# Aplications of Maxwell's Equations

(Parte 1 de 7)

Applications of Maxwell’s Equations

John F. Cochran and Bretislav Heinrich

Simon Fraser University, Burnaby, B.C., Cnada

Contents

 1.1 Fundamental Postulates 2 1.2 Maxwell’s Equations 8 1.2.1 Deflnition of the Free Charge Density, ρf 10 1.2.2 Deflnition of the Free Current Density, ~Jf 10 1.2.3 Point Dipoles 12
 Densities 20 1.3 Return to Maxwell’s Equations 2 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

1 Maxwell’s Equations 2 1.2.4 The Deflnitions of the Electric and the Magnetic Dipole

 2.1 Introduction 30

2 Electrostatic Field (I) 30 i

CONTENTS i

 Field 32 2.2 The Scalar Potential Function 34
 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
 ization Vector 43 2.4 The Tangential Components of ~E 4 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

2.1.1 Dipole Moment Density as a Source for the Electric 2.2.1 The Particular Solution for the Potential Function given 2.3.3 Discontinuity in the Normal Component of the Polar-

 3.1 Introduction 69 3.2 Soluble Problems 74 3.2.1 (1) Orthogonal Systems 74 3.2.2 The Method of Images 92 3.2.3 Two-dimensional Problems 102 3.3 Electrostatic Field Energy 105 3.3.1 Generalized Capacitance Coe–cients 107 3.3.2 Electrostatic Forces 109 3.3.3 The Maxwell Stress Tensor 115 3.4 The Field Energy as Minimum 116 3.5 Appendix(A). The Onsager Problem 118

3 Electrostatic Field (I) 69

 4.1 Introduction 122 4.2 The Law of Biot-Savart 126
 4.3.1 A Long Straight Wire 128 4.3.2 A Long Straight Wire Revisited 130 4.3.3 A Circular Loop 132 4.3.4 The Magnetic Field along the Axis of a Solenoid 133 4.3.5 The Magnetic Field of an Inflnite Solenoid 134 4.3.6 The Field generated by a Point Magnetic Dipole 136 4.3.7 A Long Uniformly Magnetized Rod 138 4.3.8 A Uniformly Magnetized Disc 140 4.4 A Second Approach to Magnetostatics 141 4.4.1 An Inflnitely Long Uniformly Magnetized Rod 143 4.4.2 A Thin Disc Uniformly Magnetized along its Axis 144 4.4.3 A Uniformly Magnetized Ellipsoid 146 4.4.4 A Magnetic Point Dipole 147

CONTENTS i

 5.1 Introduction: Sources in a Uniform Permeable Material 150 5.2 Calculation of ofi-axis Fields 152 5.3 A Discontinuity in the Permeability 156 5.3.1 A Permeable Sphere in a Uniform Magnetic Field 157
 Magnetic Field 158 5.3.3 A Point Magnetic Dipole Near a Permeable Plane 159
 rent of I Amps 164 5.4 The Magnetostatic Field Energy 166 5.5 Inductance Coe–cients 168 5.6 Forces on Magnetic Circuits 170 5.6.1 Forces on a Magnetic Dipole 174 5.6.2 Torque on a Magnetic Dipole 177 5.6.3 Forces on a Solenoid 179 5.7 The Maxwell Stress Tensor 183

5 The Magnetostatic Field (I) 150 5.3.2 An Inflnitely Long permeable Cylinder in a Uniform 5.3.4 A Wire Parallel with an Interface and carrying a Cur-

 6.1 Introduction 185 6.2 B-H Curves 188 6.2.1 Measuring the B-H Loop 197 6.3 Digital Magnetic Recording 200

CONTENTS iv

 7.1 Introduction 206 7.2 Time Dependent Maxwell’s Equations 207 7.3 A Simple Radio Antenna 212 7.4 An Electric Dipole Radiator 217 7.5 A Point Magnetic Dipole 221 7.6 A Moving Point Charge in Vacuum 2

7 Time Dependent Electromagnetic Fields. 206

 8.1 Introduction 229 8.2 Poynting’s Theorem 229 8.3 Power Radiated by a Simple Antenna 234 8.4 A Non-Sinusoidal Time Dependence 238 8.5 Scattering from a Stationary Atom 240

8 E.M. Fields and Energy Flow. 229

 9.1 Introduction 246 9.2 Phasors 251 9.3 Elliptically Polarized Plane Waves 254 9.4 Gaussian Light Beams 259 9.4.1 The Fourier Transform of a Gaussian 263
 eqn.(9.26) 264

9 Plane Waves (I). 246 9.4.2 Integrals that are Required in the Fourier Transform,

 10.1 Normal Incidence 267 10.2 Boundary Conditions 273 10.2.1 The Tangential Components of the Electric Field 273 10.2.2 The Tangential Components of the Magnetic Field 275 10.2.3 The Normal Component of the Field B 275 10.3 Application of the Boundary Conditions to a Plane Interface 276 10.4 Re°ection from a Metal at Radio Frequencies 279 10.5 Oblique Incidence 284 10.5.1 S-polarized Waves 285 10.5.2 P-polarized Waves 289 10.5.3 Oblique Incidence on a Lossless Material 292 10.6 Example: Copper 292
 10.8 Metals at Radio Frequencies 301 10.8.1 S-polarization 302 10.8.2 P-polarization 304

