© 1995, 2001 J.C.Ạ Wevers Version: November 13, 2001
Dear reader,
This document contains a 108 page IẤTEX 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/7~ johanw/index.htm1, where
also a Postscript version is availablẹ
If you find any errors or have any comments, please let me know I am always open for suggestions and :hỊ he phưsies £ larỵ This document is Copyright 1995, 1998 by J.C.Ạ Wevers All rights are reserved Permission to use, copy
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The Physics Formulary is made with teTEX and I4TRX version 2.09 It can be possible that your IÃIEX version has problems compiling the filẹ The most probable source of problems would be the use of large bezier
Trang 3Contents Contents I Physical Constants 1 1 Mechanics 2 1.1 Point-kinetics in a fñxed coordinate sysfem ee 2 1.1.1 Definitions 2 ee 2 1.1.2” Polarcoordinates ee es 2
[2 Relaive moion ee ee eee 2
1.3 Point-dynamics in a fixed coordinate system 2 ee es 2
1.3.1 Force, (angularymomentum andenergy 2 0 eee ee ee 2
1.3.2 Conservative force fields 0 QẶ Q Q Q Q HQ Và vV 3 IS: 90 “e“6 << 4 .a (Š .ẻ.Íằ.Í.đa Í.ÍÝŸÝŸ AĂš< 3
13.4 OrbialequaHlons Q Q Q Q Q Q Q Q nu n cv Q à k VN Và 3
l.25 The vilial(h@OrEm QC HO QO Q HQ HQ KV ky KV Ko 4 1.4 Point dynamics in a moving coordinate system 1 Ặ Q Q Q Q Q ee 4
1.4.1 Apparentforces 2 aa AC iaaA 4 1.4.2 TensornofaHOn ee 5 1.5 Dynamics of masspoIntcolleCOnS Q QO Q Q Q Q Q Q Q V 22 5 1.51 Thecenreofmass ee 5 1.5.2 Collisions HẠ Ta aa 5 l6 Dynamics ofriglidbodles Q Q Q Q0 Q On HQ HQ Q v KV k Và 6 1.6.1 MomentofInertia ee ee 6 1.6.2 PrincIpalaxes Q Q LH HQ n HQ ng vn ng cv Q à KV VN V kia 6 1.63 Timedependencẹ CO Q CO Q LH ng Hà v v VY và 6 1.7 Variational Calculus, Hamilton and Lagrangemechancs 6 1/71 VariaionalCalculus QẶ Q Q Q 0Q Q SH HQ và 6 1.72 Hamiltonmechancs Ặ Q Q Q Q Qua 7
1.7.3 Motion around an equllibrium, linearlzatlon - 7
1.7.4 Phase space, Liouville°sequalon c c c Q Q vn V22 7 1.75 Generating functions 2 c c c c Q Q kg n n và k v v NV và 8 2 Electricity & Magnetism 9 2.1 The Maxwellequations 6 Q Q Q Q Q Q ng n n Q v V v v NV Và 9
2.2 Force and potenHal c Q Q Q Q Q Q HQ Q Q n ng v v k k k k k k va 9
2.3 Gaugetransformations eee ee ee ee ee ee ee eee eee ee es 10 24 Energy of the electromagnetic field ee na 10 2.5 ElectromagnetCWAV©S Q Q Q Q LH Q Q n ng Q à vn vn k k v k k VN V kia 10 2.5.1 Electromagnetic waves in vacuum 1 ee 10 2.5.2 ElectromagnetiC waves In maff€T Q Q Q Q2 1] 2.6 Multipoles 2 ẽằẽ.aTC II
2/7 ElectrICCUTTEIS QC 0Q Q LH HQ Q n Q n Q à vn N k k k VN k NV kia 1]
2.8 Depolarizing feld Q Q Q HO Q HQ HQ nu cv k v V k k NV Và 12
Trang 4II Physics Formulary by ir J.C.Ạ Wevers 3 Relativity 13 3.1 Special relativity 2 ee 13 3.1.1 The Lorentz transformation 20 0 ee ee 13 3.1.2 Redandblue shift 2 0.0 ee ee 14 3.1.3 The stress-energy tensor and the fieldtensor 14 3.2 _Generalrelativity .-.0.% 14 3.2.1 Riemannian geometry, the Einsteintensor 2.2.0+00085 14 3.2.2 Thelineelement 0 0 2 ee 15 3.2.3 Planetary orbits and the perihellonshíÍt - 16 3.2.4 The trajectoryofaphotOn c c k Q k Q Q Quà VY và ky v 17 SN Ê CÀ (70/00: 210.2: aRDỤAIiaiiaiẳiẳiiiiiiaadaẠ 17 “9` 90920 SKhKN II IIIaIaAIaẠ.aNẶNaa 17 4 Oscillations 18 CN i09: s 9 vì) (0) 18 4.2 Mechanic osclIllalOnS ee 18 4.3 ElectricoscIlalOns Q Q Q Q Q Q Q n Q Q nu kg cv cv kg v kg k Và 18
4.4 Waves In long conducfOrS QC CO Q Q Q SH HQ Q ng và KV xà 19 45 Coupled conductors and transformers .Ặ Q Q Q Q Q S Q Q Q s 19 4.6 Pendulums ee 19 5 Waves 20 5.1 The waveequation ., , .-0- 20 5.2_ Solutions ofthe wave eQualOon c Q c Q k Q Q Q Q Q n v và v Và xà 20 “NT Ô `“ n6Hđj.Ằ ee 20 5.2.2 SpherlcalWaVes Q Q Q Q Q Q nu cu Hà cv và k VN k k KV va 21 53.23 CylindricalWaVves c Q Q Q kg Quà Q à kg k k k k va 21 5.2.4 The general solutionin one dimension 0 0 2 eee ee ee ee 21 5.3 The statonary phasemethod Ặ QC Q HQ HQ VY xà 21 5.6 Non-linear Wave €eQuatOnS Q Q Q Q LH Q n n Quà Qà k N v NV Và Kia 23 6 Optics 24 6.1 The bending ofligh( QC QC Q HQ HH n nà Q à k NV và 24 6.2 Paraxial geometricalOpCS Ặ Q Q Q LH Q HH n V v v g v v.v va 24 6.2.1 Lenses “Mi L 24 „3n 9 TT eee eee eee 25 6:2.3—PrmeipaLplanesS 7c.“ 25 6.2.4 Magnificaton c Q Q Q Q Q nh Q n n v vn cv k VN vn kia 25 63 Matrixmethods QC CO QC Q HQ Q HQ gà Q à k N k v VN k k k Và 26
i9 Hạ ạẽäẽẼĂẽ IỊ 26
6.5 Reflection and transmIsSiIOn Ặ Q Q Q Q Q Q Q Q Q Q vn v vv và xà 26 1x3: 9 08 ad ad a (Ẽ(ẽ.:.: aaa 27
6.7 Prisms anđispersiOn ee 27
66 Diffraction ee 28
6.9 Special optical effects ee ee ee 28
6.10 The Fabry-Perot interferometer 2 Ặ c Q Q Q Q Q Q Q HQ v22 v2 29
7 Statistical physics 30 71 Degreesoffreedom c Q Q Q Q Q Q Q Q n Hạ nà cv v kg v k k VN vn va 30 7.2 The energy distributonfunctlon c c c c Q Q Q kg và và Và và 30 7.3 Pressureonawall, ee 31 7.4 The equation of state 2 ee 31
7.5 Collisions between molecules 2 0.0 ee 32
Trang 5
Physics Formulary by ir J.C.Ạ Wevers IH 7.6 Interaction between molecules Q Q Q Q Q Q Q Q kg Q.2 và 32 8 Thermodynamics 33 8.1 Mathematical introduction 2 kg kg n nà và và va 33 82 Definiions c Q Q Q Q HQ nu cv Q à q à v kg k k à k N k k kia 33 68.3 ThermalheatCcapaClỵ c Q Q Q LH Q HQ ng ng ng cv Q à VN k N k kia 33
8.4 The laws ofthermodynamics Ặ c Q Q Q Q Q HS Q1 v2 v2 34
8.5 State functions and MaxwellrelalOns c Q Q Q2 v2 34 5 “~°° ` E TRE l l lh LIiaaa a I"ặNẵNIHAaaạa aaT 35 Dư ri 00/2 Sẽ Mạ ee 36 8.8 PhasetranSIiiONS Ặ CO QC Q Q Q kg ng ng và k N cv v k v k v k va 36 8.9 Thermodynamic potentlal Ặ Q Q Q Q Q Q Q HQ nà v1 v v2 v2 37 6.10 lIdealmixfuires QC QC Q On Q HQ nà Q à cv kg k v k V VN va 37 8.11 Conditions forequilibrium ., 37 8.12 Statistical basis for thermodynamics 2 ee Q Q Q Q Q Q Q Q Q Q vV 38 8.13 Application to otherSySfMS QC QC Q Q Q Q Q HQ n và Và v k V và 38 9 Transport phenomena 39 9] Mathematicalintoducion c Q Q Q Q Q kg Q1 1v v23 v2 39 LNÔo 2à 00) 8 6 ằạ(ạ .aaãa Hạ 39 93 Bernoullisequalons c Q Q Q kg Q và v.v VN k VN VY va 4] 9.4 Characterising of fows by dimensionless numbers, 2 4] 9.5 Tube flows 1 42 ` 6i 02A1 ẽHH ă HH Á AT 42
97 Boundary layers c Q Q Q HQ HQ Q n n ng cà Q à k k k v VN k k k kia 43 9.7.1 Flow boundary layers Q Q Q Q Q Q HQ Q Q.1 và 43 9.7.2 Temperature boundary Ïlay€ers Ặ c Q Q Q Q Q và Y2 43
9.8 Heatconductancẹ ee 43
9.9 Turbulence 1 ee 44
