Equations in Physics By ir. J.C.A. Wevers Contents Contents I Physical Constants 1 1 Mechanics 2 1.1 Point-kinetics in a fixed coordinate system ......................... 2 1.1.1 Definitions ...................................... 2 1.1.2 Polar coordinates ................................... 2 1.2 Relative motion ........................................ 2 1.3 Point-dynamics in a fixed coordinate system ........................ 2 1.3.1 Force, (angular)momentum and energy ....................... 2 1.3.2 Conservative force fields ............................... 3 1.3.3 Gravitation ...................................... 3 1.3.4 Orbital equations ................................... 3 1.3.5 The virial theorem .................................. 4 1.4 Point dynamics in a moving coordinate system ...................... 4 1.4.1 Apparent forces .................................... 4 1.4.2 Tensor notation .................................... 5 1.5 Dynamics of masspoint collections ............................. 5 1.5.1 The center of mass .................................. 5 1.5.2 Collisions ....................................... 6 1.6 Dynamics of rigid bodies ................................... 6 1.6.1 Moment of Inertia .................................. 6 1.6.2 Principal axes ..................................... 6 1.6.3 Time dependence ................................... 6 1.7 Variational Calculus, Hamilton and Lagrange mechanics ................. 7 1.7.1 Variational Calculus ................................. 7 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 Generating functions ................................. 8 2 Electricity & Magnetism 9 2.1 The Maxwell equations .................................... 9 2.2 Force and potential ...................................... 9 2.3 Gauge transformations .................................... 10 2.4 Energy of the electromagnetic field ............................. 10 2.5 Electromagnetic waves .................................... 10 2.5.1 Electromagnetic waves in vacuum .......................... 10 2.5.2 Electromagnetic waves in mater ........................... 11 2.6 Multipoles ........................................... 11 2.7 Electric currents ....................................... 11 2.8 Depolarizing field ....................................... 12 2.9 Mixtures of materials ..................................... 12 I II Equations in Physics by ir. 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Wevers 3 Relativity 13 3.1 Special relativity ....................................... 13 3.1.1 The Lorentz transformation ............................. 13 3.1.2 Red and blue shift .................................. 14 3.1.3 The stress-energy tensor and the field tensor ................... 14 3.2 General relativity ....................................... 14 3.2.1 Riemannian geometry, the Einstein tensor ..................... 14 3.2.2 The line element ................................... 15 3.2.3 Planetary orbits and the perihelium shift ..................... 16 3.2.4 The trajectory of a photon ............................. 17 3.2.5 Gravitational waves ................................. 17 3.2.6 Cosmology ...................................... 17 4 Oscillations 18 4.1 Harmonic oscillations ..................................... 18 4.2 Mechanic oscillations ..................................... 18 4.3 Electric oscillations ...................................... 19 4.4 Waves in long conductors .................................. 19 4.5 Coupled conductors and transformers ........................... 19 4.6 Pendulums .......................................... 19 5 Waves 20 5.1 The wave equation ...................................... 20 5.2 Solutions of the wave equation ............................... 20 5.2.1 Plane waves ...................................... 20 5.2.2 Spherical waves .................................... 21 5.2.3 Cylindrical waves ................................... 21 5.2.4 The general solution in one dimension ....................... 21 5.3 The stationary phase method ................................ 21 5.4 Green functions for the initial-value problem ....................... 22 5.5 Waveguides and resonating cavities ............................. 22 5.6 Non-linear wave equations .................................. 23 6 Optics 24 6.1 The bending of light ..................................... 24 6.2 Paraxial geometrical optics ................................. 24 6.2.1 Lenses ......................................... 24 6.2.2 Mirrors ........................................ 25 6.2.3 Principal planes ................................... 25 6.2.4 Magnification ..................................... 25 6.3 Matrix methods ........................................ 25 6.4 Aberrations .......................................... 26 6.5 Reflection and transmission ................................. 26 6.6 Polarization .......................................... 27 6.7 Prisms and dispersion .................................... 27 6.8 Diffraction ........................................... 28 6.9 Special optical effects ..................................... 28 6.10 The Fabry-Perot interferometer ............................... 29 7 Statistical physics 30 7.1 Degrees of freedom ...................................... 30 7.2 The energy distribution function .............................. 30 7.3 Pressure on a wall ...................................... 31 7.4 The equation of state ..................................... 31 7.5 Collisions between molecules ................................ 32 Equations in Physics by ir. J.C.A. Wevers III 7.6 Interaction between molecules ................................ 32 8 Thermodynamics 33 8.1 Mathematical introduction .................................. 33 8.2 Definitions ........................................... 33 8.3 Thermal heat capacity .................................... 33 8.4 The laws of thermodynamics ................................ 34 8.5 State functions and Maxwell relations ........................... 34 8.6 Processes ........................................... 35 8.7 Maximal work ......................................... 36 8.8 Phase transitions ....................................... 36 8.9 Thermodynamic potential .................................. 37 8.10 Ideal mixtures ......................................... 37 8.11 Conditions for equilibrium .................................. 37 8.12 Statistical basis for thermodynamics ............................ 38 8.13 Application on other systems ................................ 38 9 Transport phenomena 39 9.1 Mathematical introduction .................................. 39 9.2 Conservation laws ...................................... 39 9.3 Bernoulli’s equations ..................................... 41 9.4 Caracterising of flows with dimensionless numbers .................... 41 9.5 Tube flows ........................................... 42 9.6 Potential theory ....................................... 42 9.7 Boundary layers ....................................... 43 9.7.1 Flow boundary layers ................................ 43 9.7.2 Temperature boundary layers ............................ 43 9.8 Heat conductance ....................................... 44 9.9 Turbulence .......................................... 44 9.10 Self organization ....................................... 44 10 Quantum physics 45 10.1 Introduction in quantum physics ............................. 45 10.1.1 Black body radiation ................................. 45 10.1.2 The Compton effect ................................. 45 10.1.3 Electron diffraction .................................. 45 10.2 Wave functions ....................................... 45 10.3 Operators in quantum physics ............................... 45 10.4 The uncertaincy principle ................................. 46 10.5 The Schr¨odinger equation ................................. 46 10.6 Parity ............................................ 46 10.7 The tunnel effect ...................................... 47 10.8 The harmonic oscillator .................................. 47 10.9 Angular momentum .................................... 47 10.10 Spin ............................................. 48 10.11 The Dirac formalism .................................... 48 10.12 Atom physics ........................................ 49 10.12.1 Solutions ...................................... 49 10.12.2 Eigenvalue equations ................................ 49 10.12.3 Spin-orbit interaction ................................ 49 10.12.4 Selection rules .................................... 50 10.13 Interaction with electromagnetic fields .......................... 50 10.14 Perturbation theory ..................................... 50 10.14.1 Time-independent perturbation theory ...................... 50 10.14.2 Time-dependent perturbation theory ....................... 51 IV Equations in Physics by ir. J.C.A. Wevers 10.15 N-particle systems ..................................... 51 10.15.1 General ....................................... 51 10.15.2 Molecules ...................................... 52 10.16 Quantum statistics ..................................... 52 11 Plasma physics 54 11.1 Introduction .......................................... 54 11.2 Transport ........................................... 54 11.3 Elastic collisions ....................................... 55 11.3.1 General .......................................