Electricity and Magnetism

Undergraduate Lecture Notes in Physics

Teruo Matsushita

Electricity

and

Magnetism

New Formulation by Introduction of

Superconductivity

Undergraduate Lecture Notes in Physics

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Teruo Matsushita

Electricity and Magnetism

New Formulation by Introduction

of Superconductivity

123

Teruo Matsushita

Department of Computer Science &

Electronics

Kyushu Institute of Technology

Iizuka, Fukuoka, Japan

Original Japanese edition, Shin Denjiki-gaku, By Teruo Matsushita, Copyright c (2004), Published by

CORONA PUBLISHING CO., LTD., 4-46-10, Sengoku, Bunkyo, Tokyo, Japan

ISSN 2192-4791 ISSN 2192-4805 (electronic)

ISBN 978-4-431-54525-5 ISBN 978-4-431-54526-2 (eBook)

DOI 10.1007/978-4-431-54526-2

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Preface

Electromagnetism is an important subject in today’s physics. The number of

textbooks on electromagnetism is much larger than those on other subjects. This

is because abstract concepts are frequently used and therefore it is not easy

for students to come to a complete understanding of electromagnetism, although

various phenomena are concisely described with mathematics. For this reason many

textbooks have been published to assist students to understand electromagnetism

better. Why, then, is a new textbook on electromagnetism necessary now?

Electromagnetism is a classical subject that was almost completely formulated in

the nineteenth century. However, concerning its theoretical description, there is still

room for further progress. In addition, textbooks are required to describe their topics

adequately within a limited space. Therefore, there is also room for improvement in

textbooks from the technical point of view.

In principle, there is a beautiful formal analogy between static electric and

magnetic phenomena, as will be shown in this textbook. However, the analogy is

not necessarily perfect in existing textbooks because of the lack of an important

concept. Electric materials are classified into conductors and dielectric materials,

but only magnets are studied as magnetic materials. While it is known that

electric phenomena in dielectric materials and magnetic phenomena in magnets are

analogous to each other, no one has discussed magnetic materials that correspond to

electric conductors. However, we have to note superconductors. In a superconductor

a current flows on its surface to shield the inside against an external magnetic field,

so that the magnetic flux density B is zero in the superconductor. This is analogous

to the electric phenomenon of a conductor in an external electric field. That is, an

electric charge appears on its surface to shield the inside against an external electric

field, so that the electric field E is zero in the conductor. This is one of the remarkable

analogies in the present E–B analogy.

Thus, the introduction of the superconductor into electromagnetism, which

has not yet been tried systematically, seems to be quite useful for understanding

electromagnetism. That is, the analogy between electricity and magnetism can be

completed by the introduction of the superconductor. There can be various ways of

v

vi Preface

education without such a comprehensive analogy, and this is another reason why

many textbooks on electromagnetism have appeared.

From another point of view, superconductivity is a general phenomenon that

appears in many single elements and most metallic compounds, if the cases of pres￾surization and thin films are included. The intrinsic property of superconductivity,

the breaking of Ohm’s law, may seem to be peculiar. However, superconductivity

is a purely physical phenomenon that can be derived from minimizing free energy.

In contrast, the empirical Ohm’s law associated with energy dissipation cannot be

derived theoretically, and electromagnetic theory is incomplete for other current￾carrying materials in this sense.

Usually students learn about static magnetic energy after they study electromag￾netic induction. One of the appreciable advantages of using a superconductor is

the direct derivation of magnetic energy as mechanical work done by magnetic

force, similar to the electric energy resulting from the electric force. This is because

the magnetic flux is conserved in a superconducting circuit disconnected from any

electric sources. As a result, the electromagnetic induction can be predicted for a

usual electric circuit using the relationship between the energy and magnetic force.

In electromagnetism the magnetic moment of a magnet caused by spins and

orbital motions of electrons is described using a virtual magnetizing current.

However, the magnetic moment of a superconductor comes from a real current

flowing in it. Hence, the introduction of the superconductor is also beneficial with

regard to persuasion of the appropriateness of the virtual magnetizing current.