Trang 6IV Physics Formulary by ir J.C.Ạ Wevers 10.15 N-particle systems 2 QC Q Q Q Q n ng nu nà Q à k vn NV NV k k k Kà 51 10.15.1 General 2 ad IlIaiaiH( 51 10.152 Molecules QC Q Q HQ nu ng V kg NV kia 52 10.16 Quanfum stafSHCS ee 52 11 Plasma physics 54 1t.¥ Introduction ee ee eee ee ee Roe Roe ea 54 I0 91 1 am TRAẶẠ ga 54 11.3 Elasticcollislons CO Q0 Q Q Q HQ HQ nu ng k v v VN k k k Và 55 11.3.1 General 55 11.3.2 The Coulomb Interaction Ặ Q Q Q Q Q2 56 11.3.3 The induced dipole Interactlon co Q2 56 11.3.4 The centre of mass system 2 ee ee ee 56 IỊ3.5 Scattering oflight ee ee ee 56 11.4 Thermodynamic equilibrium and reversibility .2 0000004 57 11.5 Inelastic collisions 2 Q Q Q Q Q n ng nu k Q v VN k k k Và 57 11.5.1 Types ofcollisions Q QQ Q Q Q Q Q Q Q vn A v V Và va 57 11.5.2 TOSSS€CHODS QC QC Q Q Q Q n H n Q u Q à k n k v ki k k NV V k V na 58 11.6 Radiation TC IliaaasnnaaA 1a a a (l(ẽ(.: qaaaaa .ẼX& 58 11.7 The Boltzmamn transporfequaHon ‹- ‹ c c k Q k K 59 IỊ8 Collision-radiative models ˆ co ee ee ee 60 1Ị9 Waves In plasmaÌS Q Q LH HQ HQ ng ng cv Q à k k k NV và 60
12 Solid state physics 62
12.1 Crystal structurẹ 2 2 ee 62
12.2 Crystal binding 2 Q Q Q Q Q Q Q ng cv vn ngà v k k k k k k va 62
12:3-CrystaLvibratlONS—z 7C cm ee eee ee 63
12.3.1 Á lattice withone type OfatOTS—-—- 77 7c c7 cv r c r ng na rẽ zẽ mẽ šn ng 63
123.2 A lattice with two typesOfafOmS ee 63
123.3 Phonons Q Q Q Q Q Q HQ HQ Quà Q v kg k V k V kia 63 12.3.4 Thermal heat capacity 2 Ặ Q Q Q LH HQ nà V v k k Và 64
12.4 Magnetic field In the solid stat€ Ặ Q Q Q Q Q Q Q S Q Q Q v vv2 65
12.441 Dielectrlcs QC QO Q Q HQ HQ ng ng ng cv k k k k VN k Và 65 12.4.2- ParamagnetIS—.—.—.— 7 cốc c c Q g c na ca mẽ NT 65 12.43 FerromagnetsM —.”.”.”.”.”.”.” cu kh na Ta kh Tang 65 12.5 Free electron Fermløas Q Q Q QO Q Q Q Q Q HQ vn k k N k Và 66 12.5.1 Thermal heat capacity 2 Ặ Q Q Q LH HQ nà VN k Và 66 12.5.2 Electric conductance 1 2 Q Q Q Q Q Q nu v vv Và va 66 12.5.3 The Hall-effect 2 ee ee 67 12.5.4 Thermal heatconductVIỵ ee 67 12.6 Energy bands ee ee ee ee ee ee ee 67 12.7 SemiconductOFS ee ee ee ee es 67 12.8 Superconductivity 2 ha a Hạ 68 128.1 DescrIpHOn Q Q Q Q HQ Q HQ ng ng cv Q à k k VN VY và 68 12.8.2 The Josephsoneffect Ặ Q Q Q Q Q Q HQ nà và và va 69 12.8.3 Flux quantisation In a superconductingring 69 12.8.4 Macroscopic quantum Interferencẹ c2 70 12.8.5 The London equaton c Q Q k Q Q Q1 và Vy v 70
12.8.6 The BCS model .0 0.00.0 0000 02 ee ee ee ee ee 70
Trang 7Physics Formulary by ir J.C.Ạ Wevers V 13 Theory of groups 71 13.1 Introduction 1 ee 71 13.1.1 Defnition ofagØroup c c c Q Q Q Q vn và vn v va 71 13.12 The Cayleytablẹ ee 71
13.1.3 Conjugated elements, subgroups and classes 71 13.1.4 Isomorfism and homomorfism; representations 72
13.1.5 Reducible and irreducible representations 2 ee 72
13.2 The fundamental orthogonality theorem 2 0 ee Q Vy Và 72 13.2.1 Schur’slemma 2 Q0 Q On HQ n n v V v V k k V KV 72 13.2.2 The fundamernrtal orthogonalitytheorem Ặ.Ặ 72
13.23 Charactr c Q Q Q Q Q HH ng cu cà gà vn k VN NV NV và Kia 72
13.3 The relation with quantutmmechamcs Ặ Ặ Ặ Ặ KV S 73
13.3.1 Representations, energy levels and degeneracy 0.000 ee eee 73
13.3.2 Breaking of degeneracy by a perturbation 73
13.3.3 The construction of a base function ., 73 13.3.4 The direct product ofrepresenfaiÓS Q2 74 13.3.5 Clebsch-Gordan coeffclens ee 74 13.3.6 Symmetric transformations of operators, irreducible tensor operators 74 13.3.7 The Wigner-Eckarttheorem Ặ Q Q Q Q2 75 13.4 Continuous ørfOUPS Q Q Q Q Q Q Q n HQ ng cv gà k k ki v VN v k k Và 75 13.4.1 The 3-dimensional transÏatlon øroup c Q2 75 13:4.2- The -3-dimenstonalrotation-ÐØFOUD—z z cc c ccrcẽẽ rẽ vn: 7Ð 13.5 The group SO(3) ee 77 13.6 Applications to quantunmechancs Ặ Q Q Q Q Q Q Q Q2 v2 77
13.6.1 Vectormodel for the ađition of angular momentum 4 77 13.6.2 Irreducible tensor operators, matrixelements and selectionrules 78
13.7 Applications to particle physics 2 CO QC Q Q Q Q k n V Q n v vn v k k Và xà 79
—————————— l4 Nưclcar physies St
14.1 Nuclear forces———— eee ST 14.2 The shape of the nucleus 1 QO Q Q Q Q kg ng vn và k2 v2 82 14.3 Radioactive ẨeCaVỵ Q Q Q Q Q Q HQ HQ cu nu k à a k v k N k v VN v k k Và Kà 82 14.4 Scattering and nuclearreaclÓS c Q Q Q Q k Q Q Q vn và và v và và 83 14.4.1 Kineticmodel Ặ ee 83
14.4.2 Quantum mechanical model for n-p scaftering 83
14.4.3 Conservation of energy and momentum in nuclear reactions 84 14.5 Radiation dosimetrỵ ee ee ee ee 84 15 Quantum field theory & Particle physics 85 15.1 Creation and annihilaton OperafOFS Ặ kg Q2 v2 85 15.2 Classical and quanumflelds Ặ Q Q Q Q Q Q Q22 85 15.3 The interaction picturẹ 2 CO QC QO Q Q Q Q HQ ng ng cv Qà k k kg V kia S6 15.4 Real scalar field in the Interactlon pICfUT€ ee V S6
15.5 Charged spin-0 particles, conservatlon ofchargẹ c Q2 87
15.6 Field functions for spin-š I10i1095M.ẮỐẮỐỒỀẮẮẮằằằẰằẶằằŒœœẶœẶœẶ.Ặ—-.<ẹmmeéẹé6éáéẶé 87
15.7 Quantization of spin=s fields ee eee 88
15.8 Quantization of the electromagnetic field 0 0.0 ee ee 89
15.9 Interacting fñields and the S-mafrlXx Ặ ee 89
15.10 Divergences and renormalilzaion Ặ c c k Q Q Q22 2 90 15.11 Classification of elementary particles Ặ Q2 v2 90 15.12 PandCP-violaton ee 92 15.13 The standard model 2 ee ee 93 15.13.1 The electroweak theory 2 c Q Q Q Q Q nà và và Vy V 93
15.13.2 Spontaneous symmetry breaking: the Higgs mechanism 94
Trang 8VI Physics Formulary by ir J.C.Ạ Wevers 15.133 Quantumchromodynamics Ặ Q2 94 15.14 Pathintegrals QC Q QO Q Q HQ HQ HQ 1 nà cv k v V k k NV kia 95 15.15 Unification and quantum øraVy c Q Q Q Q Q Q Q Q Q v1 v1 v3 v2 95 16 Astrophysics 96 16.1 Determination of distances 2 ee 96
16.2 Brightness and magnitudes Ặ QC Q KH SE vn và v2 x2 96
16.3 Radiation and stellaratmospheres ee 97
Trang 9Physical Constants | Name | Symbol Value Unit | Number + 7 3.14159265358979323846 Number e e 2.71828182845904523536 rh Euler’s constant y= dim (= 1/k— n(n) = 0.5772156649 —` k=1 Elementary charge € 1.60217733 - 10~19 C Gravitational constant G, 6.67259 - 10—1! m#kg—!s~2
Fine-structure constant a = ẻ /2hceo 1/137
Speed of light in vacuum C 2.99792458 - 108 m/s (def) Permittivity of the vacuum £0 8.854187 - 101“ F/m
Permeability of the vacuum to 47 - 1077 H/m (4meo)~1 8.0870 - 109 Nm?C~? Planck’s constant h 6.6260755 - 10-34 Js Dirac’s constant h =h/2n 1.0545727 - 10734 Js Bohr magneton Up = eh/2me 9.2741 - 10 Am2 Bohr radius ao 6.52918 A Rydberg’s constant Ry 13.595 eV Electron Compton wavelength | Ace = h/mec 2.2463 - 10~12 m
Proton Compton wavelength | Àcp = h/mpc 1.3214: 1015 m
Reduced mass of the H-atom | py 9.1045755- 1073! kg
Stefan-Boltzmann’s constant o 5.67032 - 1078 Wm—?K~*
Wien’s constant kw 2.8978 - 1073 mK
Molar gasconstant R 8.31441 J/mol
