1000 Solved Problems in Classical Physics Ahmad A Kamal 1000 Solved Problems in Classical Physics An Exercise Book 123 Dr Ahmad A Kamal Silversprings Lane 425 75094 Murphy Texas USA anwarakamal@yahoo.com ISBN 978-3-642-11942-2 e-ISBN 978-3-642-11943-9 DOI 10.1007/978-3-642-11943-9 Springer Heidelberg Dordrecht London New York © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: eStudio Calamar S.L Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to my Parents Preface This book complements the book 1000 Solved Problems in Modern Physics by the same author and published by Springer-Verlag so that bulk of the courses for undergraduate curriculum are covered It is targeted mainly at the undergraduate students of USA, UK and other European countries and the M.Sc students of Asian countries, but will be found useful for the graduate students, students preparing for graduate record examination (GRE), teachers and tutors This is a by-product of lectures given at the Osmania University, University of Ottawa and University of Tebriz over several years and is intended to assist the students in their assignments and examinations The book covers a wide spectrum of disciplines in classical physics and is mainly based on the actual examination papers of UK and the Indian universities The selected problems display a large variety and conform to syllabi which are currently being used in various countries The book is divided into 15 chapters Each chapter begins with basic concepts and a set of formulae used for solving problems for quick reference, followed by a number of problems and their solutions The problems are judiciously selected and are arranged section-wise The solutions are neither pedantic nor terse The approach is straightforward and step-by-step solutions are elaborately provided There are approximately 450 line diagrams, onefourth of them in colour for illustration A subject index and a problem index are provided at the end of the book Elementary calculus, vector calculus and algebra are the prerequisites The areas of mechanics and electromagnetism are emphasized No book on problems can claim to exhaust the variety in the limited space An attempt is made to include the important types of problems at the undergraduate level It is a pleasure to thank Javid, Suraiya and Techastra Solutions (P) Ltd for typesetting and Maryam for her patience I am grateful to the universities of UK and India for permitting me to use their question papers; to R.W Norris and W Seymour, Mechanics via Calculus, Longmans, Green and Co., 1923; to Robert A Becker, Introduction to Theoretical Mechanics, McGraw-Hill Book Co Inc, 1954, for one problem; and Google Images for the cover page My thanks are to Springer-Verlag, vii viii Preface in particular Claus Ascheron, Adelheid Duhm and Elke Sauer, for constant encouragement Murphy, Texas November 2010 Ahmad A Kamal Contents Kinematics and Statics 1.1 Basic Concepts and Formulae 1.2 Problems 1.2.1 Motion in One Dimension 1.2.2 Motion in Resisting Medium 1.2.3 Motion in Two Dimensions 1.2.4 Force and Torque 1.2.5 Centre of Mass 1.2.6 Equilibrium 1.3 Solutions 1.3.1 Motion in One Dimension 1.3.2 Motion in Resisting Medium 1.3.3 Motion in Two Dimensions 1.3.4 Force and Torque 1.3.5 Centre of Mass 1.3.6 Equilibrium 1 3 6 10 12 13 13 21 26 35 36 44 Particle Dynamics 2.1 Basic Concepts and Formulae 2.2 Problems 2.2.1 Motion of Blocks on a Plane 2.2.2 Motion on Incline 2.2.3 Work, Power, Energy 2.2.4 Collisions 2.2.5 Variable Mass 2.3 Solutions 2.3.1 Motion of Blocks on a Plane 2.3.2 Motion on Incline 2.3.3 Work, Power, Energy 2.3.4 Collisions 2.3.5 Variable Mass 47 47 52 52 53 56 58 63 64 64 68 75 77 95 ix 788 Problem Index 10.35 Latent heat of fusion determination 10.36 Mean specific heat and specific heat at midpoint 10.37 Equilibrium temperature from the mixture of liquid ( A, B), (B, C) and (A, C) 10.38 Rise of temperature of the bullet in its collision with a block of wood (a) fixed and (b) free to move 10.39 Rise in temperature due to water fall 10.40 Rise in temperature due to fall of a lead piece from a given height on a nonconducting slab 10.41 Work done on a gas during an adiabatic compression 10.42 A thermodynamic cycle 10.43 In an adiabatic process of a monatomic ideal gas P V 5/3 = const 10.44 Work done on a gas during an isothermal compression 10.45 P − V diagram for a sequence of thermodynamic processes 10.46 Number of degrees of freedom for gas molecules 10.47 Efficiency of a heat engine 10.48 Temperatures of the source and sink of a heat engine 10.49 Air pressure at a given attitude h, number density of gas molecules and p at h/2 10.50 P − V diagram for the Carnot cycle and the Sterling cycle 10.51 Entropy change for isobaric and isochoric processes 10.52 Change of values of thermodynamic parameters in reversible isothermal expansion 10.53 Internal energy, heat, enthalpy, work and Gibbs function 10.54 Shear modulus of a material 10.55 Stress, strain and Young’s modulus of a wire 10.56 Isothermal elasticity and adiabatic elasticity 10.57 Poisson’s ratio from the ratio of Young’s modulus and the rigidity modulus 10.58 Young’s modulus from the depression of a wire caused by attaching known load at midpoint of a horizontal wire 10.59 Speed of an object released from a catapult 10.60 Maximum angular speed of an object attached to the end of a wire whirled in a horizontal plane 10.61 Maximum length of a wire which will not break