It should be noted that the definition of magnetization is different for magnets

and superconductors. That is, magnetization comes directly from the magnetization

M in magnets, while it comes from the magnetic field H in superconductors. This

arises from the difference in the origin of the magnetic moments. According to

the definition used for magnets, superconductors are classified as non-magnetic

materials. On the other hand, the analogous electric phenomena are electrostatic

shielding in conductors and electric polarization in dielectrics. These are similar

electric shielding mechanisms caused by electric charges and polarization charges,

but the above-mentioned different terms are used. Such comparison between

electricity and magnetism is also useful for education.

The final merit of the introduction of superconductors is application of the

analysis method of electromagnetic phenomena in superconductors. The continuity

equation of magnetic flux used for superconductors is useful for estimating the

velocity of magnetic flux lines under a magnetic field varying with time. This

enables us to unify the magnetic flux law and the motional law for electromagnetic

induction, which usually have been treated separately.

The purpose of this textbook is to show the remarkable analogy between

static electric phenomena, described in Part I, and static magnetic phenomena,

described in Part II. Hence, a comparison between the corresponding chapters in

each part, such as Chap. 2 on conductors and Chap. 7 on superconductors, will

assist in understanding electromagnetism. Dynamic electromagnetic phenomena are

described in Part III.

Preface vii

I would like to express my sincere acknowledgment to Prof. Klaus Lueders at the

Berlin Free University for the useful discussion we had. In addition, I would like to

thank Tomoko Onoue, Etsuko Shirahasi, and Kaori Ono for assistance in making

electronic files and drawing electronic figures.

Iizuka, Fukuoka, Japan Teruo Matsushita

Contents

Part I Static Electric Phenomena

1 Electrostatic Field .......................................................... 3

1.1 Electric Charge in Vacuum .......................................... 3

1.2 Coulomb’s Law....................................................... 4

1.3 Electric Field ......................................................... 7

1.4 Gauss’ Law ........................................................... 11

1.5 Electric Potential ..................................................... 16

1.6 Electric Dipole........................................................ 23

Exercises ..................................................................... 29

2 Conductors .................................................................. 33

2.1 Electric Properties of Conductors ................................... 33

2.2 Special Solution Method for Electrostatic Field .................... 41

2.3 Electrostatic Induction ............................................... 46

Exercises ..................................................................... 51

3 Conductor Systems in Vacuum............................................ 55

3.1 Coefficients in Conductor System ................................... 55

3.2 Capacitor.............................................................. 60

3.3 Electrostatic Energy .................................................. 65

3.4 Electrostatic Force.................................................... 70

Exercises ..................................................................... 73

4 Dielectric Materials......................................................... 75

4.1 Electric Polarization .................................................. 75

4.2 Electric Flux Density ................................................. 81

4.3 Boundary Conditions................................................. 85

4.4 Electrostatic Energy in Dielectric Materials ........................ 92

Exercises ..................................................................... 95

ix

x Contents

5 Steady Current.............................................................. 99

5.1 Current ................................................................ 99

5.2 Ohm’s Law ........................................................... 101

5.3 Microscopic Investigation of Electric Resistance ................... 103

5.4 Fundamental Equations for Steady Electric Current................ 106

5.5 Electromotive Force .................................................. 112

5.6 Kirchhoff’s Law ...................................................... 114

Exercises ..................................................................... 117

Part II Static Magnetic Phenomena

6 Current and Magnetic Flux Density ...................................... 123

6.1 Magnetic Flux Density by Current .................................. 123

6.2 The Biot–Savart Law................................................. 125

6.3 Force on Current ..................................................... 128

6.4 Magnetic Flux Lines ................................................. 133

6.5 Ampere’s Law ........................................................ 134

6.6 Vector Potential....................................................... 139

6.7 Small Closed Current ................................................ 144

6.8 Magnetic Charge ..................................................... 146

Exercises ..................................................................... 152

7 Superconductors............................................................ 155

7.1 Magnetic Properties of Superconductors............................ 155

7.2 Special Solution Method for Magnetic Flux Density ............... 163

7.3 Magnetization ........................................................ 167

Exercises ..................................................................... 174

8 Current Systems ............................................................ 177

8.1 Inductance ............................................................ 177

8.2 Coils .................................................................. 183

8.3 Magnetic Energy ..................................................... 188

8.4 Magnetic Force ....................................................... 193

Exercises ..................................................................... 196

9 Magnetic Materials......................................................... 201

9.1 Magnetization ........................................................ 201

9.2 Magnetic Field........................................................ 208

9.3 Boundary Conditions................................................. 211

9.4 Magnetic Energy in Magnetic Material ............................. 220

9.5 Analogy Between Electric and Magnetic Phenomena .............. 222

Exercises ..................................................................... 226