Avogadro’s constant Na 6.0221367 - 1023 mol! Boltzmann’s constant k= R/Na 1.380658 - 107? J/K Electron mass Me 9.1093897 - 10734 kg Proton mass Mp 1.6726231 - 107?" kg Neutron mass Mn 1.674954 - 10727 kg Elementary mass unit mụ = 1sm(lC)_ 1.6605656- 107?” kg Nuclear magneton ON 5.0508 - 10-2” J/T
Diameter of the Sun Do 1392 - 108 m
Mass of the Sun Mo 1.989 - 10°° kg
Rotational period of the Sun To 25.38 days Radius of Earth Ra 6.378 - 108 m Mass of Earth MA 5.976 - 1022 kg
Rotational period of Earth Ta 23.96 hours
Trang 10Chapter 1 Mechanics 1.1 Point-kinetics in a fixed coordinate system 1.1.1 Definitions The position 7, the velocity v and the acceleration @ are defined by: 7 = (x, y, z), UV = (“,y, 2), @ = (2, ¥, Z) The following holds:
s(t) = so + / jăt)|dt; F(t) =F + / ăt)dt; ot) =% + / ăt)dt
When the acceleration is constant this gives: u(t) = vo + at and s(t) = 589 + vot + sat For the unit vectors in a direction _L to the orbit €; and parallel to it €, holds: „ Vv ar 3 Vv, 4 et ỳy —C—= ỳy — —C€n ; Cạn —= —— li]: dsp li For the curvature k and the radius of curvature p holds: £_ đã _ d7 _ |d¿ ~ ds ds? | da ¬ Ik| 1.1.2 Polar coordinates Polar coordinates are defined by: x = rcos(@), y = rsin(@) So, for the unit coordinate vectors holds: er = 6&6, €9 = —0E, The velocity and the acceleration are derived from: 7 = ré,., U = 7é, + rÖểp, đ = (— r02)ể„ + (2+0 + rÖ)£) 1.2 Relative motion WX VQ œ2
For the motion of a point D w.r.t a point Q holds: rp = 7Q + with QĐ = 7b — ?Q and œ = 6 Further holds: a = Ø ” means that the quantity is defined in a moving system of coordinates In a moving system holds: U=UQtv'+0x 7’ and@=GQ+ G'+ Ax 7’ +26 xV-Gx (Wx 7") with lở x (ở x #)| =2”,
1.3 Point-dynamics in a fixed coordinate system
1.3.1 Force, (angular)momentum and energy
Newton’s 2nd law connects the force on an object and the resulting acceleration of the object where the mo-
mentum is given by p = mv:
> dm m=const > — Mma
Trang 11Chapter 1: Mechanics 3
Newton’s 3rd law is given by: Faction = —Freaction: actio
For the power P holds: P = W = F` For the total energy W/, the kinetic energy 7' and the potential energy U holds: W =T'+; T= —-U wth7= smụ
The kick § is given by: S=Ap= / Fdt
2 2
The work A, delivered by a force, is A =| F ds = Jr cos(a)ds
1 1
The torque T is related to the angular momentum L: 7 = L =7x F; and L="x p= mi x F, |L| = mr2w The following equation is valid:
OU _ Ø8
Hence, the conditions for a mechanical equilibrium are: ` F, = 0and 5 `7; = 0
The force of friction is usually proportional to the force perpendicular to the surface, except when the motion starts, when a threshold has to be overcome: Firic = f - Fnorm - €t-
1.3.2 Conservative force fields A conservative force can be written as the gradient of a potential: Fons = —VỤ From this follows that Vx F =0 For such a force field also holds: T1 fF ds=0 => U=Up~ [ Fas To
Trang 124 Physics Formulary by ir J.C.Ạ Wevers
Kepler’s orbital equations
Ina force field F = kr~?, the orbits are conic sections with the origin of the force in one of the foci (Kepler’s 1st law) The equation of the orbit is: ( 0) = c2 +2 = (0 — cả r0) 1 +ecos(Ø — 6q) ` 0ø + =—ez) with L2 2 2W L? e ( k
(=———: £ =l†+—z.m-=l —:;: d=—_—s=.- G2 Mtex Œ2u3 Mệ„ a I-e2 2W
a is half the length of the long axis of the elliptical orbit in case the orbit is closed Half the length of the short axis is b = Val ¢ is the excentricity of the orbit Orbits with an equal ¢ are of equal shapẹ Now, 5 types of orbits are possible:
1 k < Oande = 0: acirclẹ
2k <Oand0<<¢< 1: an ellipse
3 k < Oandeé = 1: a parabolẹ
4 k < 0and e > 1: ahyperbole, curved towards the centre of forcẹ
5 k > Oande > 1: ahyperbole, curved away from the centre of forcẹ
Other combinations are not possible: the total energy in a repulsive force field is always positive so ¢ > 1
If the surface between the orbit covered between ¢; and f2 and the focus C around which the planet moves is
Ăt, t2), Kepler’s 2nd taw is
Le
Ăt,,t2) (ti, t2) = <i (t2 — #) = —(te -t
Kepler’s 3rd law is, with T' the period and Mi ¢ the total mass of the system: T? _ Ar?
ae GMiot
1.3.5 The virial theorem The virial theorem for one particle is: 2 dU k UP = — 1 -P) =L(rm —)ì=+ i — (mv-r) =0 => (T) SF ') 5 (> 7) 57 (U) if U on The virial theorem for a collection of particles is: / — — \ (T)=-3( SO Frit D0 Fafa ) \ particles pairs /
These propositions can also be written as: 2Exin + Epot = 0
1.4 Point dynamics in a moving coordinate system
1.4.1 Apparent forces
The total force in a moving coordinate system can be found by subtracting the apparent forces from the forces working in the reference frame: F’’ = F’ — F,pp The different apparent forces are given by:
Trang 13Chapter 1: Mechanics 5 1.4.2 Tensor notation Transformation of the Newtonian equations of motion to x* = x(x) gives: dt Ô#8 dt’ The chain rule gives: đa* - i | (3s TÌ _ Ox az? = d#° d (55) di dt de® dt\da® at) 088 d2 dt di (05 So: dÖz2 _ Ô Ôã2d8t _ Ø3.” dữ dt 028 Ơ#Ơz8 dt Ơ#8Ơ9#Y dt This leads to: d2xe _ Ox" d2z8 1 72% dz? (dz? dt2 Oz dt? ~~ OxPOzT dt \ dt Hence the Newtonian equation of motion 2œ arr ig me will be transformed into: fda" de | BY de dt J +LT4 —— _-})=Ƒ“ dx? dx¥ \ dả dx? The apparent forces are taken from he origin to the effect side in the way T°; — —_
1.5 Dynamics of masspoint collections
1.5.1 The centre of mass
The velocity w.r.t the centre of mass Ris given by v — R The coordinates of the centre of mass are given by:
Bom THÊ
The motion within and outside the centre of mass can be separated:
pete ee Fete Dee Pet
=“oOutside “OUtSIqdC ; 4~TNSIGdeE /THSIqd€
D=MUn; Fext =MGmn; Fig = pu
1.5.2 Collisions
is constant, and T’ = sm”2 is constant The changes in the relative velocities can be derived from: S = Ap =
[t(Wast — Ubefore) Further holds ALc =CB x §, ? | S =constant and L w.r.t B is constant
Trang 146 Physics Formulary by ir J.C.Ạ Wevers 1.6 Dynamics of rigid bodies 1.6.1 Moment of Inertia The angular momentum in a moving coordinate system is given by: => => L= Tà , 7? +7 — lỰ TỊ T2
where I is the moment of inertia with respect to a central axis, which is given by:
or, in the continuous case: m I[=— [ av = [dm Ves J Further holds: -
L; = Iw; ; Lii = I; ) Li; = = lj¡ = = = may
Steiner”s theorem 1s: Ïw.rt.b = Iw.rt.c + m(DM)? if axis C || axis D || Object | J || Object | I |
Cavern cylinder I =mR? Massive cylinder = 3mR?
Disc, axis in plane disc throughm | J = +mR? Halter [= 3 wR?
Cavern sphere I = 3mR? Massive sphere I = 2mR?
