Electromagnetism - Theory and Applications

1015 Pages·2016·7.45 MB·English

by  Ashutosh Pramanik

Second Edition
ELECTROMAGNETISM
Theory and Applications
y
z const
const
r1
const
const
w1
o x
r
a a
w
z
const
Ashutosh Pramanik
(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:2)(cid:5)(cid:12)(cid:13)(cid:8)
(cid:5)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18) (cid:19)(cid:20)(cid:21) (cid:9)(cid:22)(cid:22)(cid:23)(cid:24)(cid:25)(cid:19)(cid:26)(cid:24)(cid:16)(cid:20)(cid:27)
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:3)(cid:7)(cid:9)(cid:10)(cid:9)(cid:5)(cid:6)
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:4)(cid:8)(cid:9)(cid:10)(cid:2)(cid:11)(cid:2)(cid:12)(cid:13)(cid:14)
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7)(cid:8)(cid:9)(cid:3)(cid:10)(cid:9)(cid:11)(cid:3)(cid:4)(cid:12)(cid:3)(cid:4)(cid:13)(cid:14)(cid:6)(cid:9)(cid:15)(cid:6)(cid:16)(cid:6)(cid:13)(cid:4)(cid:17)(cid:18)(cid:9)(cid:13)(cid:19)(cid:20)(cid:9)(cid:21)(cid:6)(cid:22)(cid:6)(cid:7)(cid:3)(cid:12)(cid:5)(cid:6)(cid:19)(cid:14)(cid:9)(cid:21)(cid:23)(cid:22)(cid:23)(cid:16)(cid:23)(cid:3)(cid:19)(cid:24)(cid:9)(cid:25)(cid:26)(cid:27)(cid:28)
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:9)(cid:13)(cid:4)(cid:5)(cid:4)(cid:8)(cid:14)(cid:15)(cid:9)(cid:16)(cid:14)(cid:5)(cid:14)(cid:17)(cid:4)(cid:8)(cid:18)
(cid:2)(cid:3)(cid:4)
(cid:29)(cid:18)(cid:6)(cid:9)(cid:30)(cid:19)(cid:23)(cid:22)(cid:6)(cid:4)(cid:16)(cid:23)(cid:14)(cid:23)(cid:6)(cid:16)(cid:9)(cid:3)(cid:10)(cid:9)(cid:25)(cid:23)(cid:4)(cid:5)(cid:23)(cid:19)(cid:31)(cid:18)(cid:13)(cid:5)(cid:9)(cid:13)(cid:19)(cid:20)(cid:9)(cid:28)(cid:6)(cid:6)(cid:20)(cid:16)
(cid:2)(cid:19)(cid:4)(cid:20)(cid:4)(cid:14)(cid:8)(cid:21)(cid:22)(cid:9)(cid:23)(cid:5)(cid:17)(cid:6)(cid:5)(cid:4)(cid:4)(cid:8)(cid:9)(cid:14)(cid:5)(cid:24)(cid:9)(cid:25)(cid:4)(cid:21)(cid:26)(cid:27)(cid:8)(cid:4)(cid:8)(cid:18)
(cid:2)(cid:3)(cid:4)
(cid:21) ! (cid:9)"(cid:13)(cid:19)(cid:20)(cid:18)(cid:23)(cid:9)(cid:21)(cid:23)(cid:16)(cid:14)(cid:23)(cid:19)(cid:31)#(cid:23)(cid:16)(cid:18)(cid:6)(cid:20)(cid:9)$(cid:23)(cid:16)(cid:23)(cid:14)(cid:23)(cid:19)(cid:31)(cid:9)%(cid:4)(cid:3)(cid:10)(cid:6)(cid:16)(cid:16)(cid:3)(cid:4)(cid:24)(cid:9)&&(cid:29)(cid:9)(cid:25)(cid:3)(cid:5)’(cid:13)(cid:8)
PHI Learning Private Limited
New Delhi-110001
2009
(cid:10)(cid:11)(cid:10)(cid:12)(cid:13)(cid:2)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:10)(cid:13)(cid:19)(cid:20)(cid:15)(cid:21)(cid:5)(cid:13)(cid:22)(cid:10)(cid:14)(cid:2)(cid:23)(cid:5)(cid:16)(cid:18)(cid:24)(cid:5)(cid:16)(cid:25)(cid:25)(cid:11)(cid:19)(cid:12)(cid:16)(cid:13)(cid:19)(cid:14)(cid:18)(cid:20)(cid:26)(cid:5)(cid:27)(cid:28)(cid:29)(cid:5)(cid:10)(cid:29)(cid:4)
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:4)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:11)(cid:13)(cid:14)(cid:15)
