CLASSIC TEXTS SERIES Problems in GENERAL PHYSICS ARIHANT PRAKASHAN (Series), MEERUT ARIHANT PRAKASHAN (Series), MEERUT All Rights Reserved © PUBLISHER No part of this publication may be re-produced, stored in a retrieval system or by any means, electronic, mechanical, photocopying, recording, scanning, web or otherwise without the written permission of the publisher. Arihant has obtained all the information in this book from the sources believed to be reliable and true. However, Arihant or its editors or authors or illustrators don’t take any responsibility for the absolute accuracy of any information published and the damage or loss suffered thereupon. All disputes subject to Meerut (UP) jurisdiction only. ADMINISTRATIVE & PRODUCTION OFFICES Regd. Office 'Ramchhaya' 4577/15, Agarwal Road, Darya Ganj, New Delhi -110002 Tele: 011- 47630600, 43518550; Fax: 011- 23280316 Head Office Kalindi, TP Nagar, Meerut (UP) - 250002 Tel: 0121-2401479, 2512970, 4004199; Fax: 0121-2401648 SALES & SUPPORT OFFICES Agra, Ahmedabad, Bengaluru, Bhubaneswar, Bareilly, Chennai, Delhi, Guwahati, Hyderabad, Jaipur, Jhansi, Kolkata, Lucknow, Meerut, Nagpur & Pune. ISBN 978-93-5176-256-0 Printed & Bound By Arihant Publications (I) Ltd. (Press Unit) For further information about the books published by Arihant, log on to www.arihantbooks.com or e-mail at [email protected] PREFACE This book of problems is intended as a textbook for students at higher educational institutions studying advanced course in physics. Besides, because of the great number of simple problems it may be used by students studying a general course in physics. The book contains about 1900 problems with hints for solving the most complicated ones. For students convenience each chapter opens with a time-saving summary of the principal formulas for the relevant area of physics. As a rule the formulas are given without detailed explanations since a student, starting solving a problem, is assumed to know the meaning of the quantities appearing in the formulas. Explanatory notes are only given in those cases when misunderstanding may arise. All the formulas in the text and answers are in SI system, except in Part Six, where the Gaussian system is used. Quantitative data and answers are presented in accordance with the rules of approximation and numerical accuracy. The main physical constants and tables are summarised at the end of the book. The Periodic System of Elements is printed at the front end sheet and the Table of elementary Particles at the back sheet of the book. In conclusion, the author wants to express his deep gratitude to colleagues from MIPhI and to readers who sent their remarks on some problems, helping thereby to improve the book. I.E. Irodov CONTENTS Preface 3 A Few Hints for Solving the Problems 6 Notation 7 Part One Physical Fundamentals of Mechanics 1 1.1. Kinematics 1 1.2. The Fundamental Equation of Dynamics 10 1.3. Laws of Conservation of Energy, Momentum, and Angular Momentum 19 1.4. Universal Gravitation 33 1.5. Dynamics of a Solid Body 36 1.6. Elastic Deformations of a Solid Body 37 1.7. Hydrodynamics 50 1.8. Relativistic Mechanics 55 Part Two Thermodynamics and Molecular Physics 62 2.1. Equation of the Gas State. Processes 2.2. The First Law of Thermodynamics. Heat Capacity 65 2.3. Kinetic Theory of Gases. Boltzmann’s Law and Maxwell’s Distribution 69 2.4. The Second Law of Thermodynamics. Entropy 75 2.5. Liquids. Capillary Effects. 81 2.6. Phase Transformations 83 2.7. Transport Phenomena 87 Part Three Electrodynamics 92 3.1. Constant Electric Field in Vacuum 92 3.2. Conductors and Dielectrics in an Electric Field 98 3.3. Electric Capacitance. Energy of an Electric Field 105 3.4. Electric Current 112 3.5. Constant Magnetic Field. Magnetics 123 3.6. Electromagnetic Induction. Maxwell’s Equations 134 3.7. Motion of Charged Particles in Electric and Magnetic Fields 147 Part Four Oscillations and Waves 153 4.1. Mechanical Oscillations 153 4.2. Electric Oscillations 167 4.3. Elastic Waves. Acoustics 175 4.4. Electromagnetic Waves. Radiation. 180 Part Five Optics 186 5.1. Photometry and Geometrical Optics 186 5.2. Interference of Light 197 5.3. Diffraction of Light 202 5.4. Polarization of Light 213 5.5. Dispersion and Absorption of Light 220 5.6. Optics of Moving Sources 223 5.7. Thermal Radiation. Quantum Nature of Light 226 Part Six Atomic and Nuclear Physics 232 6.1. Scattering of Particles. Rutherford-Bohr Atom 232 6.2. Wave Properties of Particles. Schrodinger Equation 237 6.3. Properties of Atoms. Spectra 243 6.4. Molecules and Crystals 250 6.5. Radioactivity 256 6.6. Nuclear Reactions 260 6.7. Elementary Particles 265 Answer and Solutions 269 Appendices 353 1. Basic Trigonometrical Formulas 355 2. Sine Function Values 356 3. Tangent Function Values 357 4. Common Logarithms 358 5. Exponential Functions 360 6. Greek Alphabet 362 7. Numerical Constants and Approximations 362 8. Some Data on Vectors 362 9. Derivatives and Integrals 363 10. Astronomical Data 364 11. Density of Substance 364 12. Thermal Expansion Coefficients 365 13. Elastic Constants, Tensile Strength 365 14. Saturated Vapour Pressure 365 15. Gas Constants 366 16. Some Parameters of Liquids and Solids 367 17. Permittivities 367 18. Resistivities of Conductors 367 19. Magnetic Susceptibilities of Para and Diamagnetics 368 20. Refractive Indices 368 21. Rotation of the Plane of Polarization 369 22. Work Function of Various Metals. 