under its own weight when it hangs freely 10.62 Capillary rise 10.63 Volume of air bubble at depth 100 m – gas equation 10.64 Energy released in coalescing of droplets 10.65 Work done in blowing a soap bubble to a larger size 10.66 A hollow vessel with a small hole of radius r is immersed to a known depth when water just penetrates into vessel, to find r 10.67 Depth of water column at which an air bubble is in equilibrium 10.68 Inadequate capillary tube length 10.69 Effect of charge on soap bubble 10.70 Radius of bigger bubble when two bubbles coalesce Problem Index 789 Chapter 11 Electrostatics 11.1 (a) Force between two charges Position of neutral point (b) E and F of proton at the position of electron in H atom 11.2 (a) Tension in the thread when a charged ball hangs in an electric field and (b) Vb − Va positive or negative 11.3 Electric potential along the axis of a charged circular loop 11.4 Potential energy of four charges at the corners of a square and their stability 11.5 V at the surface of a charged liquid drop, when two such drops coalesce, V at the surface of new drop 11.6 A charged pendulum bob is in equilibrium in a horizontal electric field Tension and angle of the thread with the vertical 11.7 An infinite number of charges, each equal to q are placed at x = 1, 2, 4, 8, units V and E at x = 11.8 In prob (11.7) what will be V and E if consecutive charges have opposite sign 11.9 A charged particle is released from rest on the axis of a fixed oppositely charged ring To show that the particle has SHM and to find time period of oscillation 11.10 Three charges each of value q are at the corners of an equilateral triangle and the fourth one at the centre It Q = −q, charges move toward or away from the centre Value of Q for the charge to be stationary 11.11 Two identically charged spheres are suspended by strings of equal length The strings make an angle of 30◦ with each other When immersed in a liquid the angle remains the same To find the dielectric constant of the liquid 11.12 One charge is placed at one corner of a square and another at the center Work done to move the charge from the centre to an empty corner 11.13 A charged pith ball suspended by a thread is deflected by a known distance by a horizontal electric field To find E 11.14 Suspension of a charged oil drop under electric field and gravity 11.15 Equal amount of charge on the earth and the moon to nullify gravitational attraction 11.16 An energy output of 10−5 J results from a spark between insulated surfaces at P.D × 106 V To find q transferred and number of electrons flowed 11.17 E at a given distance along the axis of a charged rod 11.18 E along the axis of a charged disc 11.19 Electronic charge by Millikan’s oil drop method 11.20 Total charge on a circular wire which has cos2 θ charge density dependence 11.21 Strength of electric force compared to gravitation force in H atom 11.22 Charges on two spheres given their combined charge and force of repulsion at the given distance 790 Problem Index 11.23 Four charges are placed at the four corners of a square as in the diagram To find E and its direction 11.24 E on the perpendicular bisector of a thin charged rod 11.25 A thin non-conducting charged rod is bent to form an arc of circle and subtends an angle θo at the centre of circle To find E at the centre of circle 11.26 In prob (11.3) to find the distance at which E is maximum 11.27 E and V at the centre of a charged non-conducting hemispherical cup 11.28 E varies as 1/r for an electric quadrupole 11.29 Electric and gravitational forces between two bodies each of mass m and charge q will be equal if q/m = 8.6 × 10−10 /kg 11.30 Two equally charged spheres are suspended from the same point by silk thread of the same length To find the rate at which charge leaks out given the relative velocity of approach 11.31 A long charged thread is placed on the axis of a charged ring with one end coinciding with the centre of the ring To find force of interaction 11.32 A very long wire with charge density λ is placed on the x-axis with one end coinciding with the origin To calculate E from the y-axis 11.33 E from the potential φ = cxy 11.34 To show that the locus of zero potential points for two fixed charges is a circle 11.35 Two identical rings charged to Q and Q coaxially placed at a fixed distance To find the work done in moving a charge q from the centre of one ring to that of the other 11.36 To find x if the potentials at (0, 2a) and P2 (x, 0) are equal and to find the potential 11.37 Value of Q if the interaction energy of three charges +q1 , +q2 and Q placed at the vertices of a right-angled isosceles triangle is zero 11.38 Work done in assembling the charges at the four corners of a square as in the diagram 11.39 Total potential energy of a charged sphere 11.40 A linear quadrupole 11.41 Charge that can be placed on a sphere for the given field strength and the corresponding V 11.42 Speed of electron when it approaches a positive charge 11.43 For the linear quadrupole of prob (11.40) for r >> d, E(r ) ∝ 1/r 11.44 Force and acceleration of electron when it passes through a hole in a condenser plate 11.45 Potential on the axis of a charged disc and the limiting case of x >> R 11.46 Kepler’s third law of motion is applicable to electron in H atom 11.47 Equal charges are placed on four corners of a square To calculate F on one of them due to other three 11.48 E on the perpendicular bisector