Physics Formulary - By ir. J.C.A.Wevers.pdf

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physics.dvi
Physics Formulary
By ir. J.C.A. Wevers
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1995, 2001 J.C.A. Wevers
Version: November 13, 2001
Dear reader,
This document contains a 108 page L A T E X file which contains a lot equations in physics. It is written at advanced
undergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics and
often needs to look up equations.
This, and a Dutch version of this file, can be obtained from the author, Johan Wevers
( johanw@vulcan.xs4all.nl ).
It can also be obtained on the WWW. See http://www.xs4all.nl/˜johanw/index.html ,where
also a Postscript version is available.
If you find any errors or have any comments, please let me know. I am always open for suggestions and
possible corrections to the physics formulary.
This document is Copyright 1995, 1998 by J.C.A. Wevers. All rights are reserved. Permission to use, copy
and distribute this unmodified document by any means and for any purpose except profit purposes is hereby
granted. Reproducing this document by any means, included, but not limited to, printing, copying existing
prints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause,
unless upon explicit prior written permission of the author.
This document is provided by the author “as is”, with all its faults. Any express or implied warranties, in-
cluding, but not limited to, any implied warranties of merchantability, accuracy, or fitness for any particular
purpose, are disclaimed. If you use the information in this document, in any way, you do so at your own risk.
The Physics Formulary is made with teT E Xand A T E X version 2.09. It can be possible that your L A T E Xversion
has problems compiling the file. The most probable source of problems would be the use of large bezier
curves and/or emT E X specials in pictures. If you prefer the notation in which vectors are typefaced in boldface,
uncomment the redefinition of the
n vec command in the T E X file and recompile the file.
Johan Wevers
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Contents
Contents
I
Physical Constants
1
1 Mechanics 2
1.1 Point-kineticsinafixedcoordinatesystem............................ 2
1.1.1 Definitions ........................................ 2
1.1.2 Polarcoordinates..................................... 2
1.2 Relativemotion.......................................... 2
1.3 Point-dynamics in a fixed coordinate system . .......................... 2
1.3.1 Force, (angular)momentum and energy . . . . . . ................... 2
1.3.2 Conservativeforcefields................................. 3
1.3.3 Gravitation ........................................ 3
1.3.4 Orbitalequations..................................... 3
1.3.5 Thevirialtheorem .................................... 4
1.4 Point dynamics in a moving coordinate system . . . . . . ................... 4
1.4.1 Apparentforces...................................... 4
1.4.2 Tensornotation...................................... 5
1.5 Dynamicsofmasspointcollections................................ 5
1.5.1 Thecentreofmass .................................... 5
1.5.2 Collisions ......................................... 5
1.6 Dynamics of rigid bodies . . . .................................. 6
1.6.1 MomentofInertia .................................... 6
1.6.2 Principalaxes....................................... 6
1.6.3 Timedependence..................................... 6
1.7 Variational Calculus, Hamilton and Lagrange mechanics . ................... 6
1.7.1 VariationalCalculus ................................... 6
1.7.2 Hamilton mechanics . .................................. 7
1.7.3 Motion around an equilibrium, linearization . . . . ................... 7
1.7.4 Phase space, Liouville’s equation . . .......................... 7
1.7.5 Generatingfunctions................................... 8
2 Electricity & Magnetism 9
2.1 TheMaxwellequations...................................... 9
2.2 Forceandpotential ........................................ 9
2.3 Gaugetransformations ...................................... 10
2.4 Energyoftheelectromagneticfield................................ 10
2.5 Electromagneticwaves ...................................... 10
2.5.1 Electromagneticwavesinvacuum............................ 10
2.5.2 Electromagneticwavesinmatter............................. 11
2.6 Multipoles............................................. 11
2.7 Electriccurrents.......................................... 11
2.8 Depolarizingfield......................................... 12
2.9 Mixturesofmaterials....................................... 12
I
II
Physics Formulary by ir. J.C.A. Wevers
3R la v ty 13
3.1 Specialrelativity ......................................... 13
3.1.1 TheLorentztransformation ............................... 13
3.1.2 Redandblueshift .................................... 14
3.1.3 Thestress-energytensorandthefieldtensor....................... 14
3.2 Generalrelativity ......................................... 14
3.2.1 Riemanniangeometry,theEinsteintensor........................ 14
3.2.2 Thelineelement ..................................... 15
3.2.3 Planetaryorbitsandtheperihelionshift ......................... 16
3.2.4 The trajectory of a photon . . .............................. 17
3.2.5 Gravitationalwaves.................................... 17
3.2.6 Cosmology ........................................ 17
4 Oscillations 18
4.1 Harmonic oscillations . . . . . .................................. 18
4.2 Mechanic oscillations . . . . . .................................. 18
4.3 Electric oscillations . . . . . . .................................. 18
