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

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

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

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 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

Bohr's postulates are: (1) The electron revolves around the nucleus only in certain fixed circular orbits called stationary states, in which it does not radiate energy. (2) Only those orbits are allowed for which the angular momentum of the electron is an integral multiple of h/(2 pi), that is, m v r = n h/(2 pi) (quantisation of angular momentum). (3) Energy is emitted or absorbed only when the electron jumps from one orbit to another, the energy of the emitted or absorbed photon being h f = E2 - E1.

m v r = n h/(2 pi)

Energy of the electron in n = 3: E3 = -13.6/9 = -1.51 eV. Energy in n = 2: E2 = -13.6/4 = -3.4 eV. The energy of the emitted photon = E3 - E2 = -1.51 - (-3.4) = 1.89 eV. This corresponds to the H-alpha line of the Balmer series (visible red light).

E(photon) = E(higher) - E(lower)

The binding energy is the energy equivalent of the mass defect. BE = (mass defect in u) x 931 MeV = 0.0304 x 931 = 28.3 MeV. Thus the binding energy of the helium nucleus is about 28.3 MeV, and the binding energy per nucleon = 28.3/4 = 7.1 MeV.

BE (MeV) = (delta m in u) x 931

The radioactive decay law states that the rate of decay of a radioactive sample is directly proportional to the number of undecayed nuclei present at that instant: N = N0 e^(-lambda t), where lambda is the decay constant. The half-life T is related to it by T = 0.693/lambda. After each half-life, half of the sample remains, so after 3 half-lives the fraction remaining = (1/2)^3 = 1/8 of the original sample.

N = N0 (1/2)^(t/T)

A rectifier converts alternating current (AC) into direct current (DC). In a half-wave rectifier, a single diode is connected in series with the AC source and a load resistor. During the positive half-cycle of the AC input, the diode is forward biased and conducts, so current flows through the load. During the negative half-cycle, the diode is reverse biased and does not conduct, so no current flows. As a result, output is obtained only during one half of each cycle, giving a pulsating DC. Its efficiency is low because half the input is wasted.

A full-wave rectifier converts both halves of the AC input into DC. It uses two diodes with a centre-tapped transformer and a load resistor. During the positive half-cycle, one diode is forward biased and conducts while the other is reverse biased; during the negative half-cycle, the second diode conducts while the first does not. In both half-cycles, the current through the load flows in the same direction, so output is obtained during the whole cycle. This gives a smoother, more efficient pulsating DC than a half-wave rectifier.

A logic gate is a digital circuit that gives an output based on its inputs following a logical rule (using binary 0 and 1). OR gate (output Y = A + B): Y = 1 if any input is 1; for inputs (0,0),(0,1),(1,0),(1,1) the outputs are 0,1,1,1. AND gate (Y = A.B): Y = 1 only if both inputs are 1; outputs are 0,0,0,1. NOT gate (Y = not A): it has a single input and inverts it, so input 0 gives output 1 and input 1 gives output 0.

OR: Y = A + B; AND: Y = A.B; NOT: Y = not A