(cid:16)(cid:8)(cid:17)(cid:18)(cid:18)(cid:19)(cid:8)(cid:20)(cid:21)(cid:8)(cid:9)(cid:22)(cid:23)(cid:8)(cid:24)(cid:25)(cid:11)(cid:10)(cid:13)(cid:14)(cid:13)(cid:26)(cid:8)(cid:9)(cid:10)(cid:14)(cid:27)(cid:11)(cid:6)(cid:25)(cid:8)(cid:24)(cid:14)(cid:12)(cid:14)(cid:6)(cid:25)(cid:28)(cid:29)(cid:8)(cid:30)(cid:25)(cid:31)(cid:8) (cid:25)!(cid:4)(cid:14)"(cid:8)(cid:2)!!(cid:8)(cid:10)(cid:14)(cid:26)(cid:4)(cid:6)(cid:3)(cid:8)(cid:10)(cid:25)(cid:3)(cid:25)(cid:10)(cid:27)(cid:25)(cid:28)"(cid:8)(cid:30)(cid:7)(cid:8)#(cid:11)(cid:10)(cid:6)(cid:8)(cid:7)$(cid:8)(cid:6)(cid:4)(cid:14)(cid:3)(cid:8)(cid:20)(cid:7)(cid:7)(cid:15)
(cid:12)(cid:11)(cid:21)(cid:8)(cid:20)(cid:25)(cid:8)(cid:10)(cid:25)#(cid:10)(cid:7)(cid:28)(cid:5)%(cid:25)(cid:28)(cid:8)(cid:14)(cid:13)(cid:8)(cid:11)(cid:13)(cid:21)(cid:8)$(cid:7)(cid:10)(cid:12)(cid:29)(cid:8)(cid:20)(cid:21)(cid:8)(cid:12)(cid:14)(cid:12)(cid:25)(cid:7)(cid:26)(cid:10)(cid:11)#(cid:4)(cid:8)(cid:7)(cid:10)(cid:8)(cid:11)(cid:13)(cid:21)(cid:8)(cid:7)(cid:6)(cid:4)(cid:25)(cid:10)(cid:8)(cid:12)(cid:25)(cid:11)(cid:13)(cid:3)(cid:29)(cid:8)(cid:31)(cid:14)(cid:6)(cid:4)(cid:7)(cid:5)(cid:6)(cid:8)#(cid:25)(cid:10)(cid:12)(cid:14)(cid:3)(cid:3)(cid:14)(cid:7)(cid:13)(cid:8)(cid:14)(cid:13)
(cid:31)(cid:10)(cid:14)(cid:6)(cid:14)(cid:13)(cid:26)(cid:8)$(cid:10)(cid:7)(cid:12)(cid:8)(cid:6)(cid:4)(cid:25)(cid:8)#(cid:5)(cid:20)!(cid:14)(cid:3)(cid:4)(cid:25)(cid:10)"
(cid:19)(cid:20)(cid:30)(cid:18)(cid:31)(cid:7) !(cid:31)!"(cid:31)(cid:27)(cid:9)#(cid:31)#(cid:6)$(cid:8)(cid:31)$
&(cid:4)(cid:25)(cid:8)(cid:25)’#(cid:7)(cid:10)(cid:6)(cid:8)(cid:10)(cid:14)(cid:26)(cid:4)(cid:6)(cid:3)(cid:8)(cid:7)$(cid:8)(cid:6)(cid:4)(cid:14)(cid:3)(cid:8)(cid:20)(cid:7)(cid:7)(cid:15)(cid:8)(cid:11)(cid:10)(cid:25)(cid:8)(cid:27)(cid:25)(cid:3)(cid:6)(cid:25)(cid:28)(cid:8)(cid:3)(cid:7)!(cid:25)!(cid:21)(cid:8)(cid:31)(cid:14)(cid:6)(cid:4)(cid:8)(cid:6)(cid:4)(cid:25)(cid:8)#(cid:5)(cid:20)!(cid:14)(cid:3)(cid:4)(cid:25)(cid:10)"
%&’()(cid:5)(cid:25)*&(cid:28)(&(cid:28)+(cid:5),(cid:20)-./(cid:28)(cid:29)(cid:5)(cid:10)(cid:29)&(&/(cid:28)0 (cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:4)(cid:4)(cid:4)(cid:5) (cid:5)(cid:5)(cid:5)(cid:5)(cid:4)(cid:4)(cid:4)(cid:5)(cid:5)(cid:5) (cid:5)(cid:5)(cid:5)(cid:5)(cid:4)(cid:4)(cid:4)(cid:5)(cid:5)(cid:5) (cid:5)(cid:5)12(cid:28)-(cid:26)(cid:5)(cid:27)(cid:9)(cid:9)(cid:7)
(cid:9)(cid:5)(cid:20)!(cid:14)(cid:3)(cid:4)(cid:25)(cid:28)(cid:8)(cid:20)(cid:21)(cid:8)(cid:2)(cid:3)(cid:7)(cid:15)(cid:25)(cid:8)("(cid:8))(cid:4)(cid:7)(cid:3)(cid:4)(cid:29)(cid:8)(cid:9)(cid:22)(cid:23)(cid:8)(cid:24)(cid:25)(cid:11)(cid:10)(cid:13)(cid:14)(cid:13)(cid:26)(cid:8)(cid:9)(cid:10)(cid:14)(cid:27)(cid:11)(cid:6)(cid:25)(cid:8)(cid:24)(cid:14)(cid:12)(cid:14)(cid:6)(cid:25)(cid:28)(cid:29)(cid:8)*+,-(cid:29)(cid:8).(cid:7)(cid:13)(cid:13)(cid:11)(cid:5)(cid:26)(cid:4)(cid:6)(cid:8).(cid:14)(cid:10)%(cid:5)(cid:3)(cid:29)
(cid:30)(cid:25)(cid:31)(cid:8) (cid:25)!(cid:4)(cid:14)+//(cid:18)(cid:18)(cid:18)/(cid:8)(cid:11)(cid:13)(cid:28)(cid:8)(cid:9)(cid:10)(cid:14)(cid:13)(cid:6)(cid:25)(cid:28)(cid:8)(cid:20)(cid:21)(cid:8)0(cid:11)(cid:21)(cid:8)(cid:9)(cid:10)(cid:14)(cid:13)(cid:6)(cid:8)(cid:9)(cid:11)%(cid:15)(cid:8)(cid:9)(cid:10)(cid:14)(cid:27)(cid:11)(cid:6)(cid:25)(cid:8)(cid:24)(cid:14)(cid:12)(cid:14)(cid:6)(cid:25)(cid:28)(cid:29)(cid:8)(cid:30)(cid:25)(cid:31)(cid:8) (cid:25)!(cid:4)(cid:14)+//(cid:18)(cid:18)/1"