of a dipole For x >> d/2, E ∝ 1/r 11.49 Application of Gauss’ law to an infinite sheet of charge σ of the sheet by deflection of a charged mass hanging by a silk thread Problem Index 791 11.50 Application of Gauss’ law to calculate E inside and outside a charged sphere 11.51 In prob (11.50) to find E on the surface of charged sphere with cavity 11.52 In prob (11.50) E ∝ r for r < R and that V (0) = 32 V (R) 11.53 E in three regions of a non-conducting charged sphere 11.54 E for two regions of a long charged cylinder 11.55 Net charge within the sphere’s surface given E σ from E on a football field Total electric flux 11.56 Derivation of Coulomb’s formula from Gauss’ law Electric flux through spherical surface concentric with a charged sphere 11.57 Application of Gauss’ law to find E in the three regions of two charged concentric spherical shells 11.58 Two insulated spheres positively charged at large distance are brought into contact and separated by the same distance as before To compare force of repulsion before and after contact 11.59 Maximum charge a sphere can withstand given the breakdown voltage 11.60 (a) Capacitance of a conducting sphere and (b) U when two charged spheres are connected by a wire and wire is removed 11.61 When two spherical charged conductors are brought in contact and separated σ ∝ 1/r 11.62 To find V if E on balloon is given Pressure in balloon which would produce the same effect Total electrostatic energy of the balloon 11.63 A soap bubble of radius R1 when charged expands to radius R2 To derive an expression for the charge 11.64 E in the three regions when an insulating shell is charged to specified charged density 11.65 (a) Electrostatic field is conservative and (b) U when a charged soap bubble collapses to a smaller size 11.66 Form of E generated by a long charged cylinder Speed of electron circling around the axis of the cylinder 11.67 To find σ for non-conducting and conducting infinite sheets 11.68 Capacitance of a parallel plate capacitor Modification of C when a thin metal is introduced 11.69 Values of C1 and C2 , given their combined values in series and parallel arrangement 11.70 Energy in two capacitors: (a) singly; (b) in series; and (c) in parallel, given P.D 11.71 U when a charged air capacitor is submerged in oil of given εr 11.72 Value of K when A, d and C are given 11.73 Resulting voltage when a charged capacitor is connected to an uncharged one 11.74 To find the equivalent capacitance of capacitors in the given arrangement 11.75 To calculate q, V and U for three capacitors connected in series to a battery of 260 V 792 Problem Index 11.76 Two capacitors are charged to a battery and connected in parallel Find P.D of the combination if (a) positive ends are connected and (b) positive end is connected to negative terminal of the other 11.77 Two capacitors are charged to P.D V1 and V2 and connected in parallel U when (a) positive ends are joined and (b) positive end of one is joined to negative end of the other 11.78 Effect of dielectric on V , E, q, C and U , when the battery (a) remains connected and (b) is disconnected 11.79 In prob (11.78) dependence of the given quantities on the distance of separation of plates 11.80 Force of attraction between the plates of a parallel plate capacitor 11.81 n identical droplets each of radius r and charge q coalesce to form a large drop To find relations of radius, C, V , σ and U for the large drop and droplet 11.82 Half of√the stored U E of a cylindrical capacitor of radii a and b lies within a radius ab of the cylinder 11.83 Capacitor of capacitance C1 withstands maximum voltage V1 and C2 withstands maximum voltage V2 Maximum voltage that the system of C1 and C2 can withstand when connected in series 11.84 Application of Gauss’ law to calculate the capacitance of Geiger–Muller counter 11.85 Capacitance of a capacitor formed by two spherical metallic shells 11.86 For two concentric shells the capacitance reduces to that of a parallel plate capacitor in the limit of large radii 11.87 In the R – C circuit shown to find (i) time for charge to reach 90% of its final value; (ii) U stored in the capacitor at t = τ ; and (iii) Joule heating in R at t = τ 11.88 In prob (11.87) number of time constants after which energy in capacitor will reach half of equilibrium value 11.89 Capacitance of a parallel plate capacitor whose plates are slightly inclined 11.90 and 11.91 In the given arrangements of capacitor to find P.D, q and U in the capacitors 11.92 To find the effective capacitance between two points in the given arrangement of capacitors 11.93 To obtain an expression for q(t) for an R − C circuit 11.94 For the given R − C circuit, to find battery current at t = and t = ∞ when switch is closed To find the current through R when switch is open after a long time 11.95 A charged capacitor is discharged through a resistance To find U , i, Vc at given time, τ and equation for t when q drops to half of its value 11.96 Charge q is uniformly distributed in a sphere of radius R (i) To find div E inside the sphere; (ii) electric force on a proton at r < R; and (iii) work done on proton to move it from infinity to a point at r < R 11.97 The electric displacement D is uniform in a parallel plate capacitor To obtain E(x) for a non-uniform relative permittivity, using Gauss’ law 11.98 To obtain the differential Gauss’ law for gravitation Problem Index 793 Chapter 12 Electric Circuits 12.1 Effective