methodology of teaching basic electricity the magnetic flux linkage..... 69 6.3 the own and the...

Peter Kitak, Tine Zorič

METHODOLOGY OF TEACHING BASIC

ELECTRICITY Minimal Set of Basic Laws

E-publication (draft)

Maribor, 2013

II

Title of the publication:

METHODOLOGY OF TEACHING BASIC ELECTRICITY

Minimal Set of Basic Laws

Type of the publication:

E-publication

Author: doc. dr. Peter Kitak

prof. dr. Tine Zorič

Expert review: prof. dr. Igor Tičar, univ. dipl. ing. el., UM FERI

high. lect. mag. Andrej Orgulan, univ. dipl. ing. el., UM FERI

Cveto Štandeker, univ. dipl. ing. el., pedagogical counsellor

Language review: -

Publishing:

Faculty of Electrical Engineering, Computer and Information Sciences , University of Maribor

Available: www.

CIP – Catalogue record of publications University Library Maribor

Copyright 2013, Faculty of Electrical Engineering, Computer Sciences and Informatics, University of Maribor, Institute for Fundamentals and Theory in Electrical Engineering

III

Table of Contents

1 INTRODUCTON ........................................................................................................................................ 1

2 METHODICAL RULES ................................................................................................................................ 5

3 ELECTROSTATIC FIELD ........................................................................................................................... 11

3.1 BASIC LAW SET FOR ELECTROSTATIC FIELDS ........................................................................................................... 12 3.1.1 Coulomb's Law .................................................................................................................................... 12 3.1.2 Electric flux density ............................................................................................................................. 14 3.1.3 The electric voltage and the electric potential.................................................................................... 15

3.2 THE INFLUENCE OF THE SUBSTANTIAL PROPERTIES OF DIELECTRICS ON AN ELECTROSTATIC FIELD ....................................... 18 3.2.1 The polarisation of dielectrics and relative dielectric constant .......................................................... 18 3.2.2 The constitutional laws of electrostatic field in the integral form ...................................................... 19

3.3 KIRCHHOFF'S LAWS OF ELECTROSTATIC FIELDS ....................................................................................................... 20 3.3.1 Kirchhoff's law of a closed loop .......................................................................................................... 21 3.3.2 Kirchhoff's law of a closed surface ...................................................................................................... 21

3.4 LOCATIONS OF DIELECTRICS AND CAPACITORS CIRCUITS ........................................................................................... 22 3.4.1 Configuration of dielectrics ................................................................................................................. 22 3.4.2 Capacitor circuits ................................................................................................................................ 24

3.5 THE ENERGY OF ELECTROSTATIC FIELD AND ITS ENERGY DENSITY ................................................................................ 25 3.6 THE MINIMAL SET OF BASIC LAWS FOE ELECTROSTATIC FIELDS ................................................................................... 27

4 THE ELECTRIC CURRENT FIELD ............................................................................................................... 29

4.1 THE ANALOGY WITH THE ELECTROSTATIC FIELD ...................................................................................................... 30 4. 2 THE DIFFERENCES BETWEEN THE ELECTROSTATIC AND ELECTRIC CURRENT FIELD ........................................................... 31 4. 3 THE MECHANISM OF ELECTRIC CURRENT IN METALLIC CONDUCTORS .......................................................................... 32 4.4 THE OHM'S LAW IN THE INTEGRAL FORM .............................................................................................................. 35 4.5 THE JOULE'S LAW IN THE DIFFERENTIAL FORM ....................................................................................................... 37 4.6 THE JOULE'S LAW IN THE INTEGRAL FORM ............................................................................................................. 38 4.7 THE GEOMETRIC PLACING OF CONDUCTING SUBSTANCES AND RESISTOR CIRCUITS ......................................................... 38 4.8 THE STRUCTURAL CONNECTEDNESS OF BASIC QUANTITIES AND LAWS IN THE ELECTRIC CURRENT FIELD ............................... 41 4.9 THE MINIMAL SET OF BASIC LAWS IN THE ELECTRIC CURRENT FIELD ............................................................................. 41

5 THE DC MAGNETIC FIELD ....................................................................................................................... 43

5.1 BASIC LAWS OF MAGNETIC FIELDS FROM THE VIEWPOINT: CAUSE - CONSEQUENCE ........................................................ 45 5.1.1 The magnetic field strength ................................................................................................................ 45

5.2 THE MAGNETIC FLUX DENSITY ............................................................................................................................. 48 5.3 THE MAGNETIC FLUX Φ AND THE MAGNETIC FLUX LINKAGE Ψ ................................................................................... 50 5.4 THE FORCES ON MOVED ELECTRIC CHARGES IN MAGNETIC FIELDS ............................................................................... 51 5.5 MAGNETIC PHENOMENA IN MAGNETIC SUBSTANCES ............................................................................................... 54

5.5.1 The explanation of ferromagnetism ................................................................................................... 54 5.5.2 The magnetization curve .................................................................................................................... 55 5.5.3 The configuration of magnetic substances ......................................................................................... 58 5.5.4 Magnetic circuits ................................................................................................................................ 60

6 THE INDUCED ELECTRIC FIELD ............................................................................................................... 63

6.1 TIME DEPENDENT MAGNETIC FIELDS .................................................................................................................... 64 6.1.1 The electromagnetic induction ........................................................................................................... 64

6.2 THE MAGNETIC FLUX LINKAGE ............................................................................................................................ 69 6.3 THE OWN AND THE MUTUAL INDUCTANCES ........................................................................................................... 71

IV

6.4 INTERDEPENDENCE OF ELECTRIC CURRENTS AND ELECTRIC VOLTAGES ON PASSIVE ELEMENTS OF ELECTRIC CIRCUITS .............. 72 6.5 THE ENERGETIC CONSIDERATIONS IN MAGNETIC FIELDS ............................................................................................ 73

6.5.1 The magnetic field energy .................................................................................................................. 74 6.5.2 The magnetic field energy density ...................................................................................................... 75 6.5.3 The forces on borders of flux tubes ..................................................................................................... 77

6.6 THE STRUCTURAL LINKING OF BASIC MAGNETIC FIELD .............................................................................................. 78

7 MAXWELL'S FIELD EQUATIONS ............................................................................................................. 81

7.1 THE PHYSICAL BACKGROUND OF MAXWELL'S FIELD EQUATIONS ................................................................................. 82 7.1.1 The theorems which are the basis for Maxwell's field equations ....................................................... 82

7.2 THE APPLICATION OF GAUSS' AND THE STOKES' THEOREM IN THE DERIVATION OF MAXWELL'S FIELD EQUATIONS ................ 84

8 CONCLUSION ......................................................................................................................................... 91

LITERATURE .................................................................................................................................................... 92

V

The topmost artistry of a teacher is his ability to awake in their pupils the enjoyment in a creative diction and knowledge.

Einstein Albert

1

1 INTRODUCTON

Nothing will never become the reality if we are not able to prove it with an experiment – even a proverb is not a proverb, before it is proved through life.

Keats John

METHODOLOGY OF TEACHING BASIC ELECTRICITY - minimal set of basic laws

2

We are living in an era of explosive expansion of human knowledge, in a period of ten years its total volume nearly doubles. When Newton was still able to master the entire area of mathematics and physics, today for mastering every broader area of science teams of experts are required.

Technical sciences represent a very important part of entire human knowledges, its most important duty is to meet the increasing needs of humanity. Parallel with increased importance of technical sciences also the significance of technical education is gaining more and more importance, because it enables the transfer of achievements and knowledge to coming generations and ensures the future development of technical sciences. Therefore it is a serious concern that in spite of extraordinary development of technical sciences no adequate development is noted in the theory of technical education.

Although the pedagogy presents the general rules and requirements, which are valid for all areas of education, they are not sufficient for preparation and realisation of technical education. Just to state the rules and requirements is not enough, if the performer do not know how to implement them in a special area of technical education. In Slovenia special didactics for all natural sciences in area of pedagogical education are implemented, therefore it is not understandable and harmful that no comprehension is found for special didactics of fundamental subjects in fields of technical education.

Common conviction, that the synthesis of teacher's professional knowledge and consideration of general pedagogical rules and requirements already ensures successful technical education, is incorrect and misleading. Technical education must be easily scanned, the way which leads from experiment to fundamental law must be clearly seen. It must give participants knowledge and instructions, how to realise this knowledge in his professional work and which tools are for this necessary. Technical knowledge, which we do not know to verify physical phenomena with measurement or calculation, is for a professional technician a useless and unproductive knowledge. To transfer such usable knowledge is in case of nonexistent special didactics of fundamental technical subjects a very hard, if not impossible task.

Nonexistent special didactics of basic technical subjects are only one of obstacles at performance of a long-termed and successful technical education. At an explosive growth of technical knowledge the available number of hours for basic technical subjects are becoming less and less. Therefore it is necessary very careful to define following decisions:

a) Where are the limits, where a successful mastering of basic technical laws turns into instructions or recipes for solving of technical problems.

b) It is necessary to prepare a methodology of teaching basic technical subjects. From it the red thread that links the explanation and derivation of basic laws must be clearly seen. As basic laws only those laws are considered, which are valid for the entire field of technical applications.

c) The application of basic laws in a specific branch must be included into teaching programs of special branches.

The arguments in precedent passage are the motive, in account of it we will try in a series of chapters to present a first outline of methodology as a part of special didactics of basic electricity. We use the term “first outline” because we are convinced, that all conclusions could be still complemented and improved. So is this on WEB-pages

Introduction

3

presented first outline on one side the invitation to interested technical educators to take part in forming the special didactics of fundamental technical subjects and on the other side a challenge to pedagogues to take part in forming of special didactics for fundamental technical subjects or at least not obstruct them.

In a sequel of chapters we will present our idea of Methodology of teaching Basic Electricity in extent that is common for all branches of electrical engineering. We will divide the e-teaching materials into three WEB-pages:

a) Methodology 1

In e-teaching materials Methodology 1 we will present the minimal extent of derivation and explaining of basic rules valid for all parts of electrical engineering. From them the structural linkage of basic quantities should be seen, the formal and substantial analogy between basic department of basic electricity should be exploited to full extent. Those e-teaching materials will include following chapters:

1. Introduction will present reasons for publishing those e-teaching materials.

2. Methodical rules will contain all rules and procedures used at explanation and derivation of basic laws. These rules are in principle valid for all areas of technical knowledge.

The basic rules of electricity will be contained in four chapters.

3. Electrostatic fields will contain the explanation and derivation of basic laws caused in substances without movable electric carriers by standstill electric charges or dc voltages.

4. Current fields will contain the explanation and derivation of basic laws caused in substances with movable electric charges by electric voltage or respectively by them caused electric field in conductors.