To the revered memory
of
my parents
whose encouragement and support for my professional career
made this book possible
Contents
Preface xxiii
Preface to the First Edition xxv
0 Vector Analysis 1–41
0.1 Introduction 1
0.2 Vectors and Vector Algebra 1
0.2.1 Addition and Subtraction of Vectors 2
0.2.2 Components of a Vector 3
0.2.3 Multiplication of Vectors 3
0.2.3.1 The scalar product of vectors 3
0.2.3.2 The vector product of vectors 4
0.2.3.3 Triple products 6
0.3 Three Orthogonal Coordinate Systems 7
0.3.1 Rectangular Cartesian Coordinates (x, y, z) 7
0.3.2 Cylindrical Polar Coordinate System (r, f, z) 8
0.3.3 Spherical Polar Coordinate System (r, q, f) 8
0.4 Vector Calculus 9
0.4.1 Differentiation of Vectors 9
0.4.1.1 Partial differentiation 10
0.4.2 Integration—Line, Surface, and Volume Integrals 11
0.5 The Vector Operator — and Its Uses 13
0.5.1 The Gradient of a Scalar 13
0.5.1.1 Directional derivative 15
0.5.2 The Divergence of a Vector 15
0.5.3 The Curl of a Vector 17
0.6 Some Integral Theorems of Vectors 20
0.6.1 Gauss’ Theorem 20
0.6.2 Stokes’ Theorem 20
0.6.3 Green’s Theorem 20
0.6.4 Vector Analogue of Green’s Theorem 21
0.7 Applications of the Operator ‘Del’ (= —) 22
0.7.1 The Operator Div Grad 22
0.7.2 Divergence of a Vector Product 22
0.7.3 Divergence and Curl of SA 23
0.7.4 The Operator Curl Grad 24
0.7.5 The Operator —2 with Vector Operand 25
0.7.6 The Operator Grad Div 25
0.7.7 The Operator Div Curl 25
v
vi CONTENTS
0.7.8 The Operator Curl Curl 26
0.7.9 Gradient of a Scalar Product 26
0.7.10 Curl of a Vector Product 28
0.8 Types of Vector Fields 29
0.8.1 Solenoidal and Irrotational Field (Lamellar) 29
0.8.2 Irrotational but not Solenoidal Field 29
0.8.3 Solenoidal but not Irrotational Field 30
0.8.4 Neither Irrotational Nor Solenoidal Field 30
0.8.4.1 A proof of the Helmholtz theorem 30
0.9 Time Variation of Vectors 31
0.9.1 Complex Representation of Time-harmonic Vectors 32
0.9.2 Complex Representation of Rotating Vectors 33
0.9.3 Magnitudes of Vectors in Complex Representation 34
0.9.4 Complex Representation of a Vector Rotating in Cartesian Plane 35
dA ∂A
0.9.5 The Relationship between and 36
dt ∂t
0.9.6 Conversion of a Vector from One Coordinate System to Another 39
0.9.6.1 Conversion between rectangular Cartesian coordinate system and
spherical polar coordinate system 39
0.9.6.2 Conversion between rectangular Cartesian coordinate system and
cylindrical polar coordinate system 39
0.9.6.3 Conversion between cylindrical polar coordinate system and
spherical polar coordinate system 40
0.9.6.4 General comments 41
Problems 41
1 The Electrostatic Field in Free Space (in Absence of Dielectrics) 42–65
1.1 Introduction 42
1.2 The Law of Force between Charged Particles (Coulomb’s Law) 43
1.3 The Principle of Superposition 44
1.4 The Electric Force (Per Unit Charge) and the Concept of Electric Field 45
1.5 The Electric Field of Continuous Space Distribution of Charges (Gauss’ Theorem) 46
1.5.1 The Flux of E Across a Surface 48