resistance between two points A and B in the given arrangement 12.2 Change in resistance when a wire is stretched 12.3 Equivalent resistance is p for two resistors in series and q for parallel, minimum value of n where p = nq 12.4 Equivalent resistance for five resistors in the given arrangement 12.5 Effective resistance of five resistors in the given arrangement 12.6 Resistance between two terminals in the given network 12.7 Equivalent resistance between two terminals in the given network 12.8 Wheatstone bridge 12.9 Five resistors are connected in the form of a square and a diagonal To find Req across a side 12.10 Effective resistance for a network of infinite number of resistors 12.11 What equal length of an iron wire and a constantan wire of equal diameter must be joined in parallel to give equivalent resistance of ? 12.12 Temperature coefficient of resistance 12.13 A square ABCD is formed by bending a wire B and D are joined by a similar wire and a battery of negligible internal resistance is included between A and C To find Req and power dissipated 12.14 Temperature dependence of resistance 12.15 Equivalent resistance of a network in the form of a skeleton cube across the body diagonal 12.16 Brightness of two bulbs in series 12.17 Brightness of two bulbs in parallel 12.18 Three resistors in parallel are connected with two in series If a PD of 120 V is applied across the ends of circuit, to find PD drop across parallel arrangement 12.19 Maximum power delivered to the external resistor 12.20 Boiling of water by two heater coils when they are connected (a) in series and (b) in parallel 12.21 Calculation of power expended in two resistors connected in series and in parallel 12.22 Rate of energy loss in power transmission 12.23 Power dissipated in three resistors in (a) series and (b) parallel 12.24 The value of resistance in parallel with a heater so that the 1000 W heater operates at 62.5 W 12.25 Number of cells wrongly connected in a battery 12.26 Condition for maximum current in m external resistances connected to m rows of cells with each row containing n cells in series, each cell with internal resistance r 12.27 Internal resistances of two cells in series and the external resistance 12.28 Rate of heat production in resistors in series and in parallel 12.29 Charging of a battery 794 Problem Index 12.30 Comparison of power dissipated in resistance with the value for power supplied by battery 12.31 Internal resistance of a cell by potentiometer 12.32 Emf of a cell by potentiometer method 12.33 Condition for null deflection and i G /i in galvanometer of Wheatstone bridge 12.34 To find current in an ammeter in a network 12.35 Internal resistance of a battery 12.36 Internal resistance of a cell by potentiometer method 12.37 External resistance by potentiometer method 12.38 Reading in the voltmeter 12.39 Metre bridge 12.40 A moving coil meter reading up to mA to be converted into (a) 100 mA full scale and (b) 80 V full scale 12.41 Pocket voltmeter 12.42 Resistance of a galvanometer 12.43 A battery of ξ1 and r1 with a second battery of ξ2 and r2 in parallel is joined to an external resistance R To calculate i , i , P1 , P2 and P 12.44 Emf of batteries by applying the loop theorem 12.45 Application of Kirchhoff’s rules to determine currents in branches of a circuit 12.46 Currents in various resistors and PD across the cells in a network 12.47 Equivalent resistance of the circuit and power dissipated 12.48 Given the internal resistance of one cell to calculate the internal resistance of the other cell 12.49 Equivalent resistance, voltages, currents and power dissipated in a series parallel resistive circuit 12.50 Current in the circuit and terminal voltage of battery under load conditions and power dissipated in R and r 12.51 Equivalent resistance, currents, voltages and power dissipated in a seriesparallel circuit 12.52, 12.53, 12.54 Application of Kirchhoff’s rules to the circuit to produce three equations with three unknown branch currents 12.55 Application of Kirchhoff’s rules to the given circuit 12.56 Application of Kirchhoff’s rules to the circuit to calculate currents in various branches Chapter 13 Electromagnetism I 13.1 13.2 13.3 13.4 13.5 Cyclotron frequency for alpha particles Energy and oscillator frequency for p and α Mass spectrometer, velocity filter Identification of pion A particle of mass m and charge q travelling along x-axis is acted by E along y-axis The trajectory Problem Index 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14 13.15 13.16 13.17 13.18 13.19 13.20 13.21 13.22 13.23 13.24 13.25 13.26 13.27 13.28 13.29 13.30 13.31 13.32 13.33 13.34 13.35 13.36 13.37 13.38 13.39 795 The radius of a circular orbit of an electron of K = keV in B = 0.01 T Crossed E and B fields v and T for electron moving with R = 1.9 m in B = × 10−5 wb/m2 B and K for D in a cyclotron with V = × 104 V, f = MC/r and d = 1.524 m f = 11.5 MC/s, R = 30 for D K max and f for p Given B = 15,000 G and R = 50 cm, f and K for D Current from charge and time Condition for no deflection in E and B fields Radius of curvature of a particle of mass m and charge q which enters a magnetic field Separation of isotopes of uranium (a) Radius of curvature in a magnetic field and (b) emf produced in a timevarying B (a) No of electrons in charge q and (b) E to prevent a droplet from falling Time period and pitch in a magnetic field A change −q released from the plate of