5. Stationary magnetic fields will contain the explanation and derivation of basic magnetic laws caused by movable electric charges (currents).

6. Induced electric fields will contain the explanation and derivation of basic laws caused by changing magnetic fields.

7. Maxwell laws complete the list of basic laws valid for all electric and magnetic phenomena of electricity. Without them the set of basic laws in both forms, the integral and differential form is not complete. More as the application of Maxwell laws, their physical meaning is important.

8. Literature. The most important books concerning engineering pedagogy are stated and those textbooks for Basic Electricity used in formulating our conclusions.

b) In e-teaching materials Methodology 2 the application of basic and special laws in different branches of electrical engineering will be presented. Because we both are specialised in electric power systems, our contribution will be the application of basic laws in this field. Therefore we invite the experts from other branches of electrical engineering to present their view of application of basic electric laws in their field. Those WEB-pages will be open for their contribution.

c) In e-teaching materials Methodology 3 we will present some special methods, which are used to solve some special problems or in some way illuminate same physical phenomenon. Those WEB-pages will also be open for all contributions.

In order to stimulate the participation of foreign experts, we will all contributions publish in Slovene, German and English language and take care for their translations.


4

5

2 METHODICAL RULES

The knowledge, that makes it possible to measure or calculate physical phenomena, is a solid and permanent knowledge.

If that isn't the case, our knowledge is poor, foggy and short-lived.

Huygens Christiaan


6

The goals of technical education are twofold:

a) Learning and acquirement of manual and practical skills. This is the intention and goal of industrial and vocational schools. Learning of theoretical knowledge is limited to extent that is needed for successful primary goal.

b) Learning and acquirement of theoretical based knowledge in a technical profession. Manual and practical skills may be a useful addition, bur they are not a primarily goal. The content and the goal of university studies are acquirement of solid theoretical knowledge.

The middle and higher technical schools are to some extent interested on both mentioned goals, but only to some extent.

Methodical rules, we intend to elaborate, are primarily used in university studies, the procedures and ways, how to acquire manual and practical skill may interested reader find in appropriate literature.

Technique represents application of physical science in solution of technical problems. So explains basic electricity a group of physical phenomena, which are in different substances and different configurations caused by electrical charges. The electrical charges may stand still, may move with constant or varying velocity, in different substances and their configurations they may cause specific electrical phenomena.

Natural law is a mathematically formulated law of a natural phenomenon. If it is valid for all related phenomena, it is called basic law. If it is valid for only one specific phenomenon, it is called specific or derived law.

Basis and starting-point for deriving a basic law is usually an experiment. But because the experiment is always accomplished for a specific case, so is a from experiment derived law always a specific law. After evaluating experiment we have to use adequate methods to derive the basic law and later specific laws for all related phenomena.

New cognitions and laws

To bring into explanation and interpretation of some natural phenomenon new cognitions and laws, we can realise this in three possible ways:

Natural law (mathematical formulated

law of a natural phenomenon)

Basic law

(for all related phenomena)

Specific or derived law

(for only one specific phenomenon)

Methodical rules

7

a) through evaluation of the experiment we derive the law

b) using known laws we derive a new law (something new out of something known)

c) to known laws we add the cognitions from the experiment

To explain a natural phenomenon and derive the laws concerning them, we can start whether from already known laws and facts or derive them from evaluation of performed experiment. If we do not have possibility to perform the experiment, we can describe it. From specific experiment we derive the specific law. Using appropriate method, we derive basic law and later related specific laws.

Methods

By its origin (Greek) the word method means “the way to”. In our case it means the way to cognition, the way to law. We will designate method as one from subject, field and theoretical level dependent way to cognition. There existed a variety of specific methods, but most of them can be treated as a sum of basic elements, where every one of them has his own characteristics. Those basic methodical elements are:

a) analysis,

b) synthesis,

c) induction,

d) deduction

e) analogy

Analysis presents dismembering of some natural phenomenon in order to determine its essential characteristics and elements. Essential characteristics and elements are those which essentially describe the course of the natural phenomenon.

New law

Experiment

Known law

Known law

+ New law

Known law

Experiment

New law +


8

Synthesis is contrary to analysis. While we in analysis define the essential characteristics and elements of some natural phenomenon, we in synthesis from known essential characteristics and elements define the concept of the natural phenomenon and its special law.

Induction represents generalization of a special law for one specific phenomenon into a basic law valid for all related phenomena.

Deduction represents derivation of specific laws, which are valid for one specific phenomenon, from basic law, which is valid for all related phenomena,

Analogy represents explanation or derivation of law for some natural phenomenon on basis of analogy with some other natural phenomenon or its law. The analogy may be formal or conceptual or it may be both.

In most cases are the teaching methods a combination of quoted elements. When we for a method use the term induction or deduction method, it only means that we are using a method, where induction or deduction prevails.

Which teaching method we will choose for some technical area, will depend as well from mathematical proficiency of participants as well as from the level of taught matter.

We master some physical or technical area if we are able to measure or calculate all its phenomena and are able to recognise mutual interrelationship of all quantities appearing in this phenomenon.

Every physical law (basic or specific) is always a mathematical formula, which expresses one physical quantity with other related physical quantities. It is reasonable to separate all physical quantities into two groups:

diferencial quantities

Valid only in a point of treated space

Example: Point in the space between two

plates of a capacitor

Physical quantities

integral quantities

valid for all treated space

Example: Space between two plates of a capacitor

Methodical rules

9

The designation of both quantities is derived from mutual linkage of both quantities. So is density of electric current J defined as a current through a unity of a perpendicular plane:

2

d A

d m

iJ

A

and the electric current I as integral of electric current density J through a perpendicular plane A:

dA

i J A

So are electric current I and electric current density corresponding integral and differential quantities.

Differential quantities are always defined as integral quantities on unity of length, surface or volume. Somewhere names macroscopic and microscopic are used.

Most basic laws (also electrical) have two forms: integral and differential:

Integral and differential law have identical form, only the quantities are corresponding integral and differential quantities.

In basic laws which do not have two forms, integral and differential quantities may be found. Examples of this kind of law are Coulomb's nd Lorentz law.

Basic parts of electrical engineering

As a basic part of electrical engineering we define an area where all corresponding basic laws are defined. Those basic parts are:

a) Electrostatic fields include all phenomena caused by non-moving electrical charges in substances without movable carriers of electrical charges.

b) Current fields deal with electric currents, with phenomena caused in conductors, that means substances with abundance of movable carriers of electric charges in electric fields.

c) Magnetic fields deal with magnetic phenomena caused in nonmagnetic and magnetic substances by electric currents.

d) Induced electric fields deal with phenomena caused by time and/or location dependent magnetic fields.

Ohm's law

Integral form

I U G

Differential form

J E

Veljajo le v posamezni točki obravnavanega

prostora.

Primer: točka v prostoru med dvema ploščama

kondenzatorja.


10

All basic electrical laws are derived in those four chapters. We are able to understand them thoroughly if we are able asses them through calculation and/ experiment. This basic thought, told by Huygens, is the measure of quality and solidness of our knowledge.

In derivation and explanation of basic electric laws we must start in every of four chapters with this quantity, that enables us, with the use of already known laws and eventually an additional experiment, to determine all other electric quantities in this area. If we use experiment, we first derive specific law governing this specific case and with method of induction the corresponding basic law. When the corresponding basic law has been found then, with application of method of deduction, we derive specific law for all related cases. For an inside understanding of physical phenomena the analogy may be very useful, especially if it is not only formal but also inherent.

Whenever it is possible we derive the basic law out of specific law for the simplest case. The derivation is usually very simple, the conclusions are also easy to remember.

Basic laws are usually derived for some ideal conditions. In this way derived basic laws may be valid also when conditions are not ideal or linear. But from them derived specific laws are not valid in all cases. An expert worker must therefore for all specific cases know if they can be solved by procedures valid for ideal circumstances or is for the solution of a case, that is not ideal, necessary to use a special procedures and which one.

Methodology and didactics of all technical fields are dealt with in Engineering pedagogy. Engineering pedagogy is a new science, only a little more than a half of century old. The methodology of teaching basic electricity is only one part of it. Therefore we will in annex state some basic literature in this field. From them it will be evident what else must be considered, if we try in this paper discussed procedures realise in a successful teaching. In mentioned literature all social, psychological and environment factors are included, which will not be mentioned in our paper.

Content of this paper does not mean to be a textbook of Basic electricity. But it gives the order and the manner of teaching basic electricity in such a way, that the upper requirements in deriving basic laws are implemented.

11

3 ELECTROSTATIC FIELD

I picture myself an electric tube as a strained elastic band. It has the tendency to shorten its length and increase its cross-section.

Faraday Michael

Contents

3.1 BASIC LAW SET FOR ELECTROSTATIC FIELDS 3.1.1 COULOMB'S LAW 3.1.2 ELECTRIC FLUX DENSITY 3.1.3 THE ELECTRIC VOLTAGE AND THE ELECTRIC POTENTIAL 3.2 THE INFLUENCE OF THE SUBSTANTIAL PROPERTIES OF DIELECTRICS ON AN

ELECTROSTATIC FIELD 3.2.1 THE POLARISATION OF DIELECTRICS AND RELATIVE DIELECTRIC CONSTANT 3.2.2 THE CONSTITUTIONAL LAW OF ELECTROSTATIC FIELD IN INTEGRAL FORM 3.3 KIRCHHOFF'S LAWS OF ELECTROSTATIC FIELDS 3.3.1 KIRCHHOFF'S LAW OF A CLOSED LOOP 3.3.2 KIRCHHOFF'S LAW OF A CLOSED SURFACE 3.4 LOCATIONS OF DIELECTRICS AND CAPACITORS CIRCUITS 3.4.1 CONFIGURATIONS OF DIELECTRICS 3.4.2 CAPACITOR CIRCUITS 3.5 THE ENERGY OF ELECTROSTATIC FIELD AND ITS ENERGY DENSITY 3.6 THE MINIMAL SET OF BASIC LAWS FOE ELECTROSTATIC FIELDS


12

3.1 Basic law set for electrostatic Fields

3.1.1 Coulomb's Law

Starting from the demand that all new laws descend from already known laws and facts and/or experiments, it is reasonable to start deriving laws of electricity with phenomena caused by non-moving electrical charges. A space put into an electric strained state by non-moving electrical charges is called an electrostatic field.

At the beginning of any new knowledge we have to state all facts and laws we already know. In field of electrical charges those facts are:

1) There are two kinds of electrical charges: positive and negative. The name is derived from the fact, that the electrical charges sum as real numbers.

2) Each electrical charge is always an integer multiplier of the elementary electrical charge eo.