Contents xi

Part III Time-Dependent Electromagnetic Phenomena

10 Electromagnetic Induction ................................................ 231

10.1 Induction Law ........................................................ 231

10.2 Potential .............................................................. 240

10.3 Boundary Conditions................................................. 241

10.4 Magnetic Energy ..................................................... 242

10.5 Skin Effect ............................................................ 245

Exercises ..................................................................... 251

11 Displacement Current and Maxwell’s Equations ....................... 255

11.1 Displacement Current ................................................ 255

11.2 Maxwell’s Equations ................................................. 258

11.3 Boundary Conditions................................................. 261

11.4 Electromagnetic Potential............................................ 261

11.5 The Poynting Vector ................................................. 263

Exercises ..................................................................... 267

12 Electromagnetic Wave ..................................................... 271

12.1 Planar Electromagnetic Wave........................................ 271

12.2 Reflection and Refraction of the Planar Electromagnetic Wave ... 275

12.3 Energy of the Electromagnetic Wave ................................ 280

12.4 Wave Guide ........................................................... 281

12.5 Spherical Wave ....................................................... 285

12.6 Retarded Potential .................................................... 287

Exercises ..................................................................... 289

Appendix A ....................................................................... 291

A1 Vector Analysis....................................................... 291

A1.1 Scalars and Vectors ......................................... 291

A1.2 Addition of Vectors ......................................... 291

A1.3 Products of Vectors and Scalars............................ 293

A1.4 Analytic Expression of a Vector ........................... 293

A1.5 Products of Vectors ......................................... 294

A1.6 Differentiation of Vectors .................................. 295

A1.7 Gradient of a Scalar......................................... 297

A1.8 Divergence of a Vector ..................................... 297

A1.9 Rotation of a Vector ........................................ 298

A1.10 Differentiation of Products of Vectors ..................... 299

A1.11 Second Differentiation ..................................... 299

A1.12 Curvilinear Integral of a Vector ............................ 300

A1.13 Surface Integral of a Vector ................................ 301

A1.14 Gauss’ Theorem ............................................ 302

Erratum .......................................................................... E-1

xii Contents

A1.15 Stokes’ Theorem ............................................ 304

A1.16 Green’s Theorem ........................................... 306

A1.17 Cylindrical Coordinates .................................... 307

A1.18 Polar Coordinates........................................... 308

A2 Proofs ................................................................. 309

A2.1 Proof of Eq. (1.37) .......................................... 309

A2.2 Proof of Eq. (6.21) .......................................... 310

A2.3 Proof of Eq. (6.27) .......................................... 311

A2.4 Proof of Eq. (6.33) .......................................... 312

A2.5 Proof of Eq. (6.45) .......................................... 312

A2.6 Proof of Eq. (9.5) ........................................... 313

A3 Superconductivity .................................................... 314

A3.1 Phenomenological Electromagnetism ..................... 314

A3.2 Mixed State ................................................. 317

A3.3 Motion of Quantized Magnetic Flux....................... 319

A3.4 Electromagnetism and Superconductivity ................. 320

Literature ......................................................................... 323

Answers to Exercises............................................................. 325

Index ............................................................................... 381

Explanation of the figures

The upper figures show the structure of electric flux lines when a uniform electric

field is applied to a sphere of a conductor (left) and a dielectric (right), and the lower

figures show the structure of magnetic flux lines when a uniform magnetic field is

applied to a sphere of a superconductor (left) and a magnet (right). These will be

covered in Chapters 2, 4, 7 and 9, respectively. The reason why the electric flux lines

are used instead of the more important electric field lines is to emphasize the analogy

between dielectrics and magnets by showing continuous lines at the interfaces with

vacuum. In the case of electric field lines, the number of lines inside the dielectric

is smaller than that outside because of the shielding by polarization charges (see

Chapter 4). This situation is similar when we draw the magnetic field lines instead

of the magnetic flux lines for the lower right figure.

The manner of perfect shielding is different between the conductor and the

superconductor. This comes from the different nature of the corresponding fields.

Electric charges on the conductor surface absorb the lines directly, while currents

on the superconductor surface push the lines to outside.

Part I

Static Electric Phenomena