Trang 15Chapter 1: Mechanics 7 the equations of Lagrange can be derived: dOL OL dt Đải Oa;
When there are ađitional conditions applying to the variational problem dJ(u) = 0 of the type
K(u) =constant, the new problem becomes: 6J(u) — Ad K(u) = 0
1.7.2 Hamilton mechanics
The Lagrangian is given by: L = >/T (qi) — V(q) The Hamiltonian is given by: H = >) q@ipi — £ In 2 dimensions holds: £ = T — U = $m(7? + r7¢?) — U(r, ¢)
If the used coordinates are canonical the Hamilton equations are the equations of motion for the system:
dạ ag; OH ƠØ dn: ap; OH ƠØ dt An, dt 3a: at Op; at Oi Coordinates are canonical if the following holds: {q,¢;} = 0, {pi,p;} = 0, {qi,p;} = Oi; where {, } is the Poisson bracket: OAOB OAOB A, Bl = — — 14, By 2„ lấn Op, Op; OG a
g obtained by the canonical transformation x = 4, mw cos(@) and p = — mu sin
0 = arctan(—p/mwz) and I = p?/2mw + 4mwả it follows: H(0,1) = wl
The Hamiltonian of a charged particle with charge g in an external electromagnetic field is given by: 1 -»\ 2
H = — (p-¢A) + qV
2m
Thisi ivistic poi view: this is equiv
transformation of the momentum 4-vector p* — p® — qA® A gauge transformation on the potentials 42
corresponds with a canonical transformation, which make the Hamilton equations the equations of motion for the system 1.7.3 Motion around an equilibrium, linearization For natural systems around equilibrium the following equations are valid: OV 83V =——] =0: = ik Vi ith Vip =
lần ) , 0; Ví(g) = V(0) + W2yg¡q„ with Viz th ) ,
With T = $(Mixgigx) one receives the set of equations Mg + Vq = 0 If g;(t) = a; exp(iwt) is substituted, this set of equations has solutions if det(V — w2M) = 0 This leads to the eigenfrequencies of the problem:
y=
k aj M Qk
eigenvibrations
1.7.4 Phase space, Liouville’s equation
In phase space holds:
7>? 8 5 O OH O OH
Trang 168 Physics Formulary by ir J.C.Ạ Wevers If the equation of continuity, 0,0 + V - (ev) = 0 holds, this can be written as: Jo > = 0 to H}+ For an arbitrary quantity A holds: dA ay OA dot I+ 3 Liouville’s theorem can than be written as: do _ m =0; or: / pdq dq = constant 1.7.5 Generating functions Starting with the coordinate transformation: ( Qi = QilG, pi, t) one can derive the following Hamilton equations with the new Hamiltonian K: dQ; OK dP, aK dt 9 Pd a0 a ty Now, a distinction between 4 cases can be made: dF (qi, Qi, t) 1 If pig; — H = P,Q; — K(Pi, Qi, t) — 7 , the coordinates follow from: OF; OF, dF, i=; P=; K=H+—> P Ogi 3Q; + dt : đF2(q¡:-P;;t)
2 If pig; — H = —P,Q; — K(Pi, Qi, t) + lt
Trang 17Chapter 2
Electricity & Magnetism
2.1 The Maxwell equations
The classical electromagnetic field can be described by the Maxwell equations Those can be written both as differential and integral equations: > gc ` ?:)d°A = treo included V-D= Pfree G} (B-m)@A=o V:B=0 ơ -d= TC dđ Vxẫ=-< = OB ‡ 5 dt Ot _ / ‘ds = I tree,included + dt V X = Jiree + Ot For the fluxes holds: = J (D-ñ)d2A, ® = J (B-đ)d2Ạ => The electric displacement D, polarization P and electric field strength E depend on each other according to: np JeokT'
D= eg + B= coe,f, P= S> po/Vol, €r = 1+ Xe, with Xe =
The magnetic field strength H, the magnetization M and the magnetic flux density B depend on each other according to:
2
Trang 18
10 Physics Formulary by ir J.C.Ạ Wevers Here, the freedom remains to apply a gauge transformation The fields can be derived from the potentials as follows: - B=—-VV- =, B=VxA Further holds the relation: c2 = ở x # 2.3 Gauge transformations The potentials of the electromagnetic fields transform as follows when a gauge transformation is applied: A= A-Vf of ,_— — V=V+
so the fñelds # and do not changẹ This results in a canonical transformation of the Hamiltonian Further, the freedom remains to apply a limiting condition Two common choices are: p 1 OV _ 2 OL = 0 This separates the differential equations for A and V: OV = — = ° 0 1 Lorentz-gauge: V - A+ DA = —poJ
2 Coulomb gauge: V - A=0 If p = 0 and J = Oholds V = 0 and follows A from DA = 0
——————————2.4ˆ Energy ofthe electromagncficfield—E—=—=—=————————————————————————
The energy density of the electromagnetic field 1s: dW Tai w / dB + / d The energy density can be expressed in the potentials and currents as follows: e e Wmag = af J-Ad@x , we = 2 pV dex 2.5 Electromagnetic waves
2.5.1 Electromagnetic waves in vacuum
The wave equation OU (7, t) = —f (7, t) has the general solution, with e = (eoo)~ 1⁄2: ( ft —|f—T |J€) a „ TO, GT An|r — ”| If this is written as: J(?*,t) = J(7) exp(—iwt) and Ă#, t) = Ẳ) exp(—iwt) with: yl > + > yo 1 > 4m |— 7| «Ane |— 1|
A derivation via multipole expansion will show that for the radiated energy holds, if d, \ > r:
dP Rol fn eee gal?
1) = Braye || OE
The energy density of the electromagnetic wave of a vibrating dipole at a large distance is:
2 cin2 4 2cin2 4 4Iz12 —¬— tu), = ae p= oer
The radiated energy can be derived from the Poynting vector 8: S= E x H = cWé, The irradiance is the time-averaged of the Poynting vector: J = (|S |)z The radiation pressure pg is given by ps = (1 + R)|S|/c, where R is the coefficient of reflection
Trang 19Chapter 2: Electricity & Magnetism 11
2.5.2 Electromagnetic waves in matter
The wave equations in matter, with Cnat = (=w)~1⁄ ? the lightspeed in matter, are:
oO wa = a par a=
(V?= suốn BE) B=0, th
give, after substitution of monochromatic plane waves: the dispersion relation:
abu p
The first term arises from the displacement current, the second from the conductance current If & is written in
the form k := k’ + tk’ it follows that: ki =wy/5eu 1+ I+—— and k”“=œ\Jšeu —1+ I+— ; \ li (pew)? ; \ V (pew)? This results in a damped wave: E = Eexp(—k" 2-7) exp(i(k’it-7’—wt)) If the material is a good conductor, [iw the wave vanishes after approximately one wavelength, & = (1 + ?) Xp YL ft tn alac 2.0 Muitipotes 1 nr @ yo kn Ame rr nr Lr! Because =- Ss” (5) P;(cos @) the potential can be written as: V = r<~\r For the lowest-order terms this results in: NA 1 1 fa (._JTZ ® 1V1UIIODOUIC t= Us, 0 — J PGaV e Dipole: 1 = 1, ki = [ rcos(@)pdV e Quadrupole: | = 2, kz = $3>(32? — r?) 1 The electric dipole: dipole moment: p = Qleée, where é goes from @ to ©, and F= (p: V) Eoxts and W = —p: Bout: Q_ (30:7 Electric field: EB ~ —“—
4ner” \_ r x ?) The torque 1s: 7 = Ø`x Foust
Trang 2012 Physics Formulary by ir J.C.Ạ Wevers
d®
If the flux enclosed by a conductor changes this results in an induced voltage Ving = —N— If the current flowing through a conductor changes, this results in a self-inductance which opposes the original change:
Veelfind = Lạ If a conductor encloses a flux ® holds: ® = LỊ
2 +4R2-
and N the number of coils The energy contained within a coil is given by W = $LI° and L = uN?AjỊ
The capacity is defined by: C = Q/V For a capacitor holds: C = e9¢,A/d where d is the distance between the plates and A the surface of one platẹ The electric field strength between the plates is BE = a /é9 = Q/e0A where o is the surface chargẹ The accumulated energy is given by W = sCV} The current through a
¬ dV capacity is given by J = —C TE
For most PTC resistors holds approximately: R = Ro(1 + aT), where Ro = pl/Ạ For a NTC holds: R(T) = Cexp(—B/7T) where B and C depend only on the material
If a current flows through two different, connecting conductors x and y, the contact area will heat up or cool
down, depending on the direction of the current: the Peltier effect The generated or removed heat is given by: W =II,,,/t This effect can be amplified with semiconductors
The thermic voltage between 2 metals is given by: V = y(T — To) For a Cu-Konstantane connection holds: y = 0.2 — 0.7 mV/K C al Net W i À ] along a closed path holds: S° V,, = 5° 1, Ryn = 0 2.8 Depolarizing field
If a dielectric material is placed in an electric or magnetic field, the field strength within and outside the
material will change because the material will be polarized or magnetized If the medium has an ellipsoidal
Trang 21Chapter 3 Relativity 3.1 Special relativity
3.1.1 The Lorentz transformation
The Lorentz transformation (Z’, t’) = (#'(2, t), t’(Z,t)) leaves the wave equation invariant if c is invariant: ho D ¬ ả ả 9 4 ¬ lod O* Ø” Oo loo — ¬5n2 — — Ox? + Oỷ + 622 At Ax”? + Oy’? + 9z2_ c2902
This transformation can also be found when ds? = ds’* is demanded The general form of the Lorentz transformation is given by: where The velocity difference v’ between two observers transforms according to: ¬) (i + (y=) =ÿ?ø - + J ? 77 \ ) c Vy ử'= ( + (4 — VA If the velocity is parallel to the x-axis, this becomes 4’ = y, z’ = z and: v= y(x-vt), r= 7(2' +t’) Ứ=*+ (5): that 5) v= tốn LV _ , xv „— Ủa — UỊ Ifư = vé, holds: ØW =3 (TẾT , W'=(W - 0p¿) With đ = v/c the electric field of a moving charge is given by: Q (1 7 8)& E= 4meor (1 — Ø3 sin“(6))3⁄2 The electromagnetic field transforms according to: —- —- > —- = uy E Ei =+(E+0~xB) , Bao (a- T5 Cc
Length, mass and time transform according to: At, = yAto, m; = ymo, J, = lo/y, with 9 the quantities
in a co-moving reference frame and , the quantities in a frame moving with velocity v w.r.t it The proper
time 7 is defined as: dr? = ds*/c?, so Ar = At/+ For energy and momentum holds: W = m,c* = yWo,
Trang 22
14 Physics Formulary by ir J.C.Ạ Wevers
W? = mact + p?c? p = mv = ymou = Wv/c?, and pc = WB where 3 = v/c The force is defined by
F = dp/dt
4-vectors have the property that their modulus is independent of the observer: their components can change
after a coordinate transformation but not their modulus The difference of two 4-vectors transforms also as
a 4-vector The 4-vector for the velocity is given by U° = —— The relation with the “common” velocity
aT
uử := đả /dt 1s: U° = (yú, icy) For particles with nonzero restmass holds: UCU, = —c’, for particles with zero restmass (so with v = c) holds: U°U, = 0 The 4-vector for energy and momentum is given by:
p® = moU*% = (yp',iW/c) So: pap® = —mgc? = p* — W?/c?