3) The bearer of negative elemental electrical charge is an electron and the bearer of the elemental positive electric charge is a proton.

4) In matter there is always the same quantity of both electrical charges present, so all electric phenomena are always caused only by surplus of one of them. The corresponding surplus charge is often supposed to be placed in the infinity.

5) The electrical charges of the same kind repel, the electrical charges of different kind attract each others.

To understand the electric and magnetic properties of substances the knowledge of atom model is necessary. To derive of basic laws the knowledge of Bohr model of atom will be sufficient. From this the first black-white division of substances into conductors or isolators and into magnetic or non-magnetic substances will be accessible.

The first electric law was derived in 1795 by Coulomb and is therefore named Coulomb's law. Using the experiment by figure 3.1 with two opposite spherical charges +Q and –Q, he found by analysis

+Q1 -Q2

F F

Figure 3.1 Coulomb's law experiment

Electrostatic field

13

F ≈ Q1 - The Force is proportional the electric charge Q1

F ≈ Q2 - The Force is proportional the electric charge Q2

F ≈ 1/r2 - The Force is inverse proportional the square of the distance r2

F ≈ k - The Force is proportional coefficient

1

4k - space angle

F

1 - The force is inverse proportional the substantial property of the air o

For a lonely spherical charge in air (or vacuum) the quotient 0/k has the value

9

0 0

19 10

4

k [Vs/Am]

Electrostatic field of a lonely spherical charge is spreading in direction of radius into spherical angle 4 .

By this analysis Coulomb ascertained for the force between two electric point charges to be proportional to values of electric charges Q1 and Q2 and inverse proportional to square of the distance between charges r2, the space angle 4 an substantial property of

air 0 .

By synthesis he derived the special law

1 22

04

Q QF

r [N] (3.1)

To establish a basic law, we have to write the law in the form

12 1 22

04

QF Q E Q

r

generally as

F E Q [N] (3.2)

As the electric charge Q is a scalar and the force F a vector, the quantity E has to be a vector too. The vector form of equation 3.2 is

F E Q [N] (3.3)

The quantity E is called electric field strength. It is defined by its value and direction as a force acting in given point on a positive unity of electric charge +1As. For a negative

electric charge the force F and electric field strength have opposite directions.

So the expression for electric field strength at the location of the electric charge Q2

becomes


14

1

11 2

0

14

EQ

Er

(3.4)

The unit vector 11E shows into the direction of electric field strength

1E . The equation 3.4

represents a special law valid only for a lonely spherical electric charge, meanwhile the equation 3.3 represents a basic law.

Before we continue we have to explain some notions used in interpretation of vector fields.

3.1.2 Electric flux density

Because of the repelling forces between the electric charges of the same kind, all surplus electric charge is always located on the surface of electrodes. The electric charge density depends on the curvature of the electrode surface. Besides of this the electric charge density is influenced by all electric charges and metallic bodies in the vicinity.

When all electric charges and metallic bodies are so far away, that their influence on charge density may be neglected, the electric charge may be considered as a lonely electric charge. The electric charge density is for a lonely electric charge on an evenly curved surface given as

Q

A [As/m2] (3.5)

In all other cases the electric charge density has to be calculated as

limA

Q

A

(3.6)

For graphical presentation of vector fields we have to define the field lines and the flux tubes.

An electrical field line is an imaginary line, emerging from positive electric charge and ending on the corresponding negative electric charge. The tangent on the electric field line lies in the direction of electric field strength, a greater density of electric field lines means greater electric field strength.

An electric flux tube is an imaginary tube enclosing everywhere the same part of electric

flux el Q and on electrodes the corresponding electric chargesQ . For a lonely

electric charge on an evenly curved surface the electric flux density D is given by

elDA

[As/m2], (3.7)

for a lonely spherical charge as

24

QD

r [As/m2] in air (3.8)

Then the corresponding electric field strength in air is

Electrostatic field

15

2

0 04

D QE

r

[V/m] (3.9)

In isotropic dielectrics D and E are collinear vectors.

Besides spherical surface there are two evenly curved surfaces:

a) the cylinder and

b) the plaine.

Both differential quantities are for a lonely cylindrical charge given by

2 2

Q q l qD

A r l r [As/m] (3.10)

and

2

D qE

r [V/m] (3.11)

Both differential quantities for two opposite electric plane charges are given with the first part of equations 3.10 and 3.11.

The electrostatic field for an arbitrary space distribution of electric charges is completely defined if we are able to determine electric field strength in every point of space.

3.1.3 The electric voltage and the electric potential

When determined that the electric field strength is the basic differential quantity of electrostatic fields, it is a logic step to determine its corresponding integral quantities. At the same time we would like to prove that for derivation of them suffices to know the electrostatic field strength and the substance-geometric configuration of space.

There are to ways to move an electric charge:

a) We can move an electric charge (+Q) against the electrostatic field. The performed work is accumulated in increased energy content of electrostatic field.

b) The movable electric charge (+Q) can be moved by electrostatic field in direction of the electric field strength. The electrostatic field performs work on account of its energy.

For a negative electric charge the direction of movement is opposite.


16

+Q

dl

F

E

1

2

Figure 3.2 The movement of a movable electric charge in an electrostatic field

In point 1 of the electrostatic field with the electric field strength E the force F E Q is

acting on electric charge +Q. At the movement in direction dl the electrostatic field performs the work

d d d d cos dW F l Q E l Q E l Q U [J] (3.12)

the differential of electric voltage dU is defined a differential of the work performed by electrostatic field while moving positive unity of electric charge +1As in direction dl.

The voltage between the points 1 and 2 is

2

12

1

cos dU E l [V] (3.13)

For all three lonely electrical charges the corresponding electric voltages are:

2

1

12 20 0 1 2

d 1 1( )

4 4

r

r

Q r QU

r r r

[V] (3.14)

2

1

212

0 0 1

dln

2 2

r

r

rq r qU

r r

[V] in (3.15)

12

0

QU E d

A [V] (3.16)

The electrostatic field is also completely defined if we are able to calculate electric voltage between two arbitrary points of the electric field.

The analogy between electrostatic field and the earth gravitational field is helpful for physical understanding of electrostatic field. The potential energy of water mgh is besides of the mass m and the gravitational acceleration g dependent also from the distance h between the upper and lower level of water

1 2h h h [m]

In the earth gravitational field as the starting point for the heights , the sea level

0 0h [m],

has been chosen.

Electrostatic field

17

In the same way a starting point for electric potential may be chosen

0 0V [V],

the electric voltage between two points of electrostatic field can be expressed as difference of electric potentials corresponding points.

So electric potential of a point 1 in electrostatic field may be defined as the work performed by the electrostatic field by bringing a movable unit of electric charge +1As from point 1 to starting point 0.

So defined electric potentials for all three kinds of lonely electric charges are:

0

1

1 20 0 1 0

d 1 1( )

4 4

r

r

Q r QV

r r r

[V] (3.17)

0

1

01

0 0 1

dln

2 2

r

r

rq r qV

r r

[V] and (3.18)

1 0 1 0 1

0

( ) ( )Q

V x x E x xA

[V] (3.19)

The voltage between two points the can be expressed as the difference of electric potentials in corresponding points:

12 1 2U V V [V] (3.20)

The only difference between both analogies lies in them fact, that the staring point in earth gravitational field is fixed, meanwhile the starting point in electrostatic field may be arbitrary chosen. Only an earthed point has always the electric potential V0=0.

With this the first basic step in deriving basic laws for electrostatic fields has been accomplished.

From the cause – electric charge – we derived the excitation of dielectric – the electric flux density D - one from substantial property depending electrostatic field E – and at the end its corresponding integral quantity electric voltage U.

Q D E U (3.21)

At calculation of electrostatic fields also opposite direction of calculation may be used.


18

3.2 The influence of the substantial properties of dielectrics on an

electrostatic field

3.2.1 The polarisation of dielectrics and relative dielectric constant

Bringing a material dielectric into electrostatic field electrical forces are acting on electric charges inherent inside atomic structure, on protons in direction of electrical field strength and in opposite direction on electrons. As consequence of those forces the dielectric substance in electrostatic field is in an electrical strained state. It accumulates a potential energy called the electric energy. The phenomenon of electric strained dielectric is called polarisation of dielectric.

The evaluation of this phenomenon could be performed by following experiment. Two plain electrodes with surfaces A are put parallel at the distance d. In first case the dielectric between plates should be air and in second case another material dielectric.

d

U+ -

+Q0 -Q0

A

0

a)

d

U+ -

+Q0

A

b)

-Q0

Figure 3.3 A quantitative evaluation of polarisation

a) dielectric substance is air (or vacuum)

b) dielectric is another material dielectric

1. Experiment

We charge the electrodes with the source of a dc voltage U and then disconnect the source. Emptying electrodes over ballistic galvanometer we measure the electric charge Q0. Then we place between the plates another material dielectric (figure 3.3b), charge them with same voltage source U, empty them over ballistic galvanometer and measure the new electric charge Q.

The ratio of electric charges is

Electrostatic field

19

0: :1rQ Q , 1r (3.22)

2. Experiment

We charge the electrodes (Figure 3.3a) with a DC voltage U. Then we place between electrodes another material dielectric. The voltage between the plates dropped to value U'

The ratio of electric voltages is

´: 1: rU U , 1r (3.23)

The quantity r is called relative dielectric constant of material dielectric. It defines

how many times the dielectric constant of a material dielectric greater is then the dielectric constant of the vacuum.

The constitutional equation of electrostatic field in differential form, called also subsidiary Maxwell equation, has for a material dielectric the form

0 r

D

E (3.24)

It has a constant value and is called the absolute dielectric constant.

3.2.2 The constitutional laws of electrostatic field in the integral form

Capacitance

Let us do the next step. The law 3.11 we named the basic law of electrostatic field in differential form. What is the form of corresponding basic integral law? The result can be found in two ways:

a) By performing an experiment

b) By comparing their differential and integral forms.

When we measure by ballistic galvanometer the electric charge Q accumulated on electrodes at given electric voltage for a fixed configuration, we will get following results:

- at voltage U1=U on electrodes accumulated electric charge will be Q1=Q,

- at voltage U1=2U on electrodes accumulated electric charge will be Q1=2Q,

- at voltage U1=3U on electrodes accumulated electric charge will be Q1=3Q etc.

For a fixed configuration the ratio Q/U remains constant. It depend only geometrical configuration of electrodes and the dielectric constant of the isolator. It is a substance-geometrical characteristic of given configuration of electrodes, it is called capacitance C.

AsF

V

QC

U (Farad) (3.25)

A device with capacity as their most pronounced property is called a capacitor.