CONTENTS v

 1.1 Introduction 307 1.2 Strip-lines 309 1.3 Co-axial Cables 315 1.4 Transmission Lines in General 318 1.5 A Terminated Line 322 1.6 Sinusoidal Signals on a Terminated Line 330 1.6.1 Case(1). A Shorted Cable 3 1.6.2 Case(2). An Open-ended Cable 3
 tic Impedance 334 1.6.4 Case(4). A Purely Inductive Load 334 1.6.5 Case(5). A Purely Capacitive Load 334 1.6.6 Summary 335 1.7 The Slotted Line 336 1.8 Transmission Line with Losses 341

1 Transmission Lines. 307 1.6.3 Case(3). The Cable is Terminated by the Characteris-

 12.1 Simple Transverse Electric Modes 346 12.2 Higher Order Modes 356 12.2.1 TM Modes 359 12.2.2 TE Modes 362 12.3 Waveguide Discontinuities 365 12.4 Energy Losses in the Waveguide Walls 372 12.5 Circular Waveguides 375 12.5.1 TM Modes 375

List of Figures

 tor fleld 3

1.1 A cartesian co-ordinate system used to specify an electric vec-

 ~R = ~rP − ~rq 5

1.2 The flelds generated by a point charge of q Coulombs at ~rq that is stationary with respect to the observer located at ~rP. 1.3 The electric and magnetic flelds generated at the point of ob- servation P at the time t depend upon the position, the velocity, and the acceleration of the charged particle at the retarded

 ) 8
 ¢A per unit time 1

time tR = (t ¡ Rc 1.4 The x-component of the current density caused by a moving charge distribution. The charge labelled Q is representative of all charges passing through the x-oriented element of area, 1.5 Upper flgure: Sketch of a hydrogen atom in zero applied electric fleld. The nuclear charge is Q = 1:6 £ 10−19 Coulombs. The time-averaged electron charge, -Q, is distributed in a spherically symmetric cloud around the nucleus having a radius of approximately 5 £ 10¡11 m. Lower flgure: Sketch of a hydrogen atom subjected to a uniform electric fleld E0 . The displacement of the centroid of the electron charge density rel- ative to the nucleus has been exaggerated for the sake of clarity. 13 1.6 Two charges equal in magnitude but opposite in sign are separated by the vector distance ~d. By deflnition the dipole moment of this pair of charges is ~p = q~d where ~d is the vector

 uz are unit vectors directed along the x,y, and z axes 14

directed from the negative to the positive charge. ux, uy, and vi

LIST OF FIGURES vii

 cally symmetric around the dipole as axis 17
 axis and expressed as spherical polar components 18
 generates a magnetic dipole moment |~mj = qav/2 Amp-m2 19 1.10 The three commonly used co-ordinate systems 23
 normal to the surface at the element of area dS 24
 a unit vector normal to the element of surface area, dS 25

1.7 The electric fleld intensity at various positions around an electric point dipole, ~P. The electric fleld distribution is cylindri- 1.8 The electric fleld generated by a dipole oriented along the z- 1.9 A particle carrying a charge of q Coulombs and following a circular orbit of radius a meters with the speed v meters/sec 1.1 An application of Gauss’ Theorem to a volume V bounded by a closed surface S. ~A is a vector fleld, and n is a unit vector 1.12 An application of Stokes’ Theorem to a surface bounded by a closed curve C. ~dL is an element of length along curve C. n is

 charge or as a point dipole moment 31

2.1 Given a distribution of sources the electric fleld at the position of the observer, (X,Y,Z), can be calculated as the sum of the electric flelds generated by dividing the source distribution into small volume elements dV=dxdydz and treating the charges or dipole moments in each volume element as a point

2.2 A sphere of radius R0 fllled with a uniform charge density ‰0. The electric fleld at the center of the sphere is zero because the

 The two charges are separated by the distance d 39
 face of discontinuity between two materials 42
 the application of Stokes’ Theorem 45

fleld generated by element number 1 at (x,y,z) is cancelled by the fleld equal in magnitude but opposite in direction generated by the equal volume element number 2 at (-x,-y,-z). The net fleld generated by all such symmetry related pairs is zero. 3 2.3 Model for calculating the potential function for a point dipole. 2.4 Application of Gauss’ Theorem to a pill-box straddling a sur- 2.5 A rectangular loop having sides dL long and dw wide used for 2.6 Calculation of the electric fleld generated by a uniformly charged plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

LIST OF FIGURES viii

2.7 The potential function (top) and the electric fleld (bottom) generated by a uniformly charged inflnite plane carrying a charge density of σ Coulombs/m2. Ez = −@Vp @z