a capacitor in which E and B fields are set up In prob (13.19) condition that the electrons are able to reach the positive plate Current in a long wire deduced from observed B B for a wire of finite length Limiting case for infinite wire B at the centre of a conducting circular wire B at the centre of square conducting loop of side a with current i Two semicircular wires of resistance R and 4R are joined To find B at the centre of the circle B at the centre of three-fourths of a conducting circular wire (a) B from a hair pin conducting wire and (b) B at the centre of a semicircular wire B at the centre of a conducting wire in the form of a polygon Ratio of B at the centres of a conducting wire in the form of a circle and square of the same length (a) B at the common centre of circular arcs of a circuit and (b) B at the common centre of semicircular arcs of radii R1 and R2 B at the centre of a circular conducting wire plus straight portion as in the diagram B on the axis of a circular ring carrying current B inside a m long tube wound by 500 turns of wire carrying A B between parallel current-carrying wires at distance x from one of the wires B at p in a hollow copper cylinder with radii a and b (a < R < b) carrying current I Magnetic field midway between Helmholtz coils B at the centre of a charged rotating disc In prob (13.36) the magnetic field is fairly uniform B due to cylinder carrying current of 100 A at R = 1.0 m and R = mm 796 Problem Index 13.40 Application of ampere’s law to find B due to a current-carrying cylinder for r < R and r > R 13.41 To find B at the centre of two concentric arcs of radii r and 2r as in the diagram 13.42 (a) To find B distance x from the midpoint of wire of length L (b) To compare B at the centre of a loop when it is bent into (i) a square and (ii) an equilateral triangle 13.43 To calculate B midway between Helmholtz coils, given the values of N , R and I 13.44 B(r ) for the current-carrying long cylinder 13.45 (a) L of a solenoid; (b) magnetic energy; and (c) application of Faraday’s law 13.46 (a) B at a given distance from a long straight wire carrying current and (b) dB at the given values of x, y and z from I dl at the origin 13.47 B and H for a torus with and without magnetic material 13.48 Current required to circulate the earth’s core to produce the known dipolar magnetic field 13.49 Emf generated by a revolving disc midway between Helmholtz coils 13.50 Variation of voltage developed across a conductor moving with velocity v in a known field B ˆ m/s is located at x = m 13.51 A proton travelling with velocity v = (iˆ + j)10 and y = m at some instant t To find B 13.52 To find F/l between two long straight wires separated by distance d carrying i and i in opposite directions 13.53 Two parallel wires of distance d apart attract each other with a force F/l If i is given, to find i and its direction 13.54 Equilibrium between two current-carrying parallel wires separated by d vertically 13.55 Three long parallel wires carrying 20 A are placed in the same plane with equal spacing of 10 cm To find F/l for an outer wire and the central wire 13.56 To calculate the force acting on a bent wire in a uniform magnetic field as in the diagram 13.57 A rectangular coil of given dimensions is placed parallel to a long wire carrying current To find force on each segment of rectangular coil and net force on it 13.58 In a set-up similar to prob (13.57) to derive an expression for the resultant force on the coil and find its value from the given data 13.59 A loop is formed by two parallel rails, a resistor and a rod across the rails in a magnetic field A force F drags the rod at velocity v To find the current, total power and F 13.60 In prob (13.37) to calculate the magnetic moment of the disc 13.61 To show that the magnetic dipole moment of the earth can be produced by wire carrying a current of × 107 A around the magnetic equator 13.62 Energy density at the centre of a current-carrying loop 13.63 Magnetic energy density at the centre of H atom due to circulating electron 13.64 Maximum torque of a circular coil in a magnetic field Problem Index 13.65 13.66 13.67 13.68 13.69 13.70 13.71 13.72 13.73 13.74 13.75 13.76 13.77 13.78 13.79 13.80 13.81 13.82 13.83 13.84 13.85 The ratio μ/L for a charged sphere rotating with constant angular velocity Potential energy of an electric dipole is an electric field Emf induced in an expanding flexible circular wire placed in a magnetic field Emf developed between the wing tips of an aeroplane flying over earth’s magnetic field Potential difference between the centre and the outer edge of a spinning disc in the horizontal component of earth’s magnetic field A coil in the given magnetic field is suddenly withdrawn from the field and a galvanometer in series with the coil records the charge passed around the circuit To find the resistance of the coil and the galvanometer The induced emf and current in a wire loop when the magnetic field is reduced to zero in the given time Amplitude of the induced current when a square loop of wire rotates in a magnetic field A bar slides on parallel rods in a magnetic field when a current flows through the resistor To find the speed of the bar A uniform magnetic field of induction fills a cylindrical volume To calculate the emf produced at the end of a rod placed in it when B changes A square wire of length l, mass m and resistance R slides on frictionless inclined