20

To the same result we arrive by the analogy between the differential ant integral for of basic laws. From corresponding integral and differential quantities

Q D

U E

C

(3.26)

of electrostatic field and known basic law, we get the corresponding integral law by exchanging all differential quantities in differential law the corresponding integral quantities. Therefore it is

D Q

CE U

In some applications it is more practical to use instead of capacity its inverse quantity called potential coefficient F:

1 VDaraf

As

UF

C Q (3.27)

From the known dependency between the electric charge Q and the corresponding voltage between the electrodes in equations 3.14, 3.15 and 3.16, the expressions for capacity of spherical, cylindrical and plane condenser can be directly written.

3.3 Kirchhoff's laws of electrostatic fields

All integral quantities can be expressed by integral of differential quantity a) Using a line integral of the differential quantity. b) Using a surface integral of the differential quantity. c) Using a volume integral of the differential quantity.

So already we have found

2

12

1

V

d [V]

d [As] and

d [As]

A

U E l

Q D A

Q V

(3.28)

d [J]e e

V

W w V (3.29)

This expression we will derive in the next chapter.

Electrostatic field

21

All those expressions are particularly interesting if integrals are performed over a closed loop, or a closed surface.

3.3.1 Kirchhoff's law of a closed loop

The starting point of a closed loop is already the end point and the potential difference in a closed loop equal zero:

d 0s

l

E l , (3.30)

as well as in the closed loop in an electrostatic field as in a closed loop in a capacitor circuit. Using the Stokes' theorem for a closed loop in electrostatic field gives

rot 0sE , (3.31)

or for a closed loop in a capacitor circuit

g CU U (3.32)

The first expression states that electrostatic field is not a curl field, and the second that in a closed loop in a capacitor circuit the sum of source voltages is equal the sum of voltage drops.

3.3.2 Kirchhoff's law of a closed surface

Calculating the integral of electric flux density in an electrostatic field over a closed surface A two results are possible.

The first result is

dA

D A Q (3.33)

It states, that the integral of the electric flux density D over a closed surface equals the algebraic sum of enclosed electric charges. The same is valid for the following volume integral

dQ V

Using the Gauss' theorem we got the differential equivalent for the equation 3.33:

div E

(3.34)

The electrostatic field has a scalar source.

The second result

d 0A

D A (3.35)


22

implies, there is no electric charge inside of the closed surface or their algebraic sum equals zero. Using Gauss' theorem the corresponding differential equivalent states that there is non scalar source of electrostatic field inside the closed surface.

div 0D (3.36)

The electric flux entering the closed surface equals the electric flux leaving it.

3.4 Locations of dielectrics and capacitors circuits

In this chapter we would like to prove that both Kirchhoff's contain everything necessary to asses and explain phenomena connected with:

a) configuration of dielectrics in an electrostatic fields

b) and solutions of capacitor circuits.

3.4.1 Configuration of dielectrics

In electrostatic field various configurations of dielectrics with different dielectric constants are possible. So is on the figure 3.4a the transition of electrostatic field, where the border of two electric flux tubes also the border of two dielectrics represented (tangential crossing). Using the law of closed loop on the path abcda gives

1 2d 0ab cd

abcda

E l E l E l (3.37)

because the sum of both scalar products being equal zero. The longitudinal parts of path have equal length ab=cd, so

1 2E E (3.38)

At tangential crossing of electrostatic field electrostatic field strength transverses continually.

Writing the equation 3.38 in form

1 2 1 1

1 2 2 2

D D D

D, 2 1 (3.39)

Electrostatic field

23

1E

2E2

1

ab

cd

Figure 3.4 Tangential crossing of electrostatic field on the border of two dielectrics

By tangential crossing on the border of two dielectrics the electric flux densities transverse in the direct proportion of their dielectric constants.

The second possibility represents the normal crossing of electrostatic field, where the border between two dielectrics divides the electric flux tube on two consecutive parts (normal crossing), shown on figure 3.5.

d 1A

2

1

1D2Dd 2A

Figure 3.5 Normal crossing of electrostatic field on the border of two dielectrics

As closed surface contains no electric charge, the law of closed surface has the form

1 1 2 2d d d 0A

D A D A D A (3.40)

Both normal surfaces are of the same size, so equation 3.40 becomes

1 2D D (3.41)

By a normal crossing n the border of two electrostatic field the electric flux densities transverse continually.

From

1 21 1 2 2

2 1

EE E

E (3.42)

It may be concluded that at normal crossing of electrostatic field the electric field strengths transverse in inverse proportion of their dielectric constants.

At inclined crossing of electrostatic field both components the normal and the tangential transverse in accordance with their crossing laws. So the law for an inclined crossing of electrostatic field becomes

1 1

2 2

tan

tan (3.43)


24

3.4.2 Capacitor circuits

Under notion solution of a capacitor circuit, the determination of electric charges and voltages on all capacitors of a capacitor circuit is meant.

The starting points for all methods used at the solution of a capacitor circuit are both Kirchhoff's laws of closed path and closed surface.

Connecting on two parallel capacitors with capacitances C1 and C2 an external voltage source U is shown on figure 3.6

C1

+ -

C2

+ -

+ -

U1

U2

Figure 3.6 Closed path in capacitor circuit

Determining the change of electric potential in shown closed path, we get

1 2 0U U ,

So it is

1 2U U U (3.44)

The voltages on two parallel capacitors are equal.

The electric charge accumulated in two parallel capacitors is

1 2 1 2( )Q Q Q U C C U C

Two parallel capacitors can be substituted by a substitute capacitor with capacitance 1 2C C C (3.45)

From the equation 3.44 we get

1 2 1 1

1 2 2 2

Q Q Q C

C C Q C (3.46)

The electric charges on parallel capacitors are in inverse proportion of their capacitances.

Electrostatic field

25

Using on two serial connected capacitors on figure 3.7 the Kirchhoff's law of closed surface, we get

1 2Q Q Q (3.47)

The sum of voltages on two serial connected capacitors is equal the voltage of the voltage source

1 21 2

1 2

Q QQU U U

C C C (3.48)

from there the rule for the determination of substitute capacitance of two serial connected capacitors can be found

1 2

1 1 1

C C C (3.49)

As the equation 3.47 may be written in the form

1 21 1 2 2

2 1

U CU C U C

U C. (3.50)

It is evident, that the voltages on serial capacitors are in inverse proportion of their capacitances.

Although the methods for solution of capacitor circuits may be interesting for all branches of electrical engineering, it is evident that both basic laws for a closed path and a closed surface include everything necessary for their solution.

3.5 The energy of electrostatic field and its energy density

The phenomena in electrostatic field are reversible. The work spent at moving of electric charge in electrostatic field is accumulated in its electric energy. But electric energy may be again changed into work or other kinds of energies.

C1

+ -

C2

+ - + -

U1 U2

Figure 3.7 Closed surface in a capacitor circuit


26

In elastic strained dielectric potential energy is accumulated, called the energy of electrostatic field.

We

dWe

Q

U

dQ

du

Figure 3.8 Characteristic of a linear capacitor

A capacitor is an energy container of electrostatic field energy. For nearly all dielectrics (the ferroelectrics are the only exceptions) their acting characteristic, shown on figure 3.8, are linear.

Bringing at voltage u between the plates of the capacitor an additional differential electric charge dQ, the voltage increases for differential du and the energy des electrostatic field for amount

d d deW u Q C u u (3.51)

When the electric charge has the value Q that corresponding voltage between the plates of the capacitor balanced the voltage of the connected voltage source, the electric energy stored capacitor has the value

2 2

0

d2 2 2

U

e

C U Q U QW C u u

C

[J] (3.52)

The quantity of electric energy stored in a unity of volume (1m3) is called electric energy density we. Being the electric energy a integral quantity, is electric energy density its corresponding differential equivalent. Exchanging each integral quantity in equation 3.52 with its differential equivalent, the equation for electric energy density becomes:

2 2

2 2 2e e

E D E Dw f [J/m3=N/m2] (3.53)

The dimension for electric energy density equals the dimension for electric force density. This has a strong physical meaning. When Faraday suggested the notion of electric field linens and electric flux tubes, somebody in the auditory ironically asked him for his perception of lines and tubes, which can not be seen. Faraday said that he imagined them as strained elastics that tend to shorten their length and broaden their cross-section!

Electrostatic field

27

The fact that on the border of two different dielectrics in electric field an electric force density is acting, which tends to shift the border in direction of dielectric with lower dielectric constant shows, how far reaching Faraday's notion of field lines and flux tubes has been.

3.6 The minimal set of basic laws foe electrostatic fields

A minimal set of basic laws for electrostatic field contains those basic laws that enable without an additional experiment derive all application of basic laws and al corresponding special laws. Their structural linking is shown on figure 3.9

Q

CU

D

E

2e

D Ew

2e

Q UW

Q D E U

Figure 3.9 Structural linking of basic laws in electrostatic field

Besides of basic laws incorporated in shown structural linking, the universal form of the Coulomb's law and both forms of Kirchhoff's law of closed path and closed surface have to be included in this minimal set of laws .


28

29

4 THE ELECTRIC CURRENT FIELD

Every truth always goes through three phases. In the first it is ridiculed, in the second it is indignantly opposed and in the third it is treated as obvious.

Schopenhauer Alfred

Contents

4.1 THE ANALOGY WITH ELECTROSTATIC FIELD 4. 2 THE DIFFERENCES BETWEEN THE ELECTROSTATIC AND ELECTRIC CURRENT FIELD 4. 3 THE ELECTRIC CURRENT MECHANISM IN METALLIC CONDUCTORS 4.4 THE OHM'S IN THE INTEGRAL FORM 4.5 THE JOULE'S LAW IN THE DIFFERENTIAL FORM 4.6 THE JOULE'S LAW IN THE INTEGRAL FORM 4.7 THE GEOMETRIC PLACING OF CONDUCTING SUBSTANCES AND RESISTOR CIRCUITS 4.8 STRUCTURAL CONNECTEDNESS OF BASIC QUANTITIES AND LAWS IN THE ELECTRIC CURRENT FIELD 4.9 THE MINIMAL SET OF BASIC LAWS IN THE ELECTRIC CURRENT FIELD


30

4.1 The analogy with the electrostatic field

The second basic part of Basic electricity contains the phenomena caused by electric fields in a space denoted with the substantial property – specific conductance .

At the same time we would like to stress, that structural connection of basic quantities in spaces characterised by all three electric substantial properties are of the same form, so all cognitions from electrostatic fields may be useful employed in the next two ones.

A space is called the electric current field if it fulfils two conditions:

a) The space contains movable electric charges, the substance is a conductor.

b) In every point of the space an electric field strength is acting.