1.5.2 Gauss’ Theorem 48
1.5.3 An Alternative Proof of Gauss’ Theorem in Differential Form 50
1.6 Electric Potential (or Electrostatic Potential) 51
1.6.1 Electric Field and Electric Potential 54
1.6.2 Potential Function and Flux Function 55
1.6.3 Potential Field Expressed as Poisson and Laplace Equations 56
1.7 Some Useful Examples of Calculation of Fields by Gauss’ Theorem and Potentials 57
1.7.1 A Group of Charged Particles 57
1.7.2 A Hollow Charged Sphere of Radius a, and Carrying a Charge Q 58
1.7.3 Uniformly Distributed Charge on an Infinite Circular Cylinder 60
1.7.4 Infinitely Long Straight Line Charge 61
1.7.5 A Group of Parallel Line Charges 61
1.7.6 Charges Distributed Uniformly Over an Infinitely Plane Surface 62
1.7.7 Electric Dipole 63
Problems 64
CONTENTS vii
2 Conductors and Insulators in Electrostatic Field 66–96
2.1 Conductors and Insulators 66
2.2 Conductors in the Electrostatic Field 66
2.3 Relation between Electrostatic Potential and Charges on Conducting Bodies 68
2.3.1 The Case of an Isolated Conductor 68
2.3.2 The Case of Two Bodies with Equal Charges of Opposite Signs 68
2.3.3 Methods of Evaluating Capacitance (C) 69
2.4 The Behaviour of Insulators (or Dielectrics) in a Static Electric Field 69
2.4.1 The Potential and Electric Field due to an Aggregate of Dipoles (Polarization
Vector) 71
2.4.2 Charge Distribution Equivalent to a Polarized Dielectric 72
2.5 Generalized Form of Gauss’ Theorem 74
2.6 Some Physical Properties of Dielectrics 75
2.6.1 Dielectric Strength 75
2.6.2 Dielectric Relaxation 76
2.6.3 Triboelectricity 76
2.7 Capacitance: Capacitors 77
2.8 Calculation of Capacitance 78
2.8.1 Parallel Plate Capacitor 78
2.8.1.1 Two-plate tapered capacitor 79
2.8.2 Concentric Cylinders 80
2.8.3 Parallel Circular Cylinders 81
2.8.4 Wire and Parallel Plane 82
2.8.5 An Introductory Note on the Method of Images 83
2.8.6 Capacitance between Two Spheres of Equal Diameter 83
2.8.7 Capacitance between a Sphere and a Conducting Plane 84
2.8.8 Capacitances in Parallel and in Series 84
2.9 Field in a Region Containing Two Dielectric Materials: Boundary Conditions in
Electrostatics 85
2.10 Capacitors of Mixed Dielectrics and of Complex Spheres 88
2.10.1 Parallel Plate Capacitor with Mixed Dielectrics 88
2.10.2 Concentric Cylinders with Mixed Dielectrics 89
2.10.3 Concentric Spheres with Single Dielectric 90
2.10.4 Concentric Spheres with Mixed Dielectrics 92
2.10.5 Capacitance of N Conductors 94
Problems 96
3 Energy and Mechanical Forces in Electrostatic Fields 97–110
3.1 Electrostatic Forces 97
3.2 Energy of a System of Charged Conductors 97
3.3 Energy Stored in the Electric Field 98
3.3.1 An Alternative Derivation for the Field Energy 99
3.4 Forces on Conductors and Dielectrics 100
3.4.1 Forces and Pressures on Conductors 100
3.5 Electrostatic Forces on Dielectrics 102
3.6 General Method of Determining Forces in Electrostatic Fields 102
viii CONTENTS
3.7 Pressure on Boundary Surfaces 104
3.7.1 Pressure on Surface of Charged Conductors 104
3.7.2 Pressure on Boundary Surfaces of Two Dielectrics 105
3.8 Stability of the Electrostatic System (Earnshaw’s Theorem) 108
