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Electrodynamics

by Abhishek Apoorv
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Contents I Mathematical introduction I. Vectors and vector calculus 3 A. Summation convention 3 B. Derivatives 3 II. Multidimensional integration 5 A. Change of variables 5 B. Divergence theorem (Gauss) 5 C. Stokes theorem 6 III. Dirac delta 6 A. Basic definitons 6 B. Multidimensional 7 IV. Special functions 8 A. General 8 1. Oscillatory behavior 9 2. Singular points and classification of the solutions 9 3. Asymptotics, and the WKB approximation 9 B. Specific functions 10 C. Bessel functions 11 D. Asymptotes 11 1. Small x 12 E. Integer vs. non-integer n, generating function, recurrence relations and integral representation 12 1. Roots of Jν 13 2. Orthogonality 13 3. Relation to Laplacian 14 4. Modified Bessel functions 14 F. Legendre 15 0
II 1. Generating function for Legendre polynomials (LP) 15 2. Explicit expressions and recurrence relations 16 3. Relation to Laplacian 16 4. The other solution 17 Overview of fields V. Field lines; sources and curles; charge A. Field lines 19 1. Geometric meaning of the Gauss theorem 19 B. Intensity of sources and curles 19 C. Force on a charge 20 VI. Maxwell equations in free space ~ and H ~ VII. Vectors D 20 21 ~ A. Electric displacement, D ~ near the surface of a dielectric B. refraction of D ~ and E ~ C. Comparing D 21 22 23 ~ and E ~ inside a dielectric 1. ”Measuring” D ~ D. H 23 E. Maxwell equations in media 24 ~ D, ~ B, ~ H ~ F. Summary: sources and curles of E, 25 G. Beyond continuos medium 25 VIII. The Wave III 19 23 25 Electrostatics IX. Free charges 26 A. Field 26 B. Potential 27 1. Surface charge and dipole layer 1 27
C. Poisson and Laplace equations 28 D. Green’s theorem 28 E. Uniqueness of the solution 29 F. Formal properties of the Green’s function 29 G. Energy 29 X. Multipole expansion 30 A. Notations 30 B. Expansion 30 C. Comments 31 D. Dipole 32 E. Multipole expansions for fields with asimuthal symmetry 33 XI. Conductors 34 A. Elementary properties of a conductor in electrostatics 34 B. Charge near a conducting sphere 35 1. Method of images 35 2. Surface chage density 35 3. Non-zero potential or charge on the sphere 36 4. Inversion properties of Laplace equation 36 5. The Green’s function 36 ~0 C. Conducting sphere in uniform field E 37 1. Method of images 37 2. With Legendre polynomials 37 XII. Laplace equation in a semi-infinite stripe 39 A. Separation of variables 39 B. Edge 41 C. Corner 41 D. Relation to complex variables - 2D only! 42 1. Cauchy-Riemann conditions 42 2. Complex potential 42 3. Conformal mapping and solution of the problem 43 2
E. Mixed boundary conditions - thin disc 1. Charge density 46 2. Capacitance 46 3. The z-expansion 47 XIII. Dielectrics 47 A. Polarization: Elementary treatment 1. Linear dielectric IV 45 47 48 B. Charge near a flat dielectric 48 C. Dielectric sphere 49 1. a ”physicists solution” 49 2. Solution with Legendre polynomials 50 Magnetostatics XIV. Introduction 52 A. Math 52 1. Vector operations 53 2. Elliptic integrals 53 B. Motion of charges and constant currents 53 C. Macroscopic ME 53 1. Amper’s law 53 XV. Vector potential 53 A. Biot-Savarat law 53 B. Force between wire loops 53 C. Vector potential for a ring current 54 D. Expansion at large distances 54 E. Magnetic moment and gyromagnetic ratio 55 F. Torque and Larmor precession ~ G. Magnetization M 55 55 1. Scalar magnetic potential ΦM 3 55

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