3.1.2 Red and blue shift
There are three causes of red and blue shifts: N (4 s— ø c052) \ mn This can give both red- and blueshift, also | to the direction of motion 2 Gravitational redshift: Af = KM f rc
1 Motion: with &, - &, = cos(y) follows: J = 7
3 Redshift because the universe expands, resulting in ẹg the cosmic background radiation:
Ao _ Ro
a RL
3.1.3 The stress-energy tensor and the field tensor The stress-energy tensor is given by: 1 Taw = (oc? + p)UpUy + PGpv + c2 (H,„P?£ + tov FỎ Fag) The conservation laws can than be written as: V, 7” = 0 The electromagnetic field tensor is given by: OAg ÔÁa Fug = 8ˆ Ore x8
with Á„ := (Ä,šV/e) and J, := (J, icp) The Maxwell equations can than be written as:
O, FRY = Lod” , On Fu + On FL + OF yy, = 0 4A C44 dpe — = qFogủ 3.2 General relativity
The basic principles of general relativity are:
1 The geodesic postulate: free falling particles move along geodesics of space-time with the proper time T or arc length s as parameter For particles with zero rest mass (photons), the use of a free parameter is required because for them holds ds = 0 From 6 { ds = 0 the equations of motion can be derived:
d3ze a dx? dx?
as | 87 ds ds ~
Trang 23Chapter 3: Relativity 15
2 The principle of equivalence: inertial mass = gravitational mass => gravitation is equivalent with a curved space-time were particles move along geodesics
3 By a proper choice of the coordinate system it is possible to make the metric locally flat in each point
Li} Jop(®i) = Nag =diag(—1, 1,1, 1)
The Riemann tensor is defined as: Ri’ gt = VaVet" —VeVat”, where the covariant derivative 1s given
by Vja’ = dja" +T%,.a* and Vja; = Oja; — T¥,ax Here,
T° g” th Ogik OG jk 82z! 9+?
Oxi Oak Oz’
jk ~ 9 Oxk ~~ Oxi Ox! ) , for Euclidean spaces this reduces to: L7 k=
are the Christoffel symbols For a second-order tensor holds: [Va, ValT# = ReagT? + RoagTh, Veai =
Opat —Th ai + Vial, Viaij = Onaig —T4,a1; —T aj and Via = 0,0) +Ti,a"i +1} a‘ The following
holds: f2 „ = O03, — WS, +TS 03, — ToT Gy
u
The Ricci tensor is a contraction of the Riemann tensor: Rag := Ry,
The Bianchi identities are: V) Ragyy + Vv Repru + VuRagvr = 0
which is symmetric: Rag = Reạ The Einstein tensor is given by: G°% := RP — 32h, where R := RG is the Ricci scalar, for which
holds: VgGag = 0 With the variational principle 6 [(L(gy,) — Rẻ/167K)\/|g\d*z = 0 for variations Quu — Guv + OGpv the Einstein field equations can be derived: 87K _, _ STK Gap = fapy > whicirca whbich-ea e2
For empty space this is equivalent to Rag = 0 The equation Rag, = 0 has as only solution a flat spacẹ The Einstein equations are 10 independent equations, which are of second order in g,,,, From this, the Laplace equation from Newtonian gravitation can be derived by stating: gyv = Nv + hyv, where |h| < 1 In the stationary case, this results in V7ho9 = 87Ko/c?
OTK
The most general form of the field equations is: Rag — 4gagR+ Agog = ——Txs + “ † t c2 Tt
where A is the cosmological constant This constant plays a role in inflatory models of the universẹ 3.2.2 The line element Oz* ax" The metric tensor in an Euclidean space is given by: 9;; = ——— — Ox" Ox! at nords: Gs = GruvaL” a
This metric, 7, :=diag(—1, 1, 1,1), is called the Minkowski metric
The external Schwarzschild metric applies in vacuum outside a spherical mass distribution, and is given by:
2 2m\~*
ds? = (1 + =) edt? + ụ — =) dr? + rd?
Here, m := Mx«/c? is the geometrical mass of an object with mass M, and dQ? = d6? + sin? 6dỷ This
metric 1s singular for r = 2m = 2KM / c? lfan object 1s smaller than its event horizon 2m, that implies that
its escape velocity is > c, itis called a black holẹ The Newtonian limit of this metric is given by: ds? = —(14 2V)c?dt? + (1 — 2V)(dx? + dỷ + dz?)
where V = —kM/r is the Newtonian gravitation potential In general relativity, the components of g,,, are
associated with the potentials and the derivatives of g,,,, with the field strength
The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metric near
r = 2m They are defined by:
Trang 2416 Physics Formulary by ir J.C.Ạ Wevers er>2m: S ® j = -iiù l T ( + ) ¡nh t U = — — ex — | SII —— 2m P 4m 4m < r Tr t v= 1 — — ex (—) cosh | — L 2m P 4m 4m e — 2m; here, the Kruskal coordinates_are singular, which is necessary_to eliminate the coordinate singularity therẹ The line element in these coordinates is given by: 3 32m et /2m ds? = — r (dv? — dủ) + r?d?
The line r = 2m corresponds to u = v = 0, the limit x° — oo with u = v and x° — —oo with u = —v The Kruskal coordinates are only singular on the hyperbole v? — ủ = 1, this corresponds with r = 0 On the line
dv = +du holds d@ = dp = ds = 0
For the metric outside a rotating, charged spherical mass the Newman metric applies: 2ner — e° r? + ả cos? 0
ds* = 1- >—3— 2d¢2 — { ———— \dry2- (r2 2 cos? 0) d62 —
° ( r2 + a2 <5) c r2 — 2mr + a2 — ẻ r (r + a” cos )
>» 9 Qmr—ẻ)ả sin?O\ 5,, 5 2ă2mr —e7)\ 5
so I Sin® Od > J sin* O(dyp) (ccdt ụ Am + ả cos? 6 S 0d T (xạ +a2cos20 J (dy) (edt)
where m = KM /c?, a = L/Mcande = kQ/eoc’
A rotating charged black hole has an event horizon with Rg = m + Vm? — ả — ẻ
Near rotating black holes frame dragging occurs because g+, 4 0 For the Kerr metric (e = 0, a ¢ 0) then follows that within the surface Rp = m + Vm? — ả cos? 6 (de ergosphere) no particle can be at rest
3.2.3 Planetary orbits and the perihelion shift where := 1/r and h = rp =constant The term 3mu is not present in the classical solution This term can KM h? in the classical case also be found from a potential V(r) = -—— [1+ — ] r r
=Con CUMDtE Anta ĐH O On its UO O ccHi; Aan d dÌ = Ci a CHInƠD kì U Q Crh FAs OS BYU Go pe O Ca \V c npe CC ul Da On k}
theorỵ In zeroth order, this results in an elliptical orbit: uo(y) = A+ Bcos(y) with A = m/h? and B an arbitrary constant In first order, this becomes:
he orbital equation vive 5
B? B?
u(y) = A+ Becos(p — ey) +€ (4 +a Ga cos(2e))
where € = 3m/h? is small The perihelion of a planet is the point for which r is minimal, or wu maximal
This is the case if cos(y — ey) = 0 > y & 2an(1 + €) For the perihelion shift then follows: Ay = 27¢ =
621m? /h? per orbit
Trang 25Chapter 3: Relativity 17
3.2.4 The trajectory of a photon
For the trajectory of a photon (and for each particle with zero restmass) holds ds? = 0 Substituting the external Schwarzschild metric results in the following orbital equation: du (đ° — EM vu 3mwÌ =0 ap \ 7 3.2.5 Gravitational waves
Starting with the approximation g,, = "uỵ + h„„ for weak gravitational fields and the definition Riv = huy — ST he it follows that Ohi = 0 if the gauge condition Nw /8z” = 0 is satisfied From this, it follows that the loss of energy of a mechanical system, if the occurring velocities are < c and for wavelengths > the size of the system, is given by: dE CG <¬ /#&Ọ.Š? \ —— —— ` ff oe Tee (ae) with Qi; = { o(aix; — $6:j;7”)d?z the mass quadrupole moment 3.2.6 Cosmology If for the universe as a whole is assumed:
1 There exists a global time coordinate which acts as x° of a Gaussian coordinate system,
2 The 3-dimensional spaces are isotrope for a certain value of z0,
3 Each point is equivalent to each other point for a ñxed z0
then the Robertson- Walker metric can be derived for the line element: 2 x2 Rˆ( "3 3 "3 23 11 da” = =c đf”+ ———ax (ar rd) r 2 (1 1) 0 2 ( 415 For the scalefactor R(t) the following equations can be derived: 9R fet ke _ _ TRP LA and R2+kc2 8 R2 2 R2 _— 2 ° fC oc £0 t2 t} R? + kẻ _ 8r#Ko A tC where p is the pressure and @ the density of the universẹ If A = 0 can be derived for the deceleration parameter q: RR —_ 470 1= — ng — BHP
where H = R/ R is Hubble’s constant This is a measure of the velocity with which galaxies far away are
moving away from each other, and has the value ~ (75 +25) km-s~!-Mpc~? This gives 3 possible conditions
univ „MT ưu yi univ :
1 Parabolical universe: k = 0, W = 0,q = 3 The expansion velocity of the universe — 0 if t — oọ
The hereto related critical density is 0¢ = 3H?/8nk
2 Hyperbolical universe: k = —1, W < 0,q < 4 The expansion velocity of the universe remains 2
positive forever
3 Elliptical universe: k = 1,W > 0,q > 3 The expansion velocity of the universe becomes negative after some time: the universe starts collapsing
Trang 26Chapter 4 Oscillations 4.1 Harmonic oscillations
The general form of a harmonic oscillation is: U(t) = Vei*+¥) = WU cos(wt + 0),
where W is the amplitudẹ A superposition of several harmonic oscillations with the same frequency results in
another harmonic oscillation:
Nate ” My COS OG ef poy BN wt) — MCO5\Q/ cc GE}
%
with:
>> WU; sin(a;)
tan(Ø =-—_——— and 9?— 9= ng SH 2+2 HALT 8n cm —a/) WW; cos(a; — a;
For harmonic oscillations holds: Ỉ ăt)dt = ~< and — = (iw)”" ăt)
J ?J G“b
4.2 Mechanic oscillations
For a construction with a spring with constant C parallel to a damping k which is connected to a mass M, to
which a periodic force F'(t) = F'cos(wt) is applied holds the equation of motion mz = F(t) — kt — Cạ
With complex amplitudes, this becomes —mw?x2 = F — Ca — ikwx With w% = C/m follows: H' H' : ,and for the velocity holds: 4 = ———— 7 ¿VŒmỗ + k where 6 = ve 0 The quantity Z = F'/z is called the impedance of the system The quality of the system Wo W Vem is given by Q = | equency Ww a 5 Called velocity resonance frequencỵ Sis equal to wọ (ne TEsOnance @ =——— m(uậ — Ø2) + iku “1 9⁄2 +>3 Alin h 4 hÔ 8@œ@1fn hara arC QC UYy PU VV Z7\Q — ®S/ 1 Cy in ^ im Ta r› LÍ VC Z2 V TO Od sc WT Witt O
|Z (w)|V/2 In these points holds: R = X and 6 = +Q7!, and the width is 2Awg = wo/Q
The stiffness of an oscillating system is given by F'/x The amplitude resonance frequency wa is the frequency
where iwZ is minimal This is the case for wa = wo/1 — $Q?
