The Electric Force of a Current

Weber and the surface charges of resistive conductors carrying steady currents
(paperback, 239 pages; ISBN 978-0-9732911-5-5 )

Andre Koch Torres Assis and Julio Akashi Hernandes

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From the Foreword

About the Authors

Table of Contents  


From the Foreword

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Is there an interaction -some reciprocal force -between a current-carrying conductor and a stationary charge nearby? Beneath this simple question lie some remarkable misunderstandings, which are well illustrated by the fact that the answers to it commonly found in the scientific literature and also in many text books are incorrect. In case there is any uncertainty about the answer, all doubt will be eliminated by this book. It tackles the question in a brilliant and comprehensive manner, with numerous hints for relevant experiments and with impressive mathematical thoroughness. It is astonishing to learn that, as early as the middle of the 19th century, the German physicists Weber and Kirchhoff had derived and published the answer to this problem; however, their work was poorly received by the scientific community, and many rejected it as incorrect. The reasons behind this scientific setback, which are presented in detail in this book and supported with numerous quotations from the literature, represent a real treasure trove for readers interested in the history of science. It becomes clear that even in the exact science of physics people at times violate basic scientific principles, for instance by referring to the results of experiments which have never been carried out for the purpose under discussion. This book helps readers not only to develop a detailed knowledge of a seriously neglected aspect of the so-called simple electric circuit, but reminds us also that even eminent physicists can be mistaken, that mistakes maybe transferred from one text book generation to the next and that therefore persistent, watchful and critical reflection is required. A didactic comment is appropriate here. The traditional approach to teaching electric circuits based on current and potential difference is called into question by this book. When dealing with electric current one usually pictures drifting electrons, while for the terms “voltage” or “potential difference” one directly refers to the abstract notion of energy, with no opportunity for visualization. Experience shows that only few school students really understand what “voltage” and “potential difference” mean. The inevitable result of failure to understand such basic terms is that many students lose interest in physics. Those whose confidence in their understanding of science is still fragile, may attribute failure to grasp these basic concepts as due to their own lack of talent. Physics remains a popular and crucial subject, so the large numbers of students who each year study the subject implies that the search for less abstract and therefore more readily understood alternatives to traditional approaches is urgent. This book offers such an alternative. It shows that in respect to surface charges there is no fundamental difference between an electrostatic system and the flow of an electric current. It refers to recent curriculum developments concerning “voltage” and “potential difference” and presents a comprehensive survey of related scientific publications, that have appeared since the early papers by Weber and Kirchhoff. Why should we refer to drifting electrons when we teach electric current and yet not refer to drifting surface charges when teaching voltage or potential difference? The final objective of the curriculum when voltage is covered will certainly be to define it quantitatively in terms of energy. For didactic reasons, however ,it does not seem to be justifiable to omit a qualitative and more concrete preliminary stage, unless there is a lack of knowledge about the existence of surface charges. In the present market there are newly developed curriculum materials that cover basic electricity, to which the content of this book relates strongly. Comparison of the approach that this book proposes with more traditional approaches should dispel any doubts about the need for the methods that it describes. This book provides a crucial step along the path to a better understanding of electrical phenomenon especially the movement of electrons in electric circuits.

 About the Authors

Andre Koch Torres Assis was born in Brazil (1962) and educated at the State University of Campinas – UNICAMP, BS (1983), PhD (1987). He spent the academic year of 1988 in England with a post-doctoral position at the Culham Laboratory (United Kingdom Atomic Energy Authority). He spent one year in 1991-92 as a Visiting Scholar at the Center for Electromagnetics Research of Northeastern University (Boston, USA). From August 2001 to November 2002 he worked at the Institute for the History of Natural Sciences, Hamburg University (Hamburg, Germany) with a research fellowship awarded by the Alexander von Humboldt Foundation of Germany. He is the author of Weber’s Electrodynamics (1994), Relational Mechanics (1999); and (with M. A. Bueno) Inductance and Force Calculations in Electrical Circuits (2001). He has been Professor of physics at UNICAMP since 1989, working on the foundations of electromagnetism, gravitation, and cosmology.

Julio Akashi Hernandes was born in Brazil (1977) and educated at the State University of Campinas – UNICAMP, BS (1998), MS (2001), PhD (2005). He has always been keenly interested in basic physics, especially electromagnetism. He has published many papers on the electric field outside resistive wires carrying steady currents in major international journals of physics. He is Professor of physics at Universidade Bandeirante de São Paulo, Brazil.