As consequence the electric charges are moving.

To derive the basic laws of electric current field, we may start from first or second condition.

The electric current is universal defined as

( )( ) lim [A]

t

Q ti t

t, (4.1)

in case of DC currents as

[A]Q

It

(4.2)

The electric current density is universal defined as a current differential di through an orthogonal plane dA

2

A( ) lim

mA

ij t

A, (4.3)

in case of DC currents as

2

A

m

IJ

A (4.4)

The ratio between the electric field strength E as the causer and the electric current density j as the consequence depends on the substantial property of the space – the specific conductance :

As

Vm

J

E (4.5)

This relationship is known as the differential Ohm's Law.

The electric current field

31

On the length l of the conductor the change of the electric potential (the voltage drop) is

d [V]l

U E l , (4.6)

in case of a homogenous electric field as

U E l

This structural connectedness between basic quantities of the electric current fields is shown on figure 4.1.

I J E U

Figure 4.1 The structural connectedness between basic quantities of electric current fields.

The derivation could be performed in opposite direction. From given electric voltage, the electric field strength E acting on a conductor of length l is

V

m

UE

l (4.7)

For a known value of specific conductivity, the electric current density is

2

A

mJ E (4.8)

And the current in the conductor universal as

dA

I J A , (4.9)

in case of a constant cross-section as

[A]I J A (4.10)

It is interesting that historical in Basic electricity both directions of derivation were used. The structural connectedness between the electrostatic and electric current fields has the same form, even dimensional, only the unit As is exchanged with A.

4. 2 The differences between the electrostatic and electric current field

While it is beneficial to use the analogy between the electrostatic and the electric current field, we have also to be conscious of substantial differences between them. In electrostatic fields the permittivity is considered as a constant, a few ferroelectrics don't matter, but that isn't the case at conductors.


32

Although every substance with movable electric charges is defined as a conductor, because of different mechanism of electric current, there are very different kinds of conductors:

a) Metals, where the movable carriers of electric charge are free electrons.

b) Electrolytes, where the movable carriers of electric charge are positive and negative ions.

c) Ionized gasses, where the movable carriers of electric charge are free electrons, positive and negative ions.

d) Vacuum, where the movable carriers of electric charge are free electrons, created by thermionic, cold-cathode or photo emission.

e) Semiconductors, where by adding germanium, silicon etc. donors or acceptors the valence electrons or holes (defect electrons) are created.

For every kind of conducting substances we have to know the mechanism of electric current and special laws governing the movement of electric charges in the conductor. But there are some notions valid for all kinds of conductors.

So are the metals most important conductors in electric power systems, electrolytes in electrochemistry and galvanizing; ionized gases, vacuum and electric arc in circuit breakers; semiconductors in electronics etc.

A detailed treating of conductivity is a very specific knowledge. Therefore we will a little more thoroughly present only metals, though some of conclusions will be valid for all kind of conductors.

The electric field in conductors is not an electrostatic field. It isn't caused by immovable electric charges but by electric voltage sources: generators or galvanic cells. An electric voltage causes in conducting substance electric field strength, moving of electric charges – electric current.

An electric current field is completely defined by known electric field strength and substantial-geometric configuration of space.

4. 3 The mechanism of electric current in metallic conductors

The carriers of movable electric charge in metals are free electrons. For copper as one of the best metallic conductors we may suppose that all atoms lost the electron on the outmost orbit. This electron is as free electron moving in the copper's crystalline structure, its thermal energy being higher then its potential energy.

The mass and the electric charge of the electron are

31 1809,11 10 [kg] in 0,16 10 [As]em e


33

In 1 cm3 Cu there are 228,4 10n atoms of copper and the same number of free

electrons. The corresponding movable charge is:

28 18 100 3

As8,4 10 0,16 10 1,344 10

mn e

The force acting on free electron is:

0 eF e E m a [N] (4.11)

It gives the mass of the free electron the acceleration

02

m

se

ea E

m, (4.12)

into opposite direction of the electric field strength E

.

In a metallic conductor free electrons are performing two kinds of movement:

a) The movement behalf on its thermal energy,

b) and the movement caused by electric field strength.

The thermal velocities are very high, in copper at 200 C their values are approximately km/s100tv . Because of different directions and values they perform a

chaotic movement, in average with no shift. At collisions with crystalline structure of the conductor they exchange their kinetic energy and cause spreading of heath into the direction of temperature drop.

The electric field strength causes an additional translational movement of free electrons. Because the electric field accelerates the electron only between two successive collisions, their average “travelling” velocity depends on their average thermal velocity

tv , the acceleration a

of the electron and the average time of the free flight of electrons

.

The average in time covered distance s is

ts v (4.13)

The acceleration which acts during the free flight on the free electron is given by the equation 4.12. In the moment of the collision the maximal average travelling velocity of the free electron is

0maxp

e

e Ev a

m

(4.14)

The medium average travelling velocity of the free electron is

0

2p

e

e Ev

m

(4.15)


34

dl

i

A

E

pv

n

Figure 4.2 A volume element in an electric current field

Let choose in electric field a volume element in form of a cylinder with the volume:

d dV A l ,

its axis in the direction of the field. The movable electric charge in the unity of the volume dV is given with

0 0d d dQ n e V n e A l

If the movable electric charge should in the time interval dt leave the volume, the length dl should fulfil the condition:

d

dp

lv

t

The electric current I entering the plain A is

0 0

d d

d dp

Q li n e A n e A v

t t (4.16)

and the corresponding electric current density

20

02

p

e

n eiJ n e v E E

A m

(4.17)

The obtained equation is the differential form of the Ohm's law. There the expression

20

2 e

n e

m, (4.18)

is representing the specific conductance for metallic conductors. In principle this expression is valid for all kind of electric conducting substances. Specific conductance of a conducting substance depends on the fact how the product n is changing in dependence on electric field strength and temperature (or another quantity).

In pure metals the number of free electrons is practically independent from temperature, meanwhile the average free flight time is dropping. For a small temperature range the temperature dependence of specific resistivity is given with

0

1(1 ) (4.19)

And for a large temperature range with

2 3

0(1 ) (4.20)


35

So the carbon has for all temperatures up to 7290 C a negative temperature coefficient, from there on its temperature coefficient is positive. If that would not be the chance, the electric lamp with a carbon thread would not be possible.

Because every movement of electric charges may be considered as electric current, two other kind of electric current density may be mentioned.

In an electrostatic field, changing with time, the electric current through the capacitance is called the electric field current. From

( ) ( )Q t D t A ,

the electric field current density is given with

2

d ( ) d ( )( )

d d

d ( ) A

d m

p

p

Q t D ti t A J A

t t

D tJ

t

In gases because of the thermal convection the ions are moving and so they present a kind of electric current. Electrical charge in a differential of volume may be expressed as

d d dQ V A l

The electric current if this charge is moving, is given as

d d

( )d d

konv

Q li t A v A J A

t t ,

where

2

A

mkonvJ v

is called the convectional current density .

The practical influence of both mentioned density is so small, that in most practical cases it may be ignored. In the theory all three density may be considered.

4.4 The Ohm's law in the integral form

There are three possible ways to derive the Ohm's law in the integral form:

a) Using the already known laws in a simplest case.

b) Using the fact that every differential form of law has its corresponding integral form.

c) Using the knowledge that by analogy the structural dependence of basic laws have in all three substantial-geometric configurations in principle the same form.


36

For a straight conductor with a constant cross-section the electric field strength and the electric current density are given with

inU I

E Jl A

Inserting them into differential form of Ohm's law:

IJ I lA

UE U A

l

(4.21)

it can be transformed into corresponding integral form

SI A

GU l

(4.22)

The left side is the general form of the equation, the right side is valid for a straight conductor with a constant cross-section.

The same result may be gained by using the regulation b:

J I

GE U

(4.23)

or by using the analogy between the electrostatic and the electric current field.

Q I

C GU U

(4.24)

The analogy between the electrostatic and the electric current field is one of the reasons, why we prefer to use the conductivity G instead of its reversible quantity the resistivity R

1

ΩU

RG I

(4.25)

Because the length l of the conductor ant its cross-section A at a temperature change do not perceptible change, the temperature dependence of resistivity for metallic conductors for a small temperature range is given with

0

1(1 )R R

G (4.26)

and for a greater temperature range with

2 3

0(1 )R R (4.27)

Before the numeric methods for the solution of vector fields got the preference, the electrolytic tank was a very popular method for calculating capacitances from electric analogies we have in mind, when we are considering the duality of electrostatic and electric current fields.


37

4.5 The Joule's law in the differential form

Because of accelerating in electric field gained a free electron in the average time of the free flight additional kinetic energy

2 222

2 2

meel

e

m v ew E

m (4.28)

There is

m

e

ev E

m

the average maximal thermal velocity, gained by a free electron during the free flight. In the average time of free flight every free electron once collided with the crystalline structure of the conductor and transferred to it in a unit volume the following amount of kinetic energy

222

3

J

2 mel

e

new nw E

m (4.29)

The loss of electrical energy density is

22 2

3

W

2 me

w nep E E

m (4.30)

Taking into account the differential form of the Ohm's law

E

J E

We get all five expressions for the electric power density loss

2 22 2

3

W

m

E Jp JE E J (4.31)

The expression is in its form the corresponding analogous expression we derived in

electrostatic field for the electric field energy density ew . But there is a conceptual

difference between both phenomena. The phenomena in a electrostatic field are reversible, the phenomena in the electric current field are not.


38

4.6 The Joule's law in the integral form

The simplest way to derive the Joule's law in integral form is to employ the rule of the same form. From corresponding differential and integral quantities

2

3

[A/m ] [A]

[V/m] [V]

[Ωm] [Ω]

[S/m] [S]

[W/m ] [W]

J I

E U

R

G

p P

and the use of equation 4.31 we have all five forms of the integral Joule's law:

2 2

2 2 WU I

P U I G U R IR G

(4.32)

The integral form of the Joule's law can be derived also from the simplest application of equation 4.28 – from homogenous current field. By inserting in

P p V

both values

inp J E V A l

we get

WP p V J E A l U I

Both derivations are very simple.

4.7 The geometric placing of conducting substances and resistor circuits

The placing of conducting substances

Let us consider the analogous quantities of electrostatic and electric current fields

Q I

D J

C G

e

e

w p

W P


39

And the law of closed surface and closed path in electric current field,

d 1A

2

1

1J2Jd 2A

Figure 4.3 The law of a closed surface in electric current field

expressions for a normal crossing of electric current field are

1 21 2

2 1

orE

J JE

(4.33)

1E

2E 2

1

ab

cd

Figure 4.4 The law of a closed path in electric current field

The expressions for a tangential crossing of a electric current field are

1 1

1 2

2 2

aliJ

E EJ

(4.34)

In the case of the inclined crossing, the refraction law for the current field is

1 1

2 2

tan

tan (4.35)

Resistor circuits

Resistor circuits are in electricity very common and more important then capacitor circuits.