rails Magnetic field exists within the frame To show that the wire frame acquires steady velocity A copper disc spins in a magnetic field and induced emf is recorded To find B To verify that Faraday’s law is dimensionally correct Given the equation of B waves, to find emf in a coil Ratio of ξmax (television)/ξmax (radio) in a loop antenna Two rails are connected by a wire and are connected to a wire and a slider to form a loop To find F on the slider to move it with velocity v Application of Faraday’s law A wire carrying current is placed across a rectangular coil To obtain the magnetic flux and emf induced To obtain magnetic flux in a betatron The magnetic and direction of the Hall field and concentration of free electrons Mobility of the electron given Hall coefficient and electrical conductivity Chapter 14 Electromagnetism II 14.1 14.2 14.3 14.4 14.5 797 Current, P.D and energy in an RC circuit P.D and phase difference in an LR circuit Given current and energy for an inductor, to find its value To find frequency and wavelength for an LC circuit Impedance and power dissipated in an RLC circuit 798 Problem Index 14.6 L and C have equal X , at f = 600 Hz, to find X C /X L at 60 Hz 14.7 A capacitance has X C = at 250 Hz To find C and X C at 100 Hz and at 220 V, 50 Hz line 14.8 In an LR series circuit across a 12 V–50 Hz supply, i = 0.05 A flows with φ = 60◦ with V To find R and L, and to find C when connected in series to produce to φ = 14.9 i rms and i m when 0.6 H inductor is connected to 220 V–50 Hz AC line 14.10 i rms , Joule heat, Vrms in RL circuit for each component when f = 50 Hz and i rms are available 14.11 An ac applied to R = 100 given V = 0.5 Vm at t = 1/300 s To find f 14.12 To write an equation for the given LRC parallel circuit similar to that for standard equation 14.13 To verify the formula for √ velocity of light 14.14 Quantities RC, L/R and LC have units of time 14.15 Fractional decrement of the resonance frequency in an RLC circuit 14.16 Current in a damped LC circuit for low damping 14.17 RLC circuit in series 14.18 RLC circuit in parallel 14.19 Differential equation for the charge of the given RLC circuit 14.20 Solution of differential equation in prob (14.19) 14.21 X L , X C , Z , IT , φ, C R , VC , VL and f for the given RLC circuit 14.22 A 40 resistor and 50 μF capacitor in series and an AC of V–300 Hz current in the circuit 14.23 Z for the two networks involving L, R, C 14.24 Definition of electric current, current density and quantization of charge, drift speed 14.25 f for two LC circuits in series is identical with individual circuits if L C1 = L C2 14.26 Values of L and C if X C at f and X L at f are given 14.27 A condenser of 0.01 μF is charged to 100 V To find i m when condenser is connected to an inductor of L = 10 mH 14.28 An inductance of mH has resistance of Value of R and C to yield f = 500 kHz in series and Q = 150 14.29 f and Q of a parallel RLC circuit with L = mH, R = 10 and C = 0.005 μf 14.30 In the discharge of a capacitor in an RC circuit to find error on V (t), given error on R and C 14.31 (a) Time for voltage on a condenser to fall to 1/e of initial value through a resister and (b) percentage error introduced if thermal effects are ignored 14.32 Fractional half-width of resonance curve of an RLC circuit 14.33 Dielectric constant from a plane em wave equation 14.34 Speed of a Lorentz frame in which a pure magnetic field is observed 14.35 Wave equation for B wave ∂E = 14.36 To show that ∇ j + ε0 ∂ t Problem Index 14.37 14.38 14.39 14.40 14.41 14.42 14.43 14.44 14.45 14.46 14.47 14.48 14.49 14.50 14.51 14.52 14.53 14.54 14.55 14.56 14.57 14.58 14.59 14.60 14.61 14.62 14.63 14.64 14.65 14.66 14.67 14.68 14.69 14.70 14.71 14.72 14.73 14.74 14.75 799 Prediction that em waves propagate with velocity c Depth to which an em wave penetrates in aluminium To show that E = cB0 em wave equation for a conduction medium Capacitance and inductance per unit length for a coaxial cylinder E, B and S for a coaxial cable in the region between the central wire and tube To obtain the equation u B = B /2μ0 In em field u B = u E To obtain an expression for R if u B = u E in a coaxial cable Radiation energy loss of a low energy proton in a cyclotron Amplitude of an E wave at a distance from a point source Intensity, E and B0 of a laser beam Calculation of irradiance of a laser beam Time-averaged power per unit area carried by a plane em wave Irradiance of a plane em wave E × H is in the propagation direction Given the E field equation to find f , λ, v, E and polarization Equation for the B field associated with the E wave of prob (14.53) Speed of an approaching car by radar Beat frequency registered by a radar from a receding car Application of ampere’s law of a coaxial cylinder to calculate magnetic field in four regions Energy stored in the magnetic field in the large Hadron collider’s magnet Electric and magnetic field amplitude at the surface of the sun Amplitude of the electric field at the orbit of the earth Amplitude of B field at the surface of Mars and flux of radiation To show that |E|/|H | = 377 Skin depth in copper Penetration of microwave in a copper screen Derivation of formula for skin depth Energy transported per square centimetre area for a light wave having E m = 10−3 V/m Proof and interpretation of Poynting’s theorem Dispersion relation from Maxwell’s equations in dielectric To show that the identity ∇ × (∇ × E) = −∇ E + ∇(∇ · E) is true for the ˆ vector field F = x z i Use of Poynting vector to determine power flow in a coaxial cable Application of Poynting’s theorem to show that power dissipated in a conducting wire is given by i R Using Maxwell’s equations to show that the equation for a super-conductor leads to the stated equation for B Ratio of high-frequency resistance to direct resistance Derivation of the expression curl E = −∂ B/∂t using Stokes’ theorem Derivation of the expression B = μ0 (H + M) 800 Problem Index 14.76 Gauss and Ampere’s laws in free space subject to the Lorentz condition √ 14.77 To show E y = Z Hx for propagation of em wave where Z = μ0 /ε0 14.78 Given E wave’s direction of propagation and polarization of the wave Boundary condition for media of different magnetic properties 14.79 Use of boundary conditions to calculate the reflectance and transmittance of em waves at the dielectric discontinuity 14.80 To obtain an expression for reflectance for an em wave in terms of angle of incidence and refraction and the refraction indices of two dielectrics 14.81 Using the result in prob (14.80), to show that reflectance is zero when tan θ = n /n 14.82 To sketch R and T against n /n for normal incidence 14.83 (a) B is perpendicular to E; (b) B is in phase with E; and (c) B = E/c 14.84 Charge density induced on the surface of uncharged dielectric cube containing electric field 14.85 Fractional difference between phase and group velocity at λ = 5000 Å 14.86 Given the dispersion relation ω = ak , to calculate vph and vg 14.87 To show that vp vg = c2 14.88 vg = vph + kdvp /dk 14.89 vg = c/n + λc/n dn/dλ 14.90 If vph ∝ 1/λ then vg = 2vph 14.91 vg = c/ n + ω (∂n/∂ω) 14.92 Value of vp and free space wavelength for radiation to traverse a length of a rectangular waveguide in given time 14.93 1/vg = 1/vph + (ω/c) dn/dω 14.94 vg = cdv/d(nv) 14.95 vg = v for a non-relativistic classical particle 14.96 Given a relation between refractive index and λ, to calculate vg 14.97 vph , vg and λ for rectangular wave guide of given dimensions 14.98 In a rectangular guide of width a = cm, value of λ if λg = 3λ 14.99 To calculate λg and λc given a and λ 14.100 Number of states of em radiation between 5000 and 6000 Å in a cube of side 0.5 m 14.101 Possibility of AM radio waves propagating in a tunnel of given dimensions 14.102 Calculation of least cut-off frequency for TE mn wave for guide of given dimension 14.103 Variation of vph and vg of TE 01 wave in a wave guide of given dimensions 14.104 Given the wave equation for E z , to find E z and f c 14.105 Given the wave equation for Hz , to find Hz and f c Chapter 15 Optics 15.1 Fraction of light from a point source in a medium escaping across a flat surface Problem Index 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12 15.13 15.14 15.15 15.16 15.17 15.18 15.19 15.20 15.21 15.22 15.23 15.24 15.25 15.26 15.27 15.28 15.29 15.30 15.31 15.32 15.33 15.34 15.35 15.36 15.37 15.38 15.39 15.40 15.41 15.42 15.43 801 Radiation on a perfectly absorbing surface Snell’s law by Fermat’s principle Fermat’s principle applied to mirage Maximum angle of acceptance for optical fibre Angle of emergence in a prism immersed in a liquid Angle between two emerging beams from a prism Deviation of emergent light from a prism which suffers one internal reflection Focal length of system of two lenses in contact Two positions of a convex lens for real imager for a fixed source–screen separation Application of lens maker’s formula Image formation by two coaxial lenses Focal length of a glass sphere Intensity and amplitude of em waves at a given distance from a bulb Image location and its height of an object as observed by a telescope Intensity and amplitude of electric field of a laser beam Refraction matrix and translation matrix for a single lens Matrix equation for a pair of surfaces Using the result of prob (15.18) to obtain the equation for a thin lens Locus of points at constant phase difference from two coherent sources Bandwidth in double-slit experiment Fringe shift in Young’s fringes when a thin film is introduced Intensity distribution in young’s double-slit experiment Wavelength of light in Fresnel’s biprism experiment Wavelength of light with the biprism Interference fringes in a glass wedge Interference with the wedge film Radius of curvature of a lens from Newton’s rings Refractive index of liquid from Newton’s rings Newton’s rings by two curved surfaces Order for which red band coincides with the blue band in Young’s experiment Ratio of minimum and maximum intensities in reflection from two parallel glass plates Intensification of colour from reflection of white light on a thin film Minimum thickness of plate which appears dark on reflection Colour shown in reflection by thin film Minimum thickness of non-reflecting film Constructive interference in the reflected light D1 and D2 lines of sodium, Michelson interferometer Wavelength of light by Michelson interferometer Resolving power of Fabry–Perot interferometer Single slit, wavelength of light Intensity distribution of single-slit diffraction pattern Half-width for central maximum 802 Problem Index 15.44 Coincidence of two different wavelengths of different orders from a singleslit diffraction 15.45 Width of slit from position of second dark band 15.46 Missing orders for a double-slit diffraction pattern 15.47 Interference fringes within the envelope of central maximum of double-slit