 

Contents

Acknowledgments   iii

Foreword       v

Vorwort          vii

I Introduction            1

1 Main Questions and False Answers      7

1.1 Simple Questions            7

1.2 Charge Neutrality of the Resistive Wire           9

1.3       Magnetism as a Relativistic Effect   13

1.4       Weber’s Electrodynamics    . 14

1.5       Electric field of Zeroth Order; Proportional to the Voltage of the Battery; and of Second Order    . 20

2 Reasons for the Existence of the External Electric Field 23

2.1 Bending a Wire.  23

2.2 Continuity of the Tangential Component of the Electric Field      26

3 Experiments          29

3.1 Zeroth Order Electric Field         29

3.2 Electric Field Proportional to the Voltage of the Battery   30

3.3 Second Order Electric Field        42

4 Force Due to Electrostatic Induction    45

4.1 Introduction       45

4.1.1 Point Charge and Infinite Plane          45

4.1.2 Point Charge and Spherical Shell        46

4.2       Point Charge and Cylindrical Shell    46

4.3       Finite Conducting Cylindrical Shell with Internal Point Charge: Solution of Poisson’s Equation     47

4.3.1 Cylindrical Shell Held at Zero Potential      49

4.4 Infinite Conducting Cylindrical Shell with Internal Point Charge 50

4.4.1 Cylindrical Shell Held at Zero Potential      50

4.5 Infinite Conducting Cylindrical Shell with External Point Charge 52

4.5.1 Cylindrical Shell Held at Zero Potential      53

4.5.2 Thin Cylindrical Shell Held at Zero Potential         56

4.5.3 Infinite Cylindrical Shell Held at Constant Potential        58

4.6 Discussion.         61

5 Relevant Topics    65

5.1       Properties of the Electrostatic Field            65

5.2       The Electric Field in Different Points of the Cross-section of the Wire          . 66

5.3       Electromotive Force Versus Potential Difference          67

5.4       Russell’s Theorem    . 68

II          Straight Conductors 71

 

6 A Long Straight Wire of Circular Cross-section           75

6.1       Configuration of the Problem           75

6.2       Force Proportional to the Potential Difference Acting upon the Wire           . 77

6.3       Force Proportional to the Square of the Current           82

6.4       Radial Hall Effect.      84

6.5       Discussion.    86

7 Coaxial Cable        93

7.1 Introduction       93

7.2 Potentials and Fields.     94

7.3 The Symmetrical Case    97

7.4 The Asymmetrical Case  98

7.5 Discussion.         100

8 Transmission Line 103

8.1 Introduction       103

8.2 Two-Wire Transmission Line      . 103

8.3 Discussion.         108

9 Resistive Plates    113

9.1 Introduction       113

9.2 Single Plate         . 113

9.3 Two Parallel Plates         116

9.4 Four Parallel Plates        117

9.4.1 Opposite Potentials    118

9.4.2 Perfect Conductor Plate         120

10 Resistive Strip 123

10.1The Problem       123

10.2The Solution       124

10.3Discussion.        126

10.4 Comparison with the Experimental Results           128

III Curved Conductors 133

 

11 Resistive Cylindrical Shell with Azimuthal Current 137

11.1Configuration of the Problem  . 137

11.2Potential and Electric Field      138

11.3Surface Charge Densities.        141

11.4Representation in Fourier Series.        143

11.5Lumped Resistor           146

12 Resistive Spherical Shell with Azimuthal Current 151

12.1 Introduction.    151

12.2Description of the Problem      151

12.3General Solution            153

12.4Electric Field and Surface Charges      156

12.5Conclusion         . 160

13 Resistive Toroidal Conductor with Azimuthal Current 163

13.1 Introduction.    163

13.2Description of the Problem      163

13.3General Solution            166

13.4 Particular Solution for a Steady Azimuthal Current.         167

13.5Potential in Particular Cases    170

13.6Electric Field and Surface Charges      173

13.7Thin Toroid Approximation       174

13.8 Comparison of the Thin Toroid Carrying a Steady Current with the Case of a Straight Cylindrical Wire Carrying a Steady Current180

13.9 Charged Toroid without Current         181

13.10Comparison with Experimental Results 184

IV Open Questions 189

 

14 Future Prospects 191

 

Appendices 195

A Wilhelm Weber and Surface Charges 195

B Gustav Kirchhoff and Surface Charges 213

Bibliography 217
Index 236