The damping frequency wp is a measure for the time in which an oscillating system comes to rest It is given
damped oscillation (k? = 4mC) holds wp = 0 A strong damped oscillation (k? > 4mC) drops like (if
k? >> 4mC) ăt) & x exp(—t/T)
4.3 Electric oscillations
The impedance is given by: Z = R+iX The phase angle is y := arctan(X/R) The impedance of a resistor is R, of a capacitor 1/iwC and of a self inductor tw The quality of a coil is Q = wL/R The total
impedance in case several elements are positioned is given by:
Trang 27
Chapter 4: Oscillations 19
1 Series connection: V = IZ,
1 1 ⁄ ;
Leo = DSi Lot =) Li: Gy ee Q= >: Z = R(1 +iQ6)
2 parallel connection: V = IZ, 4 1 I I I = Tse TT = TT» C = C; ’ Q = ’ ; Ztet 2Ö Zỉ Lot 2Ö bị 2Ö Zo 1 +iQ6 iL 1 Here, Zp = 4/ = and wp = —— C VLC
The power given by a source is given by P(t) = V(t) - I(t), so (P), = Vor lege cos(Ad)
= 1VIcos(¢, — oj) = 3/2Re(Z) = £V?Re(1/Z), where cos(A@) is the work factor — 5 Z 4.4 Waves in long conductors dL d These cables are in use for signal transfer, ẹg coax cablẹ For them holds: Zp = Tn a ca ca da da The transmission velocity is given by v = \ / Tae ¥
4.5 Coupled conductors and transformers
For two coils enclosing each others flux holds: if ®;2 is the part of the flux originating from /2 through coil 2
which is enclosed by coil 1, than holds ®jg = Mylo, ®2; = Mo,/, For the coefficients of mutual induction M;; holds: Ma = Mai = M= kWLiLa = —— = — ~ Ni No 2 1 where 0 < k < 1 is the coupling factor For a transformer is k ~ 1 At full load holds: W lạ — 1M ~ Tị M Vo t, twhotRwoaa Vio No
The oscillation time T = 1/f, and for different types of pendulums is given by: e Oscillating spring: T = 27,/m/C if the spring force is given by F = C- Al
Trang 28Chapter 5 Waves
5.1 The wave equation
The general form of the wave equation is: Ow = 0, or:
_9 1 0?u Ø°u Pu Pu 1 0?u U—==——=—— —==——=0 02 ؇2 Ox? T Oỷ T Oz? 0u2 0t2 where u is the disturbance and v the propagation velocitỵ In general holds: v = f By definition holds: kÀ = 2z and w = 27ƒ
In principle, there are two types of waves:
1 Longitudinal waves: for these holds & || @ || & 2 Transversal waves: for these holds k || @ _L ở The phase velocity is given by vpn = w/k The group velocity is given by: dw đUgh k dn tea Ge tn + GE = oyu (1)
where n is the refractive index of the medium If vp, does not depend on w holds: vpn = vg In a dispersive
medium it 1s possible that vg Uph OF Ug Ủph, and 0y - 0y = c’ If one wants to transfer information witl
a9 > = h UY MAG VJUOUTA ¬ s nO s a Cl V Wav ^tzo
electromagnetic field propagates This velocity is often almost equal to the group velocitỵ
For some media, the propagation velocity follows from:
e Pressure waves in a liquid or gas: v = ,/«/@, where « is the modulus of compression e For pressure waves in a gas also holds: v = \/yp/oe = \/yRT/M e waves ina string: v = \/Fgpanl/m A 2 2h e Surface waves on a liquid: v = ge + “3 tanh a“ 27 6À À where h is the depth of the liquid and + the surface tension If h << À holds: 0 + V/gh 5.2 Solutions of the wave equation 5.2.1 Plane waves
In n dimensions a harmonic plane wave is defined by:
u(z,t) = 2”&cos(wt) À sin(Rz;)
¿=1
Trang 29
Chapter 5: Waves 21
The equation for a harmonic traveling plane wave is: u(Z,t) = aicos(k- #+wt + y)
If waves reflect at the end of a spring this will result in a change in phasẹ A fixed end gives a phase change of m/2 to the reflected wave, with boundary condition u(/) = 0 A lose end gives no change in the phase of the reflected wave, with boundary condition (Ou/Ox),; = 0
If an observer is moving w.r.t the wave with a velocity vops, he will observe a change in frequency: the UE — U Doppler effect This is given by: T ¬ £ 0 Uf 5.2.2 Spherical waves When the situation is spherical symmetric, the homogeneous wave equation is given by: 1 O(ru) (ru) = 0 v2 Ot? Or? with general solution: u(r, t) = ofr) +0, 010) 5.2.3 Cylindrical waves When the situation has a cylindrical symmetry, the homogeneous wave equation becomes: 1Ø?u 10 ( du\ ——— — — Ƒ'— => v2 Ot? rr Or \ Or,
This is a Bessel equation, with solutions which can be written as Hankel functions For sufficient large values
of r these are approximated by: 0 u(r, t) = te cos(k(r + vt)) Jr P wm 6 a œ co 2 5 = 2 83u(z,£) ~ o”™ a = 2 (am) u(a, t) m=0
where b„ € JR Substituting u(a,t) = Ae“(**—“*) gives two solutions w; = w;(k) as dispersion relations
The general solution is given by: oO ⁄ £ L 7 u(a, t) = | (alk) s0 + b(k)éŒ#—ez029) dk J —oco
Because in general the frequencies w; are non-linear in & there is dispersion and the solution cannot be written any more as a sum of functions depending only on x + vt: the wave front transforms
5.3 The stationary phase method
Usually the Fourier integrals of the previous section cannot be calculated exactlỵ If w;(k&) € JR the stationary
Trang 30
22 Physics Formulary by ir J.C.Ạ Wevers
5.4 Green functions for the initial-value problem
This method is preferable if the solutions deviate much from the stationary solutions, like point-like excitations Starting with the wave equation in one dimension, with V? = 0?7/0z? holds: if Q(z, 2’, t) is the solution with O 0 initial values Q(z, x’,0) = 6(a — a’) and Tem g = 0, and P(x, z’,t) the solution with initial values OP(z, x’,0 P(z,z',0) = 0 and G27) 6(a2 — x’), then the solution of the wave equation with arbitrary initial Øu(z,0
conditions f(x) = (+, 0) and g(a) = ¬ is given by:
u(a, t) -/ f(x" )Q(a, 2’, t)da’ + / g(x’) P(x, x’, t)dx P and Q are called the propagators They are defined by: Q(z,2',t) = $[6(a—a2' —vt)+d(x—2' +vt)] ‘ pl P(x, 2',t) if jr —a2’| < vt if |x—a2’|> vt OP t) Further holds the relation: Q(z, 2’, t) = OP(z,2',t) Ot
5.5 Waveguides and resonating cavities
The boundary conditions for a perfect conductor can be derived from the Maxwell equations If 7 is a unit
vector | the surface, pointed from 1 to 2, and K is a surface current density, than holds:
Tỉ - (› — Dy) = n xX (Eà — E,) = 0
n> (By— By) = ïi x (Ha = Hy) = R
=
In a waveguide holds because of the cylindrical symmetry: E(Z,t) = €(2,y)e**-“*) and B(#,t) = B(x, y)e\*?-“» , From this one can now deduce that, if B, and €, are not = 0: i OB JE, OB go Bu = Epỉ — k? vã Hư Oy ) By = mm a (kG dy ` "oe “xi 9€; OB, 9€; b= a (KE pew) y= a ee Epiw & ỞU 7 eœ2 RX OY OL
Now one can distinguish between three cases:
1 B, = 0: the Transversal Magnetic modes (TM) Boundary condition: €z|sur¢ = 0 = 0 surf OB, 2 FE, = 0: the Transversal Electric modes (TE) Boundary condition: "ôn For the TE and TM modes this gives an eigenvalue problem for €, resp 6, with boundary conditions: Q?2 2 82 = w = —y"* with eigenvalues 77 := euw? — k? Ox? + Oy
This gives a discrete solution ye with eigenvalue y7: k = \/epw? — 77 For w < we, k is imaginary
and the wave is damped Therefore, wy is called the cut-off frequencỵ In rectangular conductors the following expression can be found for the cut-off frequency for modes TE, , of TMm,n:
2
ye = ————
(m/a)? + (n/b)?
