propagation; it shows that light is a transverse wave. Brewster's law states that when unpolarised light is incident on a transparent surface at a particular angle called the polarising angle (theta_p), the reflected light is completely plane-polarised, and the refractive index of the medium is n = tan(theta_p).

n = tan(theta_p)

The photoelectric effect is the phenomenon of emission of electrons (called photoelectrons) from the surface of a metal when light of suitable frequency (usually ultraviolet or visible for some metals) falls on it. The emitted electrons carry kinetic energy. The effect occurs only when the frequency of the incident light is greater than a certain minimum value called the threshold frequency, and it provided evidence for the particle (photon) nature of light.

Einstein's photoelectric equation is h f = W0 + KE(max), where h f is the energy of the incident photon (h is Planck's constant and f the frequency of light), W0 is the work function (the minimum energy needed to eject an electron from the metal surface), and KE(max) is the maximum kinetic energy of the emitted photoelectron. It expresses conservation of energy: the photon's energy is used partly to free the electron and the rest appears as its kinetic energy. Thus KE(max) = h f - W0.

h f = W0 + KE(max)

The work function (W0) of a metal is the minimum energy required to just remove an electron from the surface of the metal without giving it any kinetic energy. The threshold frequency (f0) is the minimum frequency of the incident light below which no photoelectric emission takes place, however intense the light may be. They are related by W0 = h f0, where h is Planck's constant.

W0 = h f0

The energy of the incident photon E = hc/lambda = 1240/400 = 3.1 eV. Using Einstein's equation, the maximum kinetic energy KE(max) = E - W0 = 3.1 - 2.0 = 1.1 eV. Since the photon energy (3.1 eV) is greater than the work function (2 eV), emission occurs and the photoelectrons have a maximum kinetic energy of 1.1 eV.

KE(max) = hc/lambda - W0

(1) For a given metal, photoelectric emission occurs only if the frequency of the incident light is greater than a certain minimum value (threshold frequency), whatever the intensity. (2) The maximum kinetic energy of the emitted photoelectrons depends on the frequency of the incident light and the nature of the metal, but is independent of the intensity of the light. (3) The number of photoelectrons emitted per second (photoelectric current) is directly proportional to the intensity of the incident light. (4) The emission is instantaneous, with no measurable time lag.

The de Broglie hypothesis states that every moving particle has a wave associated with it, called a matter wave. The de Broglie wavelength is lambda = h/p = h/(m v), where h is Planck's constant, p is the momentum, m is the mass and v the velocity of the particle. In terms of kinetic energy, lambda = h/sqrt(2 m KE). This shows the dual (wave-particle) nature of matter; the wavelength is significant only for very small particles like electrons.

lambda = h/(m v)