Applying the law of a closed path in a parallel connection of two resistors, we get


40

G1

+ -

U1

+ -

I1

G2 U2

+ -

I2

I

Figure 4.5 A closed path in a resistor circuit

1 2

1 2 1 2 1 2

1 2

1 2

1 1 2

2 2 1

( )

1 1 1ali

R

U U U

I I I U G U G U G G U G

G G GR R

I G R

I G R

(4.36)

Applying the law of a closed surface in a series connection of two resistors, we get

R1

+ -U1

R2

+ -U2

I

Figure 4.6 A closed surface in a resistor circuit

1 2

1 2 1 2

1 2

1 2

1 1 2

2 2 1

( )

1 1 1ali

G

I I I

U U U I R R I R

R R RG G

U R G

U R G

(4.37)

Using the analogy between the electrostatic and the electric current field it is evident that the usage of conductivity is preferable.

Electric powers are summed irrespective of connections of elements.

From electrostatic fields as well as from electric current fields it is evident that all laws concerning the spatial arrangement of dielectrics or resistors are as a whole


41

incorporated in both laws of closed path and closed surface, which are in their integral form known as Kirchhoff's laws of closed path (knot) and closed surface.

4.8 The structural connectedness of basic quantities and laws in the electric

current field

Because of the analogy between the electrostatic and electric current field has the structural connectedness between the basic quantities and the basic laws in both fields the same form. The inner circle contains the differential and the outer circle integral quantities.

I

GU

J

E

p J E

P I U

I J E U

Figure 4.7 The structural connectedness of basic quantities in the electric current field

4.9 The minimal set of basic laws in the electric current field

The Coulomb's law is valid as well in electrostatic field as in electric current fields. The same is the case with both laws of closed path and closed surface.

The law of closed path has the same form as in electrostatic field. In an electric resistor circuits the law of a closed path has the form:


42

gi Rj j ji j j

U U I R (4.38)

It is known as Kirchhoff's loop law.

The law of the closed surface could be in electric current field presented in two ways. The first presentation is based on the fact, that in a closed surface an electric charge can not be stored, the sum of electric currents entering the closed surface must equal the sum of electric currents leaving it

1

0n

jj

I . (4.39)

Currents leaving the surface are considered positive, the one entering it negative.

Maxwell expressed the same fact in a different way. If a current should leave a closed surface, the electric charge inside must decrease. The corresponding integral law has the form

d d

d dd d

A V

QJ A V

t t (4.40)

Using the Gauss theorem, we get the corresponding differential form

d

divd

Jt

, (4.41)

It is named by Maxwell as the continuity law.

The students should already here get acquainted that electrostatic field strength sE

,

except in transient phenomena, is not electric field strength causing movement of electric charges. In time dependent magnetic fields we will get acquainted with the

dynamic or induced electric field strength iE .

43

5 THE DC MAGNETIC FIELD

We may be able to teach a student one lesson in a day, but if we are able to arouse his curiosity his learning process will last his whole lifecycle.

Bedford Clay P.

Contents

5.1 BASIC LAWS OF MAGNETIC FIELDS FROM VIEWPOINT: CAUSE - CONSEQUENCE 5.1.1 THE MAGNETIC FIELD STRENGTH

5.2 THE MAGNETIC FLUX DENSITY 5.3 THE MAGNETIC FLUX Φ AND THE MAGNETIC FLUX LINKAGE Ψ 5.4 THE FORCES ON MOVED ELECTRIC CHARGES IN MAGNETIC FIELDS 5.5 MAGNETIC PHENOMENA IN MAGNETIC SUBSTANCES

5.5.1 THE EXPLANATION OF FERROMAGNETISM 5.5.2 THE MAGNETIZATION CURVE 5.5.3 THE CONFIGURATION OF MAGNETIC SUBSTANCES 5.5.4 MAGNETIC CIRCUITS


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One of the most logical explanation and interpretation of physical phenomenon is to start with the cause and then investigate its consequences. At such access the mutual interrelation of basic quantities in a region of physical phenomena will be obvious and easy to remember.

In an electric current field we have on one side the analogy with the electrostatic field and on the other side the electric potential difference, which causes in the conductor electric current and in consequence the magnetic field. Though we will at times use both of them, from the viewpoint of cause and consequence the analogy with the electric current field is more appropriate.

In the historical progress of electricity three accessions to present the theory of magnetic phenomenon are to be found: a) All magnetic phenomena are the consequence of a relative movement of electric charges. b) The starting point is the magnetic force acting on moving electric charges in magnetic fields. c) The starting point in the investigations of magnetic fields is the magnetic flux.

It is our firm belief that from methodical and pedagogical point of view the first accession is the best. We will try to prove this.

What are the advantages of this accession:

a) It enables a successive definition of all quantities and laws of magnetic fields from its cause – a moving electric charge – up to its consequence – the magnetic flux.

b) In the same way as the electrostatic and the electric current field are two dual fields, so it is the case also for the electric current field and the magnetic field. Both have the same structural connectedness of their quantities.

c) In the same way as electric voltages are responsible for all phenomena in electric current fields, the moving electric charges are the cause for all magnetic phenomena. So a moving electric charge (electric current) may be called “the magnetic voltage”. From this point of view the magnetic field strength may be considered as magnetic voltage used up at magnetisation of space along a unity length of magnetic line.

d) The dimensional structure of electric current field quantities and the magnetic field quantities is the same, the only difference are the exchanged places of units V and A.

All three accesses allow the derivation of all magnetic field laws, only in a different order. From a logical point of view the first kind of derivation seems the best. Already at the beginning the basic postulate of magnetic fields is put in the front: “All magnetic phenomena are always a consequence of moving electric charges!”

But it is necessary already at the beginning to stress the point that in this chapter only the phenomena caused by direct currents will be discussed. The time depending electric currents cause also time dependent magnetic fields, which have a recurrent influence on the current field

The DC magnetic fields

45

5.1 Basic laws of magnetic fields from the viewpoint: cause - consequence

5.1.1 The magnetic field strength

The magnetic voltage and the magnetic field strength have in the magnetic field the same duty as electric voltage and electric field strength in electric current field. The magnetic field strength represents the magnetic voltage drop spent at magnetisation of the space on a unit length of magnetic line.

But at the same time we have to point out the differences. There are two principal differences: a) An electric field line is a bonded line, connecting the point of higher electric potential and the point of lower electric potential. The magnetic field line is always a closed line encircling the current, causing the field, in the direction of a right hand screw. b) A current in N turns of a coil magnetizes the space N-times stronger then a current in one turn. Therefore the term for a magnetic voltage may be written as

− ,I at a current in only one conductor, (5.1)

− ii

I at an algebraic sum of currents and (5.2)

− I N at a coil with N turns (5.3)

In the other interpretations of magnetic phenomena the magnetic fields caused by a current in a straight round conductor and the magnetic field caused by a current in a tight wound up coil with N turns are of great importance. The shapes of both magnetic fields are with iron fillings represented on figure 5.1.

I

B

I

a) b)

Figure 5.1 The illustration of a magnetic field :

a) In the vicinity of a straight round conductor

b) In the case of a loose wound up coil


46

The magnetic filed of a straight round conductor is a circular plain field. The magnetic filed of a tight wound up coil is a nearly homogenous magnetic field.

On a string hanged permanent magnet will in a magnetic field point in the direction of the field. The gadget equipped with a magnetic needle, with a spring and a scale, is called a magnetometer (Figure 5.2). With this gadget the magnetic field in the vicinity of a straight conductor can be investigated. We first put the magnetometer in magnetic field of a straight current conductor in such position, that the spring of the magnetometer is not strained (the north pole of the magnetic needle point into point 0 of the scale – Figure 5.2a).

Then we start to turn the pedestal of the magnetometer aside from the beginning direction. Because the magnetic field is dragging the needle back, we have to turn the pedestal for the angle 900+ , to get the needle into perpendicular position to magnetic field, where the force acting on needle has the maximal value. The value of the angle is then proportional to the magnetisation of the space – the magnetic field strength H.

IN

S

0

HI

0

NS

H

r r

a) b)

Figure 5.2 Magnetometer

a) The spring is not strained

b) The force on the spring has the maximal value

With the magnetometer we are able to form following conclusions:

- The north pole of the magnetic needle shows in the direction of the magnetic field, when the magnetic needle shows into point 0 of the scale.

- The direction of the magnetic field is orthogonal to radius r connecting the Axis od the conductor with the given point of space.

- The direction of the magnetic field is given by the right screw. When the direction of the current and the direction of the right screw coincide, the direction of the magnetic field is given with the direction of the right screw.

- The additional angle has on the same distances r the same value, therefore has the strength of magnetisation – the magnetic field strength H the same value (but a different direction). The magnetic field caused by a current in a straight round conductor is a circular plain field.


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- On the doubled distance 2r has the magnetic field strength the halved value (the additional angle is 2/ ). The magnetic field strength is inverse proportional to the distance r.

- The magnetic field strength is direct proportional to the current in the conductor. By double value of the current the magnetic field strength has the double value. If the direction of current has changed, the direction of the magnetic field strength ha changed too.

By analysis the following conclusion is evident

A

m

IH k

r (5.4)

The magnetic field of a straight round conductor is a circular plain field, so is

1

2k

The magnetic field strength

A

2 m

IH

r (5.5)

The magnetic voltage is given as a line integral of the magnetic field strength H

along a

closed path l

:

dl

H l (5.6)

It s called the Ampere' law of the magnetic field. On the same distances from the straight round conductors has the field strength constant value. Outside of a straight round conductor has the magnetic field strength the value:

2

d 2o

H r H r

, (5.7)

its value is given with 5.5.


48

I'

I

x

R

Figure 5.3 The magnetic field inside of a straight round conductor

In the interior of a straight round conductor a magnetic line encircled only a part of the current

2 2: :I I x R (5.8)

The magnetic field strength at the distance x from the axis is

2

A

2 2 mx

I I xH

x R (5.9)

In the interior of a long tight wound up coils the magnetic field strength in the interior of the coil is practically homogenous

A

m

I NI N H l H

l (5.10)

The determination of the magnetic field strength for a straight round conductor and a long tight wound up coil belong into the obligatory set of laws for stationary magnetic fields. The determination for all other forms of conductors belongs into a broader frame of magnetic fields.

