Problems in General Physics

385 Pages·2020·3.081 MB·English

by  Igorʹ Evgenʹevich Irodov

CLASSIC TEXTS SERIES
Problems in
GENERAL
PHYSICS
ARIHANT PRAKASHAN (Series), MEERUT
ARIHANT PRAKASHAN (Series), MEERUT
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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