**UFBA**

# 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

Table of Contents i List of Figures v List of Tables xxi

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)