5.2 The magnetic flux density

One of essential difference between the electrostatic and magnetic fields is:

a) In the electrostatic field the electric field strength E

is the real and the density of

electric flux D

the auxiliary electric quantity. The force acting on the electric charge is determined by the electric field strength.


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b) In the magnetic field the density of electric flux B

is the real and the magnetic field

strength H

the auxiliary magnetic quantity. The force acting on the moving electric charge is determined by the magnetic flux density.

We could state that the electric current is the cause and the density of magnetic flux the consequence of magnetising matter in a point of space. But for an essential understanding of all magnetic phenomena the recognition, that the density of the magnetic flux is always constituted by two components

0B B J [Vs/m2=T] (Tesla), (5.11)

is very important. The component

0 0B H , (5.12)

is introducing that part of the magnetic flux density, that the magnetic field strength would cause in vacuum (empty space). The constant

70

Vs4 10

Am

Is the substantial characteristic of empty space and is called the permeability of empty space.

The second component

0 0J B H , (5.13)

is called the vector of magnetic polarisation J

. It defines the contribution to the common magnetic field density, caused by moving electric charges in atomic structure.

The factor is called magnetic susceptibility and represents the multiplier of 0B

- it

shows how many times the contribution of moving electric charges in atomic structure

is bigger than 0B

.

The joint magnetic flux density is then

0 0(1 ) rB H H H , (5.14)

the quantities rand are the absolute and relative permeability.

The sign means, that the vector of magnetic polarisation may be added or subtracted.

From technical point of view, all substances belong into two groups:

a) Nonmagnetic substances, where the vector of magnetic polarisation has so small values, that it is neglected. The magnetic flux density is given with equation 5.13.

b) Magnetic substances, where the vector of magnetic polarisation is always positive, its relative permeability may reach up to 106.


50

To calculate the density of magnetic flux in nonmagnetic substances it is enough to know the magnetic field strength (5.14).

To calculate the density of magnetic flux in magnetic substances, the magnetisation curve )(HfB of the magnetic substance is to be known

From the analogy of electrostatic and magnetic fields two interesting cognitions result.

Because of

E [V/m] and H [A/m]

9

0

10

36 [As/Vm] and 7

0 4 10 [Vs/Am],

are in dimensions of electrostatic and correspondent magnetic quantities only the places of A and V exchanged.

The second interesting item is the mutual interdependence of electrostatic and magnetic substantial properties. From

16

9 7

0 0

1 36 1 km9 10 300 000

10 4 10 s

It is evident that electric and magnetic field propagate in vacuum with velocity of light.

5.3 The magnetic flux Φ and the magnetic flux linkage Ψ

The magnetic flux density B

is a differential quantity of magnetic field. Its corresponding

integral quantity is the magnetic flux through a surface A

, generally defined as

dA

B A [Vs] (5.15)

or in a case of a homogenous magnetic field and orthogonal surface as

B A B A [Vs] (5.16)

In a case of time dependent magnetic fields it is important to distinguish between the magnetic flux through one loop (given with 5.15) and the magnetic flux through a coil.

From the figure 5.1b it is evident that a turn in a coil is not linked by all magnetic lines. Generally individual turns of a coil are linked only with a part of the whole magnetic flux

i ik , 1ik

where is the total magnetic flux in the ki the linkage factor of the i-th turn. Magnetic flux linkage of the coil is then given as the sum of magnetic fluxes linked with individual turns, for a coil with N turns as


51

1

i N

ii

k k N [Vs] (5.17)

The linkage factor of the coil is the average value of the linkage factors of turns

1

i N

ii

k

kN

(5.18)

The derivation sequence of basic quantities of the magnetic field, from the cause – the moved electric charge – up to magnetic flux (magnetic flux linkage) is entirely analogous as in electrostatic fields, the only difference are the numbers N of the turns.

( ) ( )I H B

Both quantities in parenthesis are connected with the coils.

In practical problems of calculations of magnetic field the direction of solutions may go in both directions.

Although the derivation of equations was made for direct currents, respectively for evenly moved electric charges, they are valid as well for instantaneous currents or unevenly moved electric charges.

The complete structural dependency of basic quantities and basic laws of magnetic fields we are not able to present, for following reasons:

a) In electrostatic fields the substantial property dielectrics was considered as a constant. A few ferroelectrics, with )(Ef are in electrostatic fields of little

importance. In magnetic fields the causes as well as consequences of ferromagnetic substances have to be well known and taken into account.

b) The time dependent magnetic fields have a return influence on current fields and through them on their shape.

Therefore it is reasonable to present the entirely structural connectedness of basic magnetic quantities not until we are able to describe the phenomena caused by time dependent magnetic fields.

5.4 The forces on moved electric charges in magnetic fields

Amper defined the magnetic flux density with the experiment shown on figure 5.4. With a current weighing apparatus he measured the force acting on a straight current conductor in a homogenous magnetic field.


52

B

S

F

F

N

I

Figure 5.4 Ampere's experiment with a current weighing apparatus

( )F I l B [N] (5.19)

We defined magnetic flux density in a different way, so we will use this equation for calculation of the magnetic force.

Let us transform the expression into

Q

I l v t Q vt

,

The resulting form is valid for direct currents, but also for instantaneous currents I, and Q(t), the equation 5.19 becomes

( )F Q v B [N] (5.20)

To determine the direction of the magnetic force following rules may be used: a) The vector product rule (figure 5.5a), b) The rule of the left hand (figure 5.5b), c) The Faraday's rule, that magnetic lines try to shorten their length and enlarge their mutual distance (figure 5.5c).

If the conductor with the current I1 is located in a magnetic field of another straight conductor with current I2, the force with their they attract or rebound each other at the length l is:

0 1 21 2

2

I I lF I l B

a [N] (5.21)

With the equation the trajectory of an electron crossing magnetic field cold be calculated, one of the application of basic laws of magnetic fields.


53

F

B

l

B

I

F

I

a) b)

N

S

N

S

N

S

I I I

FF

c)

Figure 5.5 The determination of the direction of the magnetic force:

a) the vector product rule;

b) the rule of left hand,

c) Faraday's rule.

12F

12B

1l

aI1

21F

21B2l

I2

i

j

k

z

y

x

Figure 5.6 Magnetic force between two parallel conductors


54

5.5 Magnetic phenomena in magnetic substances

5.5.1 The explanation of ferromagnetism

Already at the introduction into magnetic fields we made the statement, that all magnetic phenomena are caused by moved electric charges, moving relative to observer. So also the permanent magnetism has to be the consequence of moved electric charges in atomic structure.

There are three kinds of moved charges in the atomic structure: a) The chaotic movement of protons in atomic nuclei. b) The circulation of electrons in orbits. c) The rotation of electrons around their axis – spin.

The chaotic movement of protons in atomic nuclei has no measurable effects.

The electron in the atomic orbit behaves in a magnetic field like a spinning top. At this movement it encircles the magnetic field in such direction, that his magnetic field weakens the magnetizing field. This phenomenon is called diamagnetism. The vector of magnetic polarisation has a negative sign, but its susceptibility is so small, that for all technical purposes it is allowed to use 1 ( 0)r .

At rotation along its orbital, every electron rotates around its axis. So it represents a small magnet, that tends to take the position of magnetizing field and strengthen it. This phenomenon is called paramagnetism. The vector of magnetic polarisation has a positive sign, but its susceptibility is again so small, that for all technical purposes it is

allowed to take 1 ( 0)r . If there are on the orbit both electrons, they rotate in

opposite directions and the paramagnetic effect can not occur.

In some substances both effects are present, (the paramagnetic is prevalent). But for all technical purposes both kind of substances are non-magnetic substances.

There are no other movable in atomic structure, therefore it is necessary to give a clear answer, what are the reasons for ferromagnetism?

The structure of natural ferromagnetic substances is polycrystalline, they are formed by a multitude of monocrystals. Inside of a monocrystal two opposite forces are acting:

a) Magnetic forces are trying to align all spin magnets inside of monocrystal in direction of one of the magnetic axes. So has every iron monocrystal six magnetic axes, their positions as the sides of a cube, two of them have opposite direction.

b) Thermal forces oppose the magnetic. At temperatures above Curie's temperature the thermal forces prevail and destroy all magnetic history.

At temperatures below Curie's temperature the magnetic forces prevail and an iron monocrystal spontaneously magnetises into six Weiss' domains, each of them a tiny magnet in direction of one of magnetic axes.


55

Every monocrystal has an even number magnetic axis, two of them in opposite direction. Every Weiss domain has the same number of atoms, so in absence of an outside magnetic field a mono crystal is magnetic neutral.

On the border of two Weiss domains spin magnets continually change their direction from one magnetic axis into the adjacent one. The zone of the transitions is called

Bloch's barrier, at a direction change of 900 its width is 925 10 m and at a direction

change of 1800 its width is 935 10 m .

5.5.2 The magnetization curve

A ferromagnetic substance without a magnetic history is magnetic neutral. For a thorough understanding of ferromagnetism the phenomena at a progressive increase of an outside magnetic field strength have to be considered.

In an iron monocrystal under the influence of an outside magnetic field in succession three processes do occur: a) An elastic shift of Bloch's barrier. b) The overturn of a whole Weiss domain into direction of the magnetic axis, nearest to the direction of the outside magnetic field strength. c) The elastic turn of Weiss domains.

If the outside magnetic field strength is increasing, the Bloch's barrier begins to shift and the Weiss domain nearest to the direction of the outside magnetic field strength is increasing. If the outside magnetic field disappears, the Bloch's barrier returns into starting position. The shifting of Bloch's barrier is a reversible process. The result of the shift is a resultant magnetic field.

A Weiss domain is in its starting position in an equilibrium with the lowest energy content. But it has also five other equilibrium positions, but with a higher energy content. If the outside magnetic field is strong enough, the whole Weiss domain turns over into magnetic axis nearest to the direction of the outside magnetic field strength. This process is called turnover of Weiss domains. Because the new position is again an non-equilibrium point, the Weiss domain will persist in it. Though a turnover of a Weiss

domain signifies a jumping increase of vector of magnetic polarisation J , because of an

immense number of monocrystals it seams like a continuous process. The turnover of Weiss domains is a non-reversible process. The process is concluded when all Weiss are in the position of magnetic axes nearest to the direction of the outside magnetic field. This point of the magnetization curve is called saturation point, the corresponding

vector of magnetic polarisation maxJ has the biggest possible value.

The final directions of magnetic axes are not always in agreement with the direction of the outside magnetic field strength, therefore will the spins magnets at still higher value of the magnetic field strength try to turn elastic into its direction. This process is called the elastic turn od Weiss domains and is again reversible.


