Contents 1 Introduction Y l.1 The Field Concept........·················· 4 1.2 Maxwell's equations:Field Equations of Motion........... 4 1.3 Heaviside-Lorentz (HL)Units 5 l.4 Physical meaning of Maxwell's equations..··....·.··. 5 1.5 Charge conservation 6 1.6 Potentials and Gauge Invariance...... > 2 Electrostatics P 2.1 Point charge and the Dirac delta function 8 2.2 Interfaces between different materials...················ 9 2.3 Uniqueness of electrostatic solutions,.Green's theorem.....···.·.·· 10 2.4 Green functions..·.·················· 11 2.5 Electrostatic energy....·.···· 12 2.6 Capacitance....··········· 12 3 Electrostatic Boundary-Value problems 13 3.1 Method of Images.............. 13 3.2 Method of Separation of Variables.... 15 3.3 Angle Differential Equations......... 20 3.4 Problems with Azimuthal Symmetry 21 3.5 Green function between two concentric spheres. 23 3.6 Conductors with a Conical Singularity 25 3.7 Cylindrical Coordinates and Bessel functions.. 26 3.8 Mathematical Properties of Bessel Functions 28 3.9 Boundary-value problems in cylindrical coordinates ........ 31 3.10 Green functions in cylindrical coordinates..... 32 3.11 A little more wisdom about Green functions 33 3.12 Electrostatics in 2 Dimensions.......... 34 4 The Multipole expansion and Dielectric Materials 39 4.1 Electric Multipoles..... 39 4.2 Electrostatics in Dielectric Materials ·。 42 4.3 Energy and Forces on Dielectrics 43 4.4 Boundary value problems with dielectrics 45 4.5 Models for Xe...··········· 46 5 Magnetostatics 50 5.1 Circular Current Loop······· 51 5.2 Magnetic Multipoles·········· 52 5.3 Magnetic Fields in Magnetic Materials·..···. 53 1 ©2010 by Charles ThornContents 1 Introduction 4 1.1 The Field Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Maxwell’s equations: Field Equations of Motion. . . . . . . . . . . . . . . . . 4 1.3 Heaviside-Lorentz (HL) Units . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Physical meaning of Maxwell’s equations . . . . . . . . . . . . . . . . . . . . 5 1.5 Charge conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Potentials and Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Electrostatics 8 2.1 Point charge and the Dirac delta function . . . . . . . . . . . . . . . . . . . 8 2.2 Interfaces between different materials . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Uniqueness of electrostatic solutions, Green’s theorem . . . . . . . . . . . . . 10 2.4 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Electrostatic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Electrostatic Boundary-Value problems 13 3.1 Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Method of Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Angle Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Problems with Azimuthal Symmetry . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Green function between two concentric spheres . . . . . . . . . . . . . . . . . 23 3.6 Conductors with a Conical Singularity . . . . . . . . . . . . . . . . . . . . . 25 3.7 Cylindrical Coordinates and Bessel functions . . . . . . . . . . . . . . . . . . 26 3.8 Mathematical Properties of Bessel Functions . . . . . . . . . . . . . . . . . . 28 3.9 Boundary-value problems in cylindrical coordinates . . . . . . . . . . . . . . 31 3.10 Green functions in cylindrical coordinates . . . . . . . . . . . . . . . . . . . . 32 3.11 A little more wisdom about Green functions . . . . . . . . . . . . . . . . . . 33 3.12 Electrostatics in 2 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 The Multipole expansion and Dielectric Materials 39 4.1 Electric Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Electrostatics in Dielectric Materials . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Energy and Forces on Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Boundary value problems with dielectrics . . . . . . . . . . . . . . . . . . . . 45 4.5 Models for χe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Magnetostatics 50 5.1 Circular Current Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Magnetic Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.3 Magnetic Fields in Magnetic Materials . . . . . . . . . . . . . . . . . . . . . 53 1 c 2010 by Charles Thorn