Trang 31Chapter 5: Waves 23
3 E, and B, are zero everywhere: the Transversal electromagnetic mode (TEM) Than holds: k =
w/e and ve = Ug, just as if here were no waveguidẹ Further k € JR, so there exists no cut-off
frequencỵ
In a rectangular, 3 dimensional resonating cavity with edges a, b and c the possible wave numbers are given
by: ky _ mn , ky = = , kp = mst This results in the possible frequencies f = vk /27 in the cavity: OG f = ụ nộ + ny + nà _— 2VWa2 b2 € For a cubic cavity, with a = b = c, the possible number of oscillating modes Ny, for longitudinal waves is given by: 4xa ƒ3 3u3 Ny = Because transversal waves have two possible polarizations holds for them: Ny = 2Nỵ
5.6 Non-linear wave equations
The Van der Pol equation is given by:
2
aa e@p(L— a„>)42
dt? dt + 2+ I wU*# L
Gx? can be ignored for very small values of the amplitudẹ Substitution of x ~ e“* gives: w = 300 ( +
2,/1— š£?) The lowest-order instabilities grow as 3£00 While z is growing, the 2nd term becomes larger and diminishes the growth Oscillations on a time scale ~ wo‘ can exist If x is expanded as = x + ex) 4 227) 4+ and this is substituted one obtains, besides periodic, secular terms ~ et If it is assumed
that there exist timescales 7,, 0 < 7 < N with 07, /0t = €” and if the secular terms are put 0 one obtains:
d {1 (dx\? da\?
dt \2 Lai) +3 jot? = eat Be (5)
This is an energy equation Energy is conserved if the left-hand side ¡s 0 If z2 > 1/, the right-hand side changes sign and an increase in energy changes into a decrease of energỵ This mechanism limits the growth of oscillations The Korteweg-De Vries equation is given by: Ou Oey Oh Ot Ox Ox poe _ 4 0x3 ~ ¬—— `¬—— non—lin dispersive
This equation is for example a model for ion-acoustic waves in a plasmạ For this equation, soliton solutions of the following form exist:
with c = 1 + Sad and ẻ = ad/(12b7)
Trang 32Chapter 6
Optics
6.1 The bending of light
For the refraction at a surface holds: n,; sin(@;) = nz sin(@,) where n is the refractive index of the material
Snell’s law is:
If An < 1, the change in phase of the light is Ay = 0, if An > 1 holds: Ay = ạ The refraction of light in a
material is caused by scattering from atoms This is described by: Nee” fj egm — we — w? — idw Jj 3] n=1+ where 7 is the electron density and f; the oscillator strength, for which holds: Ð` ƒ; = 1 From this follows j
that vg = c/(1 + (neẻ /2egmw?)) From this the equation of Cauchy can be derived: n = ap + ai /Ả More general, it is possible to expand n as: n = > 12K:
For an electromagnetic wave in general holds: = 4/#yJr The path, followed by a light ray in material can be found from Fermat’s principle: 2 2 s[a=a [9 S)„ x.: 1 1 6.2 Paraxial geometrical optics 6.2.1 Lenses
The Gaussian lens formula can be deduced from Fermat’s principle with the approximations cosy = 1 and sin y = ỵ For the refraction at a spherical surface with radius FR holds: Thy 12 T1 — T12 Vv b R where |v| is the distance of the object and |b| the distance of the imagẹ Applying this twice results in: p= 0a)
where ny is the refractive index of the lens, f is the focal length and R; and Rez are the curvature radii of both surfaces For a double concave lens holds R; < 0, Re > 0, for a double convex lens holds R; > 0 and Ry < 0 Further holds:
1
Trang 33Chapter 6: Optics 25
D := 1/f is called the dioptric power of a lens For a lens with thickness d and diameter D holds to a good approximation: 1/f = 8(n — 1)d/ D7 For two lenses placed on a line with distance d holds: Loita fo oft fe hth In these equations the following signs are being used for refraction at a spherical surface, as is seen by an incoming light ray: || Quantity | + —
R Concave surface | Convex surface
f Converging lens | Diverging lens
Đ Real object Virtual object
b Virtualimage Real image 6.2.2 Mirrors For images of mirrors holds: 1y 1-26 1L 1N f vb R Rov
vhere his the perpendicular distance from the point the igh av hits the mirror to the optical axis Spherica
aberration can be reduced by not using spherical mirrors A parabolical mirror has no spherical aberration for light rays parallel with the optical axis and is therefore often used for telescopes The used signs are: || Quantity | + | — |
R Concave mirror | Convex mirror f Concave mirror | Convex mirror Đ Real object Virtual object b Real image Virtual image
6.2.3 Principal planes
The nodal points N of a lens are defined by the figure on the right If the lens is
surrounded by the same medium on both sides, the nodal points are the same as / the principal points H The plane T the optical axis through t the © principal points Ni
0N:
alled the principal planẹ C de ribed by a ma ij d OT
distances hf; and he to the boundary of the lens holds: m11— 1 hạ =n————, hạ="m THh12 Th12 M22 —1 6.2.4 Magnification b
The linear magnification is defined by: N = ¬y
The angular magnification is defined by: Na = _ set
Qnone
where Qsys is the size of the retinal image with the optical system and Qyone the size of the retinal image
without the system Further holds: N - Na = 1 Fora telescope holds: N = fobjective/ focular- The f-number
is defined by f / Dob jective-
Trang 3426 Physics Formulary by ir J.C.Ạ Wevers
6.3 Matrix methods
A light ray can be described by a vector (na, y) with a the angle with the optical axis and y the distance to
the optical axis The change of a light ray interacting with an optical system can be obtained using a matrix multiplication:
( naq2 \ vl nị@1 \
re \ am }
where ‘Tr(M/) = 1 M is a product of elementary matrices These are:
1 Transfer along length /: Mp = ( ị in i )
2 Refraction at a surface with dioptric power D: Mp = ( ọ % )
6.4 Aberrations
Lenses usually do not give a perfect imagẹ Some causes are:
1 Chromatic aberration is caused by the fact that n = n(A) This can be partially corrected with a lens
which is composed of more lenses with different functions n;(A) Using N lenses makes it possible to obtain the same f for NV wavelengths
al-aberration is caused by second-order effects which are usually tgnore does not make a perfect lens Incomming rays far from the optical axis will more bent
N Πa
3 Coma is caused by the fact that the principal planes of a lens are only flat near the principal axis Further
away of the optical axis they are curved This curvature can be both positive or negativẹ
4 Astigmatism: from each point of an object not on the optical axis the image is an ellipse because the thickness of the lens is not the same everywherẹ
5 Field curvature can be corrected by the human eyẹ
6 Distorsion gives abberations near the edges of the imagẹ This can be corrected with a combination of positive and negative lenses
6.5 Reflection and transmission
If an electromagnetic wave hits a transparent medium part of the wave will reflect at the same angle as the
incident angle, and a part will be refracted at an angle according to Snell’s law It makes a difference whether the F field of the wave is | or || w.r.t the surfacẹ When the coefficients of reflection r and transmission t are
where Fo, is the reflected amplitude and Fo; the transmitted amplitudẹ Then the Fresnel equations are: sinl@ Q.-) — TUE —— UT) 6 —6,) — vị Ft) — "ll tan(0;+6,) ˆ mt sin(Ø¿ + 6,)
t= 2 sin(Ø;) cos(Ø;) f= 2 sin(Ø;) cos(;)
| ~ sin(Ø, + Ø;)cos(0,— 6) `“ sin(0 +6,)
The following holds: ¢; —r, = 1 and t) +r) = 1 If the coefficient of reflection A and transmission 7" are defined as (with 6; = 6,.):
1¿ cos(Ø;)
K III mịỊm— and T I; cos(6;)
Trang 35Chapter 6: Optics 27
with J = (J8 |) it follows: R+T = 1 A special case is 7) = 0 This happens if the angle between the reflected and transmitted rays is 90° From Snell’s law it then follows: tan(@;) = n This angle is called Brewster’s
anglẹ The situation with r, = 0 is not possiblẹ 6.6 Polarization 1 tRin
The polarization is defined as: P = 1, + Ta Lax + Lmin
where the intensity of the polarized light is given by J, and the intensity of the unpolarized light is given by
Lỵ Imax and [min are the maximum and minimum intensities when the light passes a polarizer If polarized
light passes through a polarizer Malus law applies: (9) = I(0) cos?(@) where @ is the angle of the polarizer
The state of a light ray can be described by the Stokes-parameters: start with 4 filters which each transmits half
ine mnftensit ne Hrst 18 iIndepenaent oF the polariZavion, the second and t(hira are near polarizers with tne
transmission axes horizontal and at +45°, while the fourth is a circular polarizer which is opaque for [-states
Then holds Si = 2h, So = 21> — 2h, S3 = 213 — 2h and S4 = 21 — 2h
The state of a polarized light ray can also be described by the Jones vector:
l2 — Epxẻ+
a ( Eoué*+ )
For the horizontal P-state holds: E = (1,0), for the vertical P-state E = (0.1), the #-state is given by E = sv2( 1,—¿) and the L-state by E= sv2( 1,2) The change in state of a light beam after passage of
optical equipment can be described as E, = M- E, For some types of optical equipment the Jones matrix Af is given by:
Homogene circular polarizor right Homogene circular polarizer left 5
Horizontal linear polarizer: ( ọ 0 ) Vertical linear polar1zer: ( " ° ) ` “7 Linear polarizer at +45° 5 ( ' ' ) 1 —I kÓ cà ° 1 Lineair polarizer at —45 5 ( T1 1 ) +- plate, fast axis vertical eit /4 ( Ị O -2 0 )
=X plate, fast axis horizontal e4 (193 4~“\ plate, Lo a) | SN OS | —¬ — 7 Pri tdi ;
A light ray passing through a prism is refracted twice and aquires a deviation from its original direction
6=6;+ 06) + a w.rt the incident direction, where a is the apex angle, 6; is the angle between the incident
angle and a line perpendicular to the surface and 6; is the angle between the ray leaving the prism and a line perpendicular to the surfacẹ When 6; varies there is an angle for which 6 becomes minimal For the refractive index of the prism now holds:
_ sin(Š (Ổmin + @)) sin(a)
Trang 3628 Physics Formulary by ir J.C.Ạ Wevers The dispersion of a prism is defined by: _ đồ _ dồ dn — dÀ_ dndÀ where the first factor depends on the shape and the second on the composition of the prism For the first factor follows: đ 2sin($œ)
dn _ co5(s (min + ơ))
For visible light usually holds đn/đÀ < 0: shorter wavelengths are stronger bent than longer The refractive
index in this area can usually be approximated by Cauchy’s formulạ
6.8 Diffraction
Fraunhofer diffraction occurs far away from the source(s) The Fraunhofer diffraction of light passing through SS
1434] 12+e +e rÍ shank
1H1U1LI DIC sits Ts GUsCrirped DỴ
I(@) — (sin(u) ° sin(Nv) ° In u sin(v)
where u = mbsin(@)/\, v = mdsin(@)/Ạ N is the number of slits, 6 the width of a slit and d the distance
between the slits The maxima in intensity are given by dsin(@) = kẠ The diffraction through a spherical aperture with radius a is described by: T@) _ (Ze ae) Ip ka sin(0)
The diffraction pattern of a rectangular aperture at distance R with length a in the x-direction and 6 in the y-direction is described by:
I(a,y) (sued ey
Ip ~ a’ 8!
