Title ALLEN Physics Module (1-6) Complete.pdf ✅

2 ALBERT EINSTEIN (1879-1955) Albert Einstein, born in Ulm, Germany in 1879, is universally regarded as one of the greatest physicists of all time. His astonishing scientific career began with the publication of three path-breaking papers in 1905. In the first paper, he introduced the notion of light quanta (now called photons) and used it to explain the features of photoelectric effect that the classical wave theory of radiation could not account for. In the second paper, he developed a theory of Brownian motion that was confirmed experimentally a few years later and provided a convincing evidence of the atomic picture of matter. The third paper gave birth to the special theory of relativity that made Einstein a legend in his own life time. In the next decade, he explored the consequences of his new theory which included, among other things, the mass-energy equivalence enshrined in his famous equation E = mc2 . He also created the general version of relativity (The General Theory of Relativity), which is the modern theory of gravitation. Some of Einstein's most significant later contributions are: the notion of stimulated emission introduced in an alternative derivation of Planck's blackbody radiation law, static model of the universe which started modern cosmology, quantum statistics of a gas of massive bosons, and a critical analysis of the foundations of quantum mechanics. The year 2005 was declared as International Year of Physics, in recognition of Einstein's monumental contribution to physics, in year 1905, describing revolutionary scientific ideas that have since influenced all of modern physics. Satyendranath Bose (1894-1974) Satyendranath Bose, born in Calcutta in 1894, is among the great Indian physicists who made a fundamental contribution to the advance of science in the twentieth century. An outstanding student throughout, Bose started his career in 1916 as a lecturer in physics in Calcutta University; five years later he joined Dacca University. Here in 1924, in a brilliant flash of insight, Bose gave a new derivation of Planck.s law, treating radiation as a gas of photons and employing new statistical methods of counting of photon states. He wrote a short paper on the subject and sent it to Einstein who immediately recognised its great significance, translated it in German and forwarded it for publication. Einstein then applied the same method to a gas of molecules. The key new conceptual ingredient in Bose's work was that the particles were regarded as indistinguishable, a radical departure from the assumption that underlies the classical Maxwell-Boltzmann statistics. It was soon realised that the new Bose-Einstein statistics was applicable to particles with integers spins, and a new quantum statistics (Fermi -Dirac statistics) was needed for particles with half integers spins satisfying Pauli's exclusion principle. Particles with integers spins are now known as bosons in honour of Bose. An important consequence of Bose-Einstein statistics is that a gas of molecules below a certain temperature will undergo a phase transition to a state where a large fraction of atoms populate the same lowest energy state. Some seventy years were to pass before the pioneering ideas of Bose, developed further by Einstein, were dramatically confirmed in the observation of a new state of matter in a dilute gas of ultra cold alkali atoms - the Bose-Eintein condensate
ALLEN Pre-Medical : Physics 3 Z:\NODE02\B0B0-BA\TARGET\PHY\ENG\MODULE_01\01-BASIC MATH IN PHY\01-THEORY.P65 E Mathematics is the supporting tool of Physics. Elementary knowledge of basic mathematics is useful in problem solving in Physics. In this chapter we study Elementary Algebra, Trigonometry, Coordinate Geometry and Calculus (differentiation and integration). 1. TRIGONOMETRY 1.1 Angle Consider a revolving line OP. P O X q Suppose that it revolves in anticlockwise direction starting from its initial position OX . The angle is defined as the amount of revolution that the revolving line makes with its initial position. From fig. the angle covered by the revolving line OP is q = ÐPOX The angle is taken positive if it is traced by the revolving line in anticlockwise direction and is taken negative if it is covered in clockwise direction. 1 ° = 60' (minute) 1 ' = 60" (second) 1 right angle = 90° (degrees) also 1 right angle = 2 p rad (radian) One radian is the angle subtended at the centre of a circle by an arc of the circle, whose length is equal to the radius of the circle. 1 rad = 180 p o » 57.3° To convert an angle from degree to radian multiply it by 180 p ° q=1rad r l=r To convert an angle from radian to degree multiply it by 180° p Illustration 1. A circular arc is of length p cm. Find angle subtended by it at the centre in radian and degree. 6cm 6cm q p cm Solution s cm rad=30° r 6 cm 6 p p q= = = As 1 rad = ° p 180 So q = p ° ́ =° p 180 30 6 BASIC MATHEMATICS USED IN PHYSICS
4 Pre-Medical : Physics ALLEN Z:\NODE02\B0B0-BA\TARGET\PHY\ENG\MODULE_01\01-BASIC MATH IN PHY\01-THEORY.P65 E Illustration 2. When a clock shows 4 o'clock, how much angle do its minute and hour needles make? (1) 120° (2) 3 p rad (3) 2 3 p rad (4) 160° Solution Ans. (1,3) From diagram angle 2 4 30 120 rad 3 p q= ́ °= °= 12 11 2 3 5 6 8 9 1 4 7 10 q 1.2 Trigonometrical ratios (or T ratios) Let two fixed lines XOX' and YOY' intersect at right angles to each other at point O. Then, (i) Point O is called origin. (ii) XOX' is known as X-axis and YOY' as Y-axis. (iii) Portions XOY, YOX' , X'OY' and Y'OX are called I, II, III and IV quadrant respectively. Consider that the revolving line OP has traced out angle q (in I quadrant) Y X' O Y' M X P 90° q in anticlockwise direction.From P, draw perpendicular PM on OX. Then, side OP (in front of right angle) is called hypotenuse,side MP (in front of angle q) is called opposite side or perpendicular and side OM (making angle q with hypotenuse) is called adjacent side or base. The three sides of a right angled triangle are connected to each other through six different ratios, called trigonometric ratios or simply T-ratios : sin q = = perpendicular MP hypotenuse OP cos q = = base OM hypotenuse OP tan q = = perpendicular MP base OM cot q = = base OM perpendicular MP sec q = = hypotenuse OP base OM cosec q = = hypotenuse OP perpendicular MP It can be easily proved that : cosec q = 1 sinq sec q = 1 cos q cot q = 1 tanq sin2q + cos2q = 1 1 + tan2q = sec2q 1 + cot2q = cosec2q Illustrations Illustration 3. Given sin q = 3/5. Find all the other T-ratios, if q lies in the first quadrant. Solution In D OMP , sin q = 3 5 so MP = 3 and OP = 5 Q OM = 2 2 (5) (3) - = 25 9- = 16 = 4 q M 3 5 O P Now, cos q = OM OP = 4 5 tan q = MP OM = 3 4 cot q = OM MP = 4 3 sec q = OP OM = 5 4 cosec q = OP MP = 5 3