Electromagnetism Course Notes (PHYS09060) 2017/18 Andreas Hermann School of Physics and Astronomy The University of Edinburgh September 2017 Contents 1 Whistlestop tour of Vector Calculus 6 1.1 Gradient . 6 1.2 Divergence and Gauss' Theorem . 7 1.3 Curl and Stokes' Theorem . 7 1.4 Laplacian . 8 1.5 Useful Identities . 8 1.6 3D Taylor expansion . 9 1.7 Important Theorem . 9 2 Revision of Electrostatics 10 2.1 Charge Density . 10 2.2 Point charges and the δ-distribution . 10 2.3 Coulomb's Law . 11 2.4 Electric Field . 12 2.5 Gauss' law for E ...................................... 13 2.6 Electrostatic Potential . 14 3 Gauss' Law 16 3.1 Conductors and Insulators . 16 3.2 Gauss' Law in differential form . 16 3.3 Using Gauss' Law . 17 1 3.4 Important δ-function identity and point charges (see tutorial 1.3) . 19 4 Poisson's Equation and Images 20 4.1 Poisson's Equation . 20 4.2 Properties of Poisson's Equation . 20 4.3 A simple example: a hollow conductor . 22 4.4 The Method of Images . 22 5 Electric Dipoles and Multipoles 24 5.1 Field of an electric dipole . 24 5.2 Dipole interaction with external Electric Field . 25 5.3 Multipole expansions . 26 6 Electrostatic Energy and Capacitors 28 6.1 Electrostatic Energy of a general charge distribution . 28 6.2 Capacitors . 29 6.3 *Finite size disc capacitors . 31 7 Magnetic force, Currents and Biot Savart Law 32 7.1 Magnetic force . 32 7.2 Current density and current elements . 32 7.3 Biot Savart Law and Magnetic Fields . 34 7.4 Magnetic Force between Currents . 34 7.5 Long straight wires . 35 8 Divergence and Curl of B; Gauss' and Amp`ere'slaws 36 8.1 div B and Gauss' Law . 36 8.2 Magnetic Dipoles . 36 8.3 curl B and Amp`ere'sLaw . 38 9 Applications of Ampere's` Law; Magnetic Vector Potential 40 9.1 Applications of Amp`ere'sLaw . 40 9.2 The Magnetic Vector Potential . 41 9.3 Poisson's equation for the vector potential . 42 9.4 Summary of 'statics . 43 10 Electromotance and Faraday's Law 44 2 10.1 Sources of steady currents . 44 10.2 Induced emf . 44 10.3 Faraday's Law . 45 10.4 Connection to the Magnetic Vector Potential . 46 10.5 Lenz's Law . 47 11 Inductance 48 11.1 Examples of Induction . 48 11.2 Mutual Inductance . 49 11.3 Self-Inductance . 50 11.4 Energy Stored in Inductors . 50 12 The Displacement Current 52 12.1 Continuity equation . 52 12.2 The Displacement Current . 52 12.3 Capacitor Paradox and Resolution . 53 12.4 Maxwell's Equations . 54 12.5 Solution of Maxwell's Equations in Free Space . 55 13 Description of Electromagnetic Waves 56 13.1 Recap of wave equations . 56 13.2 Plane Waves . 57 13.3 Electromagnetic Plane Waves . 57 13.4 Linear (Plane) Polarisation . 59 13.5 Circular Polarisation . 59 14 Electromagnetic Energy and the Poynting Vector 60 14.1 Poynting's Theorem (Griffiths 8.1.2)........................... 60 14.2 Energy of Electromagnetic Waves . 61 14.3 Energy of discharging capacitor . 62 14.4 ∗Momentum of electromagnetic radiation . 63 15 Dielectric Materials 64 15.1 Overview . 64 15.2 Dielectric Materials . 64 15.3 Electric displacement vector and Gauss' law in media . 65 3 15.4 Linear Isotropic Homogeneous Media . 66 15.5 Example: Dielectric in a Capacitor . 67 16 Magnetic Media 68 16.1 Types of magnetism . 68 16.2 The Magnetization Vector . 68 16.3 Ampere's Law in media . 70 17 Summary of EM in media; boundary conditions on fields 72 17.1 Effect of Magnetic Materials on Inductance . 72 17.2 Electromagnetism in media: summary . 72 17.3 Energy densities and Poynting Vector . 73 17.4 Boundary conditions of fields . 74 18 Examples of continuity conditions; waves in media 76 18.1 Continuity conditions: examples . 76 18.2 Waves in media . 78 18.3 Waves in conductors . 79 19 Waves in Conductors 80 19.1 Skin Depth . 80 19.2 Good and poor conductors . 81 19.3 Phase relations of fields . 82 19.4 Intrinsic Impedance . 83 20 Waves at interfaces: normal incidence 84 20.1 Summary on plane waves and interfaces . 84 20.2 Interfaces between two dielectric media . 84 20.3 General waves at interfaces: normal incidence . 86 20.4 Reflection at Conducting Surface . 87 A Supplementary Material on Types of Magnetism 88 A.1 Diamagnetism . 88 A.2 Paramagnetism . 88 A.3 Ferromagnetism . 89 A.4 Effect of Magnetic Materials on Inductance . 89 4 B Reflection at boundaries: oblique incidence 91 B.1 General Angle of Incidence . 91 B.2 Brewster's Angle . 93 B.3 Total Internal Reflection . ..