4.4 Waves in long conductors . . . .................................. 19
4.5 Coupled conductors and transformers .............................. 19
4.6 Pendulums ............................................. 19
5W s 20
5.1 Thewaveequation ........................................ 20
5.2 Solutionsofthewaveequation .................................. 20
5.2.1 Planewaves........................................ 20
5.2.2 Sphericalwaves...................................... 21
5.2.3 Cylindricalwaves..................................... 21
5.2.4 Thegeneralsolutioninonedimension.......................... 21
5.3 Thestationaryphasemethod ................................... 21
5.4 Green functions for the initial-value problem . .......................... 22
5.5 Waveguides and resonating cavities . .............................. 22
5.6 Non-linearwaveequations .................................... 23
6Op s 24
6.1 Thebendingoflight........................................ 24
6.2 Paraxialgeometricaloptics.................................... 24
6.2.1 Lenses .......................................... 24
6.2.2 Mirrors .......................................... 25
6.2.3 Principalplanes...................................... 25
6.2.4 Magnification....................................... 25
6.3 Matrix methods . ......................................... 26
6.4 Aberrations ............................................ 26
6.5 Reflectionandtransmission.................................... 26
6.6 Polarization ............................................ 27
6.7 Prismsanddispersion....................................... 27
6.8 Diffraction............................................. 28
6.9 Specialopticaleffects....................................... 28
6.10TheFabry-Perotinterferometer.................................. 29
7 Statistical physics 30
7.1 Degreesoffreedom........................................ 30
7.2 Theenergydistributionfunction ................................. 30
7.3 Pressureonawall......................................... 31
7.4 Theequationofstate ....................................... 31
7.5 Collisions between molecules . .................................. 32
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Physics Formulary by ir. J.C.A. Wevers
III
7.6 Interactionbetweenmolecules .................................. 32
8 Thermodynamics 33
8.1 Mathematical introduction . . .................................. 33
8.2 Definitions............................................. 33
8.3 Thermalheatcapacity....................................... 33
8.4 The laws of thermodynamics . .................................. 34
8.5 StatefunctionsandMaxwellrelations .............................. 34
8.6 Processes ............................................. 35
8.7 Maximalwork........................................... 36
8.8 Phasetransitions ......................................... 36
8.9 Thermodynamic potential . . . .................................. 37
8.10Idealmixtures........................................... 37
8.11 Conditions for equilibrium . . .................................. 37
8.12 Statistical basis for thermodynamics . .............................. 38
8.13Applicationtoothersystems ................................... 38
9 Transport phenomena 39
9.1 Mathematical introduction . . .................................. 39
9.2 Conservationlaws......................................... 39
9.3 Bernoulli’s equations . . . . . .................................. 41
9.4 Characterisingofflowsbydimensionlessnumbers........................ 41
9.5 Tubeflows............................................. 42
9.6 Potentialtheory.......................................... 42
9.7 Boundary layers . ......................................... 43
9.7.1 Flow boundary layers . .................................. 43
9.7.2 Temperature boundary layers . .............................. 43
9.8 Heat conductance ......................................... 43
9.9 Turbulence ............................................ 44
9.10Selforganization ......................................... 44
10 Quantum physics 45
10.1 Introduction to quantum physics . . .............................. 45
10.1.1 Black body radiation . .................................. 45
10.1.2 TheComptoneffect ................................... 45
10.1.3 Electrondiffraction.................................... 45
10.2 Wavefunctions.......................................... 45
10.3 Operators in quantum physics .................................. 45
10.4 Theuncertaintyprinciple .................................... 46
10.5 The Schr¨odingerequation.................................... 46
10.6 Parity............................................... 46
10.7 The tunnel effect ......................................... 47
10.8 The harmonic oscillator . . . .................................. 47
10.9 Angular momentum . . . . . .................................. 47
10.10Spin ............................................... 48
10.11TheDiracformalism....................................... 48
10.12 Atomic physics ......................................... 49
10.12.1 Solutions ........................................ 49
10.12.2 Eigenvalueequations .................................. 49
10.12.3 Spin-orbitinteraction .................................. 49
10.12.4 Selectionrules...................................... 50
10.13Interactionwithelectromagneticfields ............................. 50
10.14Perturbationtheory ....................................... 50
10.14.1 Time-independentperturbationtheory ......................... 50
10.14.2 Time-dependentperturbationtheory .......................... 51
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