diffraction pattern 15.48 Overlapping of fourth order with the third one in grating spectrum 15.49 Highest order seen in a grating spectrum 15.50 Missing of higher orders in grating spectrum 15.51 Missing orders in grating spectrum 15.52 Possible number of orders observed in a grating spectrum 15.53 Condition for missing order in a grating experiment 15.54 Intensity of secondary maxima relative to central maxima in single-slit diffraction 15.55 Grating with oblique incidence 15.56 Number of lines/cm in a grating 15.57 Least width of a grating to resolve D1 and D2 lines 15.58 Smallest wavelength separation that can be resolved in grating spectrum 15.59 Resolution of D1 and D2 lines in first and second orders 15.60 Length of base of a prism which can resolve D1 and D2 lines 15.61 Separation of two points on the moon by a telescope 15.62 Radius of lycopodium particles from diffraction 15.63 Fraunhofer diffraction of a circular aperture 15.64 Radii of circles on a zone plate 15.65 Phase retardation for ordinary and extraordinary rays 15.66 Application of Malus’ law 15.67 Elevation of the sun when rays are completely polarized 15.68 Polarizing angle for water–glass interface 15.69 Minimum thickness of quarter wave plate 15.70 Polarimeter experiment 15.71 Application of Malus’ law to three polarizing sheets 15.72 Inclination of a Brewster window [...]... successive distances between them being r and 2r The inner wall is 15/7 times as high as the outer walls which are equal in height The total horizontal range is nr, where n is an integer Find n [University of Dublin] 1.31 A boy wishes to throw a ball through a house via two small openings, one in the front and the other in the back window, the second window being directly behind the first If the boy stands... resistance being proportional to the square of velocity, find the loss in kinetic energy when the body has been projected upward with velocity u and return to the point of projection 1.2.3 Motion in Two Dimensions 1.26 A particle moving in the xy-plane has velocity components dx/dt = 6 + 2t and dy/dt = 4 + t where x and y are measured in metres and t in seconds (i) Integrate the above equation to obtain... Chapter 1 Kinematics and Statics Abstract Chapter 1 is devoted to problems based on one and two dimensions The use of various kinematical formulae and the sign convention are pointed out Problems in statics involve force and torque, centre of mass of various systems and equilibrium 1.1 Basic Concepts and Formulae Motion in One Dimension The notation used is as follows: u = initial velocity, v = final velocity,... their positions and speeds are as shown in the diagram If car A has a constant acceleration of 0.6 m/s2 and car B has a constant deceleration of 0.46 m/s2 , determine when A will overtake B [University of Manchester 2007] Fig 1.2 1.9 A boy stands at A in a field at a distance 600 m from the road BC In the field he can walk at 1 m/s while on the road at 2 m/s He can walk in the field along AD and on the... distance of 5 m in front of the house and the house is 6 m deep and if the opening in the front window is 5 m above him and that in the back window 2 m higher, calculate the velocity and the angle of projection of the ball that will enable him to accomplish his desire [University of Dublin] 1.32 A hunter directs his uncalibrated rifle toward a monkey sitting on a tree, at a height h above the ground and... speed is 7 m/s, find the speed in m/s at which the tip of the shadow moves √ 1.6 The relation 3t = 3x + 6 describes the displacement of a particle in one direction, where x is in metres and t in seconds Find the displacement when the velocity is zero 1.7 A particle projected up passes the same height h at 2 and 10 s Find h if g = 9.8 m/s2 1.8 Cars A and B are travelling in adjacent lanes along a straight... It then comes down and recrosses the top of the tower in time t2 , time being measured from the instant the object was projected up A second object released from √ the top of the tower reaches the ground in time t3 Show that t3 = t1 t2 1.29 A shell is fired at an angle θ with the horizontal up a plane inclined at an angle α Show that for maximum range, θ = α2 + π4 1.30 A stone is thrown from ground... imaginary and therefore unphysical 1 2 1 Kinematics and Statics v–t and a–t Graphs The area under the v–t graph gives the displacement (see prob 1.11) and the area under the a–t graph gives the velocity Motion in Two Dimensions – Projectile Motion Equation: y = x tan α − 1 gx 2 2 u 2 cos2 α (1.1) Fig 1.1 Projectile Motion Time of flight: T = Range: R = 2u sin α g u 2 sin 2α g (1.2) (1.3) u 2 sin2 α... descent However, in the presence of air resistance the final speed is less than the initial speed and the time of descent is greater than that of ascent (see prob 1.21) 1.2 Problems 3 Equation of motion of a body in air whose resistance varies as the velocity of the body (see prob 1.22) Centre of mass is defined as r cm = m i ri 1 = mi M mi r i (1.8) Centre of mass velocity is defined as Vc = 1 M... is simultaneously projected from the ground with sufficient velocity to enable it to ascend 19.6 m When and where the stones would meet 1.4 A particle moves according to the law x = A sin π t, where x is the displacement and t is time Find the distance traversed by the particle in 3.0 s 4 1 Kinematics and Statics 1.5 A man of height 1.8 m walks away from a lamp at a height of 6 m If the man’s speed