maxwell - Applications of Maxwells Equations John F....

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Unformatted text preview: Applications of Maxwells Equations John F. Cochran and Bretislav Heinrich Simon Fraser University, Burnaby, B.C., Cnada 01 December 2004 Contents Table of Contents i List of Figures v List of Tables xxi 1 Maxwells Equations 2 1.1 Fundamental Postulates. . . . . . . . . . . . . . . . . . . . . . 2 1.2 Maxwells Equations. . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Definition of the Free Charge Density, f . . . . . . . . 10 1.2.2 Definition of the Free Current Density, ~ J f . . . . . . . . 10 1.2.3 Point Dipoles. . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.4 The Definitions of the Electric and the Magnetic Dipole Densities. . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Return to Maxwells Equations. . . . . . . . . . . . . . . . . . 22 1.3.1 The curl of any gradient function is zero. . . . . . . . . 24 1.3.2 The divergence of any curl is zero. . . . . . . . . . . . . 24 1.3.3 Gauss Theorem. . . . . . . . . . . . . . . . . . . . . . 24 1.3.4 Stokes Theorem. . . . . . . . . . . . . . . . . . . . . . 24 1.4 The Auxiliary Fields ~ D and ~ H . . . . . . . . . . . . . . . . . . 25 1.5 The Force Density and Torque Density in Matter. . . . . . . . 26 1.5.1 The Force Density in Charged and Polarized Matter. . 26 1.5.2 The Torque Densities in Polarized Matter. . . . . . . . 27 1.6 The CGS System of Units. . . . . . . . . . . . . . . . . . . . . 28 2 Electrostatic Field (I) 30 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 i CONTENTS ii 2.1.1 Dipole Moment Density as a Source for the Electric Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 The Scalar Potential Function. . . . . . . . . . . . . . . . . . . 34 2.2.1 The Particular Solution for the Potential Function given the Total Charge Distribution. . . . . . . . . . . . . . . 38 2.2.2 The Potential Function for a Point Dipole. . . . . . . . 38 2.3 General Theorems. . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.1 Application of Gauss Theorem . . . . . . . . . . . . . 40 2.3.2 Boundary Condition on ~ D . . . . . . . . . . . . . . . . . 41 2.3.3 Discontinuity in the Normal Component of the Polar-ization Vector. . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 The Tangential Components of ~ E . . . . . . . . . . . . . . . . . 44 2.5 A Conducting Body. . . . . . . . . . . . . . . . . . . . . . . . 46 2.6 Continuity of the Potential Function. . . . . . . . . . . . . . . 46 2.7 Example Problems. . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.1 Plane Symmetry. . . . . . . . . . . . . . . . . . . . . . 47 2.7.2 A Spherically Symmetric Charge Distribution. . . . . . 57 2.7.3 Cylindrical Symmetry. . . . . . . . . . . . . . . . . . . 59 2.7.4 A Uniformly Polarized Ellipsoidal Body. . . . . . . . . 62 2.8 Appendix 2A. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 Electrostatic Field (II) 69 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Soluble Problems. . . . . . . . . . . . . . . . . . . . . . . . . . ....
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