Problems 109
4 Methods of Solving Electrostatic Field Problems 111–159
4.1 Introduction 111
4.2 Direct Solving of Laplace’s Equation 111
4.2.1 Introduction 111
4.2.2 Boundary Surfaces and Conditions 112
4.2.3 Coordinate Systems 113
4.2.4 Separation of Variables in a Rectangular Cartesian System 114
4.2.5 Separation of Variables in a Cylindrical Polar Coordinate System 118
4.2.6 Potential Inside a Hollow Cylindrical Ring 121
4.2.7 Separation of Variables in a Spherical Polar Coordinate System 123
4.2.8 Electric Field within a Charged Hollow Sphere 125
4.3 Green’s Function 126
4.3.1 Green’s Function for a Two-dimensional Region 130
4.3.2 Green’s Function for a Rectangular Region with Poissonian Field 131
4.3.3 Green’s Function for an Infinite Conducting Plane 134
4.4 Conformal Transformations and Complex Variables 135
4.4.1 Functions of Complex Variables and Conjugate Functions 135
4.4.2 Conformal Transformation 137
4.4.3 Complex Potential W(z) 138
4.4.4 Some Simple Examples 140
4.4.4.1 Example 1 (Parallel plate capacitor) 140
4.4.4.2 Example 2 (Two concentric cylindrical conductors) 141
4.4.4.3 Example 3 (A parallel plate capacitor taking account of the fringing
effects at the ends) 144
4.5 Method of Images 149
4.5.1 Line Charge Parallel to the Surface of a Semi-infinite Dielectric Block 151
4.5.2 Point Charge Near an Infinite Grounded Conducting Plane 152
4.5.3 Line Charge Near a Circular Boundary 155
4.5.4 Point Charge Near a Conducting Sphere 156
Problems 159
5 Approximate Methods of Solving Electrostatic Field Problems 160–195
5.1 Introduction 160
5.2 Graphical Method of Solving Electrostatic Problems 160
5.2.1 A Note on Curvilinear Squares 161
5.2.2 A Proof that the Potential Associated with a Field-plot Consisting of Curvilinear
Rectangles Satisfies the Laplace’s Equation 162
5.2.3 Plotting Technique 164
5.2.4 Two-dimensional Multi-dielectric Fields 166
5.3 Experimental Methods 168
5.3.1 Electrolytic Tank Method 168
5.3.2 Conducting Paper Analogue 170
CONTENTS ix
5.3.3 Elastic Membrane Method (Rubber Sheet Analogy) 170
5.3.4 Hydrodynamic Analogy 171
5.4 Numerical Methods 172
5.5 Finite Difference Methods 172
5.5.1 Finite Difference Representation 173
5.5.2 Basic Equations for the Square and the Rectangular Meshes 174
5.5.3 Reduction of the Field Problem into a Set of Simultaneous Equations 177
5.5.4 Computational Methods 178
5.6 Finite Element Method 186
5.6.1 Functional and Its Extremum 186
5.6.2 Functional in Two Variables 188
5.6.3 Functional for Electrostatic Fields 190
5.6.4 Functional and the Boundary Conditions 190
5.6.5 Functional Minimization 191
5.7 General Comments 194
Problems 195
6 Steady Electric Current and Electric Field 196–210
6.1 Introduction 196
6.2 Electric Current and Current Density 197
6.3 Electric Current and Electric Force 198
6.4 The Conservation of Charge (The Equation of Continuity) 199
6.5 Analogy between Electric Current and Electric Flux 200
6.6 Electromotive Force 202
6.7 Potential in the Electric Circuit 203
6.8 Ohm’s Law and Joule’s Law 204
6.9 Boundary Conditions 205
6.10 Circuit Laws 206
6.11 Series and Parallel Connection of Resistors 209