whara al —— bam
VVIILẢLIC Oa — RAL 7
When X rays are diffracted at a crystal holds for the position of the maxima in intensity Bragg’s relation: 2dsin(@) = nA where d is the distance between the crystal layers
Close at the source the Fraunhofermodel is invalid because it ignores the angle-dependence of the reflected
waves This is described by the obliquity or inclination factor, which describes the directionality of the sec-
ondary emissions: E(@) = $£o(1 + cos(@)) where is the angle w.r.t the optical axis
Diffraction limits the resolution of a system This is the minimum angle A@,,;, between two incident rays
coming from points far away for which their refraction patterns can be detected separatelỵ For a circular slit
holds: A@min = 1.22/D where D is the diameter of the slit
For a grating holds: A@min = 2A/(Nacos(6,,)) where a is the distance between two peaks and N the number of peaks The minimum difference between two wavelengths that gives a separated diffraction pattern in a multiple slit geometry is given by A\/X = nN where N is the number of lines and n the order of the pattern
6.9 Special optical effects
e Birefringe and dichroism D is not parallel with E if the polarizability P of a material is not equal in
all directions There are at least 3 directions, the principal axes, in which they are parallel This results
in 3 refractive indices n; which can be used to construct Fresnel’s ellipsoid In case ng = ng 4 nj, which happens ẹg at trigonal, hexagonal and tetragonal crystals there is one optical axis in the direction Of + Incident light rays can now be split up in two parts: the ordinary wave is linear polarized the plane through the transmission direction and the optical axis The extraordinary wave is linear polarized
Trang 37Chapter 6: Optics 29
in the plane through the transmission direction and the optical axis Dichroism is caused by a different absorption of the ordinary and extraordinary wave in some materials Double images occur when the
incident ray makes an angle with the optical axis: the extraordinary wave will refract, the ordinary will
not
e Retarders: waveplates and compensators Incident light will have a phase shift of Ay = 27d(|no —
Ne|)/Xo if an uniaxial crystal is cut in such a way that the optical axis is parallel with the front and back
extraordinary wavẹ For a quarter-wave plate holds: Ay = 7/2
e The Kerr-effect: isotropic, transparent materials can become birefringent when placed in an electric field In that case, the optical axis is parallel to Ẹ The difference in refractive index in the two directions is given by: An = \jK Ẻ, where K is the Kerr constant of the material If the electrodes have an effective length @ and are separated by a distance d, the retardation is given by: Ay = 2nK(V?/d?,
where V- is the applied -voltage—
e The Pockels or linear electro-optical effect can occur in 20 (from a total of 32) crystal symmetry classes,
namely those without a centre of symmetrỵ These crystals are also piezoelectric: their polarization
changes when a pressure is applied and vice versa: P= pd + cox The retardation in a Pockels cell is
Ay = 2rneres V/Xo where 763 is the 6-3 element of the electro-optic tensor
e The Faraday effect: the polarization of light passing through material with length d and to which a magnetic field is applied in the propagation direction is rotated by an angle @ = VBd where VY is the Verdet constant
e Cerenkov radiation arises when a charged particle with vg > v¢ arrives The radiation is emitted within
a cone with an apex angle a with sin(a@) = C/Cmedium = C/NUVq
6.10 The Fabry-Perot interferometer
Ce —+-zxÍ T> 4—~ 4 t 4 1 11,4 [| A R | A R
Por a Fabry-Ferot mterrerometer NOs m1 ñ | \ 1 5
senearal: TFT _4 PR 1 A — 1 where Tic the GkH(CIdỊ, £ | £0 T 4 — ft Wee fs Ute
transmission factor, A the reflection factor
and A the absorption factor If F' is given
by F = 4R/(1 — R)? it follows for the intensity distribution: | = aN h_{, A 1 1
TI; [| I—R} 1+ Fsin*(0) —ự | | V_-
The term [1 + #'sin?(0)] 1 := Ă@) is Source Lens “a Screen
called the Airy function Focussing lens
The width of the peaks at half height is given by y = 4/ VF The finesse F is defined as F = 3xVF The
maximum resolution is then given by Afimin = c/2ndF
Trang 38
Chapter 7 Statistical physics 7.1 Degrees of freedom
A molecule consisting of n atoms has s = 3n degrees of freedom There are 3 translational degrees of freedom,
a linear molecule has s = 3n — 5 vibrational degrees of freedom and a non-linear molecule s = 3n — 6 A
Because vibrational degrees of freedom account for both kinetic and potential energy they count doublẹ So,
for linear molecules this results in a total of s = 6n — 5 For non-linear molecules this gives s = 6n — 6 The
average energy of a molecule in thermodynamic equilibrium is (Fot) = s5kT Each degree of freedom of a
molecule has in principle the same energy: the principle of equipartition
The rotational and vibrational energy of a molecule are:
+2
Wot = s0 +1) = Bi(I+1), W%¡y = (0 + 5)hwo
The vibrational levels are excited if kT ~ hw, the rotational levels of a hetronuclear molecule are excited if kT ~ 2B For homonuclear molecules ađitional selection rules apply so the rotational levels are well coupled if kT ~ OB
797] listribution funct;
The general form of the equilibrium velocity distribution function is Plug, Vy, Vz )dvzdvydv, = P(vz)dvz + P(vy)dvy - P(vz)dvz with
P(v;)dv;, = U¿)đU¿ — alr exp 1 ub dv; a2 Ui
Trang 39Chapter 7: Statistical physics 3] 7.3 Pressure on a wall The number of molecules that collides with a wall with surface A within a time 7 is given by: 27m [ff @n = [ [ [ nAvrcos(0)P(v,0, v)duđdy HT From this follows for the particle flux on the wall: ® = +n (v) For the pressure on the wall then follows: 2m cos(0)d3N 2 3 =“—^x — =Ý“n(E đ”p - , so p gn ) | colby 3 &§
If intermolecular forces and the volume of the molecules can be neglected then for gases from p
and (E) = 2kT can be derived:
2
(; + art (V ~ bm,) = n„RT
There is an isotherme with a horizontal point of inflection In the Van der Waals equation this corresponds with the critical temperature, pressure and volume of the gas This is the upper limit of the area of coexistence
between liquid and vapor From dp/dV = 6-and d?p/dV? = 6 fottows: 7 7
8a
Tor = SER > Per = a Ver = 3bns
27b2 `
For the critical point holds: per Vin cr /RT 1 = 3, which differs from the value of 1 which follows from the
general gas law
Gases behave the same for equal values of the reduced quantities: the law of the corresponding states A virial expansion is used for even more accurate views:
(A BP) Cry
PL, Vin) = RT | om "m [3 +] “mm 7
The Boyle temperature Tz is the temperature for which the 2nd virial coefficient is 0 In a Van der Waals gas, this happens at Tg = a/ Rb The inversion temperature T, = 2Tp
The equation of state for solids and liquids is given by:
V 1 (ðV 1 (av — = Tp TT pAT — nuấp T—KkrAp=1+—(—) AT+=(=] A tư lộn), 'ÿ Âm),
Trang 4032 Physics Formulary by ir J.C.Ạ Wevers
7.5 Collisions between molecules
The collision probability of a particle in a gas that is translated over a distance dz is given by nodz, where a is
U
the cross section The mean free path is given by £ = —— with u = Vi + v3 the relative velocity between NUT
the particles If m, << mg holds: “= ,/L+ me so f= — If m1, = mz holds: 2 = UI V TH2 TU 1 nov 2 1 — This means that the average time between two collisions is given by 7 = —— If the molecules are approximated by hard 1
NOV
spheres the cross section is: ¢ = 47(D? + D3) The average distance between two molecules is 0.55n~1/°,
Collisions between molecules and small particles in a solution result in the Brownian motion For the average
motion of a particle with radius R can be derived: (x?) = $ (r?) = kT t/30nR
A gas is called a Knudsen gas if € >> the dimensions of the gas, something that can easily occur at low pressures The equilibrium condition for a vessel which has a hole with surface A in it for which holds that
L> JA/t is: 21 VT) = n2VT> Together with the general gas law follows: p+/V T1 = pa/V 1ạ
A
If two plates move along each other at a distance d with velocity w, the viscosity n is given by: F, = 7 =
The velocity profile between the plates is in that case given by w(z) = zw,/d It can be derived that 7 = 3 ol (0) where v is the thermal velocitỵ d Tạ—T The heat conductance in a non-moving gas is described by: “ =KA S5) , which results in a temper- ature profile T(z) = T, + 2(T2 —T;)/d It can be derived that k = $Cpvne (v) /Nạ Also holds: « = Cyn A better expression for & can be obtained with the Eucken correction: & = (1L + 9RẦ/(4đ„v)Cy - rị with an error <5%
7.6 Interaction between molecules
For dipole interaction between molecules can be derived that U ~ —1/r® If the distance between two
molecules approaches the molecular diameter D a repulsing force between the electron clouds appears ‘This
force can be described by Urep ~ exp(—)r) or Viep = +C5/r* with 12 < s < 20 This results in the
Lennard-Jones potential for intermolecular forces: vo=«l()-(2) with a minimum € at r = rỵ The following holds: D ~ 0.89r, For the Van der Waals coefficients a and b and the critical quantities holds: a = 5.275N% D°e, b = 1.3NaD°, kT = 1.2€ and Vin kr = 3.9NaD?