 : 50

2.8 Geometry for the application of Gauss’ Theorem to calculate the electric fleld strength generated by an inflnite, plane, uniformly charged sheet whose density is ¾ Coulombs=m2. The

 magnitude of the resulting fleld is E0 = ¾=(2†0) 51

2.9 An electric double layer consists of two inflnite plane sheets of charge densities +¾ and ¡¾ Coulombs/m2 separated by a small distance. The electric fleld is zero everywhere outside

 charged sheets 53

the double layer, but is equal to jEzj = ¾=†0 between the two 2.10 The electric fleld distribution and the corresponding potential function generated by an electrical double layer. The electric fleld intensity inside the double layer is E0 = ¡¾=†0. The jump in the potential across the double layer is ¢V = 2¾d=†0. 54

2.1 A uniformly polarized slab. The polarization density,~P0, is directed along the normal to the slab. The discontinuities in the normal component of the polarization produce efiective

 surface bound charge densities given by ¾b = ¡div(~P) 5

2.12 An inflnite slab that is uniformly polarized in plane, Px = P0. 57 2.13 The electric fleld intensity generated by a spherically symmet- ric charge distribution. The electric fleld has only a radial component because the transverse components generated by

 two equivalent charges, dq1 and dq2, cancel by symmetry 58

2.14 The electric fleld generated by a uniformly charged line lying along the z-axis. The line charge density is ‰L Coulombs/meter.

 Volts/m 59

The electric fleld is radial and has the value Er = (‰L=2…†0) (1=r) 2.15 An inflnite line of uniformly distributed point dipoles can be modelled by a uniformly charged positive line separated by a small distance d from a uniformly charged negative line. Let the charge density on the positive line be ‰L, and let the charge density on the negative line be ¡‰L, then the dipole density,

LIST OF FIGURES ix

 the body 62
 ellipse is non-uniform 63 2.18 Oblate ellipsoid of revolution 65 2.19 Cigar shaped ellipsoid of revolution 65

2.16 A uniformly polarized body having an irregular shape. The resulting surface bound charge distribution produces an electric fleld distribution that is non-uniform both inside and outside 2.17 A uniformly polarized ellipsoidal body; the polarization lies along a principle axis of the ellipse. The resulting surface bound charge distribution produces a uniform electric fleld inside the ellipse. The electric fleld distribution outside the

3.1 A charged conductor. The free charge on the conductor is given by Q = ∫R

 ~D · ~dS 72

3.2 Two plane parallel, semi-inflnite metal electrodes separated by a distance D. The electrode potentials are V1 and V2. The space between the electrodes is fllled with a material having a

 dielectric constant † 75
 of position 7

3.3 The parallel plate capacitor problem with two difierent dielectric materials. The electric fleld in each region is independent

3.4 Charge decay through a leaky capacitor. † = †r†0 is the dielectric constant for the spacer material. ¾ is the conductivity

 of the spacer material 79

3.5 A uniform cylinder, inflnitely long in the z-direction, and im-

 medium whose dielectric constant is †2 83
 presence of a uniform electric fleld Ez = E0 8

mersed in a uniform electric fleld Ex = E0. The cylinder is characterized by a dielectric constant †1. It is situated in a 3.6 An uncharged dielectric sphere, dielectric constant †1, situated in a medium characterized by a dielectric constant, †2, in the 3.7 Top flgure: a point charge located a distance d in front of an inflnite conducting metal plane. Bottom flgure: The system of charges whose electrostatic potential satisfles r2V = 0 as well as the boundary conditions for the problem posed in the top flgure. This solution is only valid in the vacuum region: in the metal V = 0. . . . . . . . . . . . . . . . . . . . . . . . . 93

LIST OF FIGURES x

 boundary conditions 96
 but of opposite sign 97
 as shown in the flgure 100

3.8 (a) The real problem: a charge q located a distance d from the interface between two uncharged dielectric media. (b) The conflguration of image charges that produce a potential that satisfles ∇2V = 0 and that can be used to satisfy the required 3.9 Equipotential surfaces for two line charges of equal strength 3.10 The real problem of a point charge located a distance d from the center of a metal sphere of radius R is shown in (a). In part (b) the real problem has been replaced by two point charges 3.1 Electrodes in the form of cylindrical hyperboloids. One pair of electrodes is held at a potential U=-1 Volt; the other pair of electrodes is held at a potential U=+1 Volt. The potential function in the space between the electrodes is given by U = x2 ¡ y2. The electric fleld components in the space between

 the electrodes are given by Ex = ¡2x and Ey = +2y 103

3.12 Electrodes of silver paint drawn on a sheet of conducting pa-

 out by means of a voltmeter connected to a pointed probe 105

per. The resistivity of the paper is much larger than that of the silver paint electrodes.The equipotential lines can be mapped 3.13 Charged conductors embedded in a linear dielectric medium.

 in the text 108

The charges are a linear function of the potentials, see eqn.(3.61) 3.14 A parallel plate capacitor. The two plates have an area A and are separated by a distance z. The charge density on each

(Parte 1 de 7)