Problems 209
7 Magnetic Field of Steady Currents in Free Space 211–244
7.1 Introduction 211
7.2 The Law of Magnetic Force between Two Small Moving Charges 212
7.3 The Concept of the Magnetic Field (The Magnetic Flux Density) 214
7.4 The Magnetic Field of an Electric Current—Biot–Savart’s Law 215
7.4.1 Magnetic Field of a Short Straight Length of Wire 218
7.4.2 Magnetic Field on the Axis of a Square Coil 220
7.4.3 Magnetic Field on the Axis of a Circular Coil 221
7.4.4 Magnetic Field on the Axis of a Short Circular Solenoid 222
7.4.5 Magnetic Flux Density of Planar Currents, at an Arbitrary Point in
Their Plane 222
7.5 The Lines of Magnetic Flux Density Vector B, and the Magnetic Flux F 223
7.6 The Law of Conservation of Magnetic Flux 224
7.7 Ampere’s Law 226
7.8 Magnetic Scalar Potential 230
7.8.1 Scalar Magnetic Potential at a Point, due to a Current Loop of any Shape 232
x CONTENTS
7.9 Distinction between B and H 233
7.10 Calculation of Magnetic Fields by Means of Potential and the Magnetic Circuit Law
(Ampere’s Law) 234
7.10.1 The Magnetic Field of Current in a Straight Circular Cylindrical Conductor
234
7.10.2 The Magnetic Field of Current in a Coaxial Cable 235
7.10.3 The Magnetic Field Inside a Cylindrical Circular Hole in a Cylindrical
Circular Conductor 237
7.10.4 The Magnetic Field of Current in a Parallel Go-and-Return Circuit 237
7.10.5 The Magnetic Field of a Toroidal Solenoid 239
7.10.6 The Magnetic Field of an Infinitely Long Solenoid 240
7.10.7 The Magnetic Flux Density on the Axis of a Circular Coil 241
7.10.8 Helmholtz Coils 241
7.10.9 The Magnetic Field of a Planar Current Sheet 242
Problems 243
8 Magnetic Field of Steady Currents in Presence of Magnetic
Materials 245–269
8.1 Behaviour of Magnetic Substances in the Magnetic Field 245
8.1.1 Introduction 245
8.1.2 Torque on a Current Loop in a Uniform Magnetic Field 245
8.1.3 Behaviour of Magnetic Materials in a Magnetic Field 248
8.2 Field of an Elementary Current Loop and of the Aggregate of the Loops 248
8.2.1 Field due to a Single Current Loop (Magnetic Dipole) 249
8.2.2 Field of Aggregates of Loops (Magnetic Moment Density Vector or
Magnetization Vector) 250
8.3 Equivalence of the Macroscopic Currents to the Magnetized Substances 251
8.4 The Generalized Form of Ampere’s Circuital Law 251
8.5 Effect of an Externally Applied Magnetic Field on Material Substances 253
8.6 Magnetic Field Intensity Vector and Its Interpretation 257
8.7 Boundary Conditions (Surfaces of Discontinuity) 258
8.8 The Magnetic Characteristics of Iron (Ferromagnetic Materials) 259
8.8.1 A Short Solid Iron Cylinder in a Straight Cylindrical Coil of Infinite
Length 260
8.8.2 B-H Curve of Iron 260
8.8.3 A Qualitative Explanation of the Hysteresis Loops 264
8.8.4 Temperature Dependence of Ferromagnetic Materials 265
8.9 Types of Iron for Specific Magnetic Applications 265
8.10 The Magnetic Circuit 266
Problems 268
9 Methods of Solving Magnetostatic Field Problems 270–310
9.1 Analytical Methods 270
9.1.1 Introduction 270
9.1.2 Method of Separation of Variables 270
9.1.2.1 Hollow cylinder in a magnetic field 270