369 23. K Band Absorption Edge 369 24. Mass Absorption Coefficients 370 25. Ionization Potentials of Atoms 370 26. Mass of Light Atoms 370 27. Half-Life Values of Radionuclides 370 28. Units of Physical Quantities 371 29. The Basic Formulas of Electrodynamics in the SI and Gaussian Systems 373 30. Fundamental Constants 375 A Few Hints for Solving The Problems First of all, look through the tables in the Appendix, for many problems cannot be 1 solved without them. Besides, the reference data quoted in the tables will make your work easier and save your time. Begin the problem by recognizing its meaning and its formulation. Make sure that 2 the data given are sufficient for solving the problem. Missing data can be found in the tables in the Appendix. Wherever possible, draw a diagram elucidating the essence of the problem; in many cases this simplifies both the search for a solution and the solution itself. 3 Solve each problem, as a rule, in the general form, that is in a letter notation, so that the quantity sought will be expressed in the same terms as the given data. A solution in the general form is particularly valuable since it makes clear the relationship between the sought quantity and the given data. What is more, an answer obtained in the general form allows one to make a fairly accurate judgement on the correctness of the solution itself (see the next item). 4 Having obtained the solution in the general form, check to see if it has the right dimensions. The wrong dimensions are an obvious indication of a wrong solution. If possible, investigate the behavior of the solution in some extreme special cases. For example, whatever the form of the expression for the gravitational force between two extended bodies, it must turn into the well-known law of gravitational interaction of mass points as the distance between the bodies increases. Otherwise, it can be immediately inferred that the solution is wrong. When starting calculations, remember that the numerical values of physical 5 quantities are always known only approximately. Therefore, in calculations you should employ the rules for operating with approximate numbers. In particular, in presenting the quantitative data and answers strict attention should be paid to the rules of approximation and numerical accuracy. Having obtained the numerical answer, evaluate its plausibility. In some cases 6 such an evaluation may disclose an error in the result obtained. For example, a stone cannot be thrown by a man over the distance of the order of 1 km, the velocity of a body cannot surpass that of light in a vacuum, etc. Notation Vectors are written in boldface upright type e.g., r, F; the same letters printed in lightface italic type (r, F) denote the modulus of a vector. Unit vectors i, j, k are the unit vectors of the Cartesian coordinates x, y, z (some times the unit vectors are denoted as ex, ey, ez) e, e, ez are the unit vectors of the cylindrical coordinates , , z, n, , are the unit vectors of a normal and a tangent. Mean values are taken in angle brackets < > e.g., <v>, <P>. Symbols , d, and in front of quantities denote : , the finite increment of a quantity, e.g., r = r2 – r1; U = U2 – U1 d, the differential (infinitesimal increment), e.g. dr.  the elementary value of a quantity e.g., A, the elementary work. Time derivative of an arbitrary function f is denoted by df/dt or by a dot over a . letter, f. Vector operator  (‘‘nabla’’). It is used to denote the following operations:  the gradient of  (grad ). the divergence of  (div ), × , the curl of  (curl ). Integrals of any multiplicity are denoted by a single sign  and differ only by the integration element; dV, a volume element, dS, a surface element, and dr, a line element. The sign  denotes an integral over a closed surface, or around a closed loop. 1 Physical Fundamentals of Mechanics 1.1 Kinematics ● Average vectors of velocity and acceleration of a point: ∆r ∆v v = , w = , …(1.1a) ∆t ∆t where∆ris the displacement vector (an increment ofaradius vector.) ● Velocity and acceleration of a point: dr dv v= , w= . …(1.1b) dt dt ● Acceleration of a point expressed in projections on the tangent and the normal to a trajectory: dv v2 w = τ,w = , …(1.1c) τ dt n R whereRis the radius of curvature of the trajectory at the given point. ● Distance covered byapoint : s=∫vdt, …(1.1d) wherevis themodulusof the velocity vector of a point. ● Angular velocity and angular acceleration of a solid body: dφ dω ω= ,β= , …(1.1e) dt dt ● Relation between linear and angular quantities for a rotating solid body: v =[ωr], w =ω2R,|w |=βR, …(1.1f) n τ whereristheradiusvectoroftheconsideredpointrelativetoanarbitrary point on the rotation axis, andRis the distance from the rotation axis. 1. AmotorboatgoingdownstreamovercamearaftatapointA;τ=60min lateritturnedbackandaftersometimepassedtheraftatadistance l =6.0kmfromthepointA.Findtheflowvelocityassumingtheduty of the engine to be constant. 2. Apointtraversedhalfthedistancewithavelocityv .Theremaining 0 partofthedistancewascoveredwithvelocityv forhalfthetime,and 1