56

c

0

a

b

elastic turn of Weiss domains

turnover od Weiss domains

elastic shift of Bloch's barriers

Bm

Br

Hk1-Hm1

Bm

Hm1

B

H

Figure 5.7 Magnetization curve and hysteresis loop

If we start in point of saturation to decrease the magnetic field strength, because of persistency of Weiss' domains the magnetizations curve describes the hysteresis loop. The hysteresis loop contains three characteristic points:

a) The saturations point, where the increase of magnetic field strength does not provide any increase in vector of magnetic polarisation.

b) The point of he remained magnetic flux density. The residual magnetic flux density, after the exciting magnetic field has been removed,

c) The coercive force. The reverse magnetic field strength required to bring the residual magnetic flux density to zero.

The area of hysteresis loop

3

Jd

mB H

,

is equal the amount of work necessary for the turnover of Weiss' domains in one cycle of hysteresis per unit volume.

Considering the shape of magnetization curve, all magnetic substances may be classified in two big groups:

a) The soft magnetic substances have a very small area of hysteresis loop. They are used in time dependent magnetic fields.

b) The hard magnetic substances have a very big area of hysteresis loop. They are used for permanent magnets.


57

Ferromagnetic substances

Fe

Fe

Co

Ni

a/r

Ft

Figure 5.8 Dependence of intern electromagnetic forces on ratio a/r

Curie temperature

At temperatures over Curie point the thermal energy destroys the magnetic structure of ferromagnetic substance. But if the temperature drops below the Curie point, the magnetizing forces again prevails, in spontaneous magnetization the Weiss domain will be again formed.

The value of magnetic force is a function of the ratio between the middle distance of two neighbouring atoms a, and the radius r of the orbit with only one electron.

The figure 5.8 represents the relationship of the three natural ferromagnetic substances: iron (Fe), cobalt (Co) and nickel (Ni) and in the table 5.1 the Curie temperatures for four natural ferromagnetic substances.

Table 5.1 The values of Curie temperature and saturated magnetic flux densities for natural ferromagnetic substances

Element Curie temperature [0C] Jsat [T] Iron (Fe) Cobalt (Co) Nickel (Ni) Gadolinium (Gd)

770 1131 358 17

2,2 1,8 0,64 2,5

The Weiss domains have more points of the potential energy balance, therefore they are sensitive to all technological processes containing blows, damages of crystal structure and temperature changes. All of them may influence on the properties of magnetic substances.


58

5.5.3 The configuration of magnetic substances

The crossing of the magnetic fields

Considering analogue quantities of current and magnetic fields

m

I

J B

U

R R

the rule of the closed path and the closed surface in magnetic field, we get for the normal crossing the rule

1 2

1 2

2 1

aliH

B BH

(5.22)

d 1A

2

1

1B2Bd 2A

Figure 5.9 The law of the closed surface

1H

2H2

1

ab

cd

Figure 5.10 The law of the closed path

For the tangential crossing the rule

1 1

1 2

2 2

aliB

H HB

(5.23)

The rule for the askew crossing of the magnetic field is

1 1

2 2

tan

tan (5.24)


59

In the case of the crossing between magnetic and non-magnetic substances, because of

1r

the magnetic field is practically perpendicular to the magnetic substance surface.

Ohm's law of magnetic circuits

The magnetic circuit is a substantial-geometric arrangement of magnetic and non-magnetic elements linking magnetic voltages and magnetic fluxes through individual elements of magnetic circuit.

The ratio of magnetic voltage and magnetic flux through element id defined as magnetic resistance+

1m

m

lR

A G (5.25)

Meanwhile the magnetic resistance of a non-magnetic element is constant, the magnetic resistance of the magnetic element I a function of magnetisation curve.

The integral law of a closed path in a magnetic circuit (figure 5.11) is

1 2

1 2

(5.26)

Rm1

+ -

+ -

+ -

1

Rm2 2

1

2

Figure 5.11 A closed path in a magnetic circuit

For a series connection of magnetic elements the law of closed surface (figure 5.12)

1 2

1 2

(5.27)

+ - + -

Rm1 Rm2

1 2

Figure 5.12 A closed surface in a magnetic circuit


60

It is evident that at employing analogy between the current and the magnetic field the usage of magnetic resistance is favourable.

The method of solving magnetic circuit with magnetic elements is the same as solving resistance circuits with non-linear resistances.

5.5.4 Magnetic circuits

Any closed conducting path in an electric current field is called an electric circuit, where a voltage source drives an electric current. Any closed path in a magnetic field is called a magnetic circuit, where a magnetic voltage drives a magnetic flux. There are many similarities and at the same time many dissimilarity between them. We must be well aware of both of them, if we try to transfer the experiences from electric circuits to magnetic circuits.

We already used the analogy between electric current field and magnetic field at the introduction to magnetic field. We stated there that an electric voltage source in electric current field is analogous to electric current (or ) as a magnetic voltage source in a magnetic field.

But that is only a part of the analogy. It goes even further:

a) The currents through in series connected resistances are even, the voltage drops on them are in the same ratio as resistances. The magnetic fluxes through in series connected magnetic elements are even, the magnetic voltage drops on them are in the same ratio as their magnetic resistances.

b) The voltage drops on parallel connected resistances are even, the ratio of currents through them is inverse ratio of their resistances. On parallel magnetic paths the magnetic voltage drops are even, in the substance with greater permeability we have higher magnetic flux density.

At the same time we must be aware of their differences. The most important of them includes the fact, that there are no magnetic isolators. We only know good and bad magnetic conductors.

Magnetization curve and magnetization characteristic

Magnetic circuits contain linear and non-linear elements. In a linear magnetic element the magnetisation curve oB H and integral correspondent mR are both linear.

Therefore it is of no importance which of them is given.

On a non-linear magnetic element the magnetization curve ( )B f H as well as it

corresponding integral equivalent the magnetization characteristic ( )f are both

non-linear. If two magnetic elements are connected in series, the magnetic flux through both of them is the same and the magnetic voltage drop the sum of both. For two parallel


61

connected magnetic elements, the magnetic voltage on both of them is the same, the incoming magnetic flux but the sum of magnetic fluxes through parallel branches.

Let consider a simple magnetic circuit containing a magnetic core with an aerial slot. The magnetization curve of the magnetic core is known. The procedure to obtain the magnetization characteristic of given magnetic circuit is following:

lFe

lae

AFe

Figure 5.13 A simple magnetic circuit with an aerial slot

We choose a number of possible magnetic fluxes i . The magnetic voltage drop in the

aerial slot is

0

, ,aeiiaei aei aei aei ae

ae

BB H H l

A

(5.28)

and the magnetic voltage drop in the magnetic core we derive from

, ( ),iFei Fei Fei Fei fei Fe

Fe

B H f B H lA

(5.29)

The necessary magnetic voltage i for the given value of the magnetic flux i is the

sum of both in series connected elements

i aei Fei aei ae Fei FeiH l H l (5.30)

Now we are able to plot the magnetization characteristic for given simple magnetic circuit (figure 5.14)


62

zrFe

Figure 5.14 The magnetization characteristic for a simple magnetic circuit with an aerial slot.

6 THE INDUCED ELECTRIC FIELD

An induced electric voltage can be realized in an only once chosen, conducting loop, when the magnetic flux through it is changing with time.

Faraday Michael

Contents

6.1 The time dependent magnetic fields 6.1.1 The electromagnetic induction 6.2 The Magnetic flux linkage 6.3 The own and the mutual inductances 6.4 Interdependence of electric currents and electric voltages on passive elements of electric circuits 6.5 The energetic considerations in magnetic fields

6.5.1 The magnetic field energy 6.5.2 The magnetic field energy density 6.5.3 The forces on borders of flux tubes

6.6 The structural linking of basic magnetic field quantities


64

Though all rules valid for direct current magnetic fields are valid also for magnetic fields

caused by time dependent electric currents, we should not forget, that time dependent

magnetic fields have a recurrent influence on current fields.

When explaining the force on a moving electric charge in a magnetic field

( )F Q v B ,

The similarity with the force on an electric charge in electrostatic field

sF Q E

offers the conclusion that an electric charge Q a crossing of magnetic field has the feeling as being exposed to an electric field with electric field strength

V

miE v B (6.1)

Because this electric field strength is a consequence of moving of an electric charge in a magnetic field it is called the dynamic or induced electric field strength. So an electric field strength in a point of space may be the sum both components.

V

ms i sE E E E v B (6.2)

The operation of majority of electric machines and devices is bound on time dependent electric and magnetic fields. Even the stationary electric and magnetic fields have to be built up, therefore it is necessary to learn the time depending magnetic fields.

6.1 Time dependent magnetic fields

6.1.1 The electromagnetic induction

The voltage between two points of induced electric field is called the induced voltage Ui. Because in the historical development the most mistakes were made just in the field of electromagnetic induction, it may be reasonable to stress its essential characteristic so exactly stated by Faraday:

“An induced voltage can be realised in one unique chosen conducting loop, if the magnetic flux through loop is changing with time!”

From the next two experiments it will be evident, that magnetic flux through a loop (or coil) may change for two reasons.

The inducted electric fields

65

a) The conducting loop (or coil) does not change, but the magnetic flux through it changes with time.

b) The magnetic flux does not change with time, but a part of conducting loop is moving.

Though in both cases the end result is a time dependent magnetic field, because of two different causes one of them sometimes is called a timely dependent and the second one a locally dependent magnetic field.

The experiment shown on figure 6.1 presents a timely dependent magnetic field.

glider movement coil 2 coil 1

Figure 6.1 The electromagnetic induction caused by a timely changing magnetic field

Moving the glider on the voltage divider the magnetic flux in the first coil is changed, the coupled magnetic flux in second coil is changed too .l

The experiment shown on figure 6.2 presents a locally dependent magnetic field.

+

-

direction of movementN

S

+

-

N

S

Figure 6.2 The electromagnetic induction caused by a locally changing magnetic field


66

The electromagnetic induction is caused by locally movement of the permanent magnet

Generally may be caused as well by a timely as locally change of magnetic field. At determination of induced voltage we must determine its instant value and its sign. The sign may be determined by Lenz's rule.

“The induced voltage has always a sign, which opposes its cause! The Lenz's rule is only electromagnetic equivalent of the mechanical rule of action and reaction.

The cross (traverse) voltage

The determination of induced electric field strength when a conductor is crossing a magnetic field is the most easily understood access in understanding induced electric fields.

Moving electric charges through a magnetic field in such a way that we are crossing magnetic field lines on charges are acting magnetic forces (6.1). If the moving substance has no movable electric charges, the polarisation occurs. The induced electric field strength is in comparison with the electrostatic very small, so polarisation may be neglected.

From the expression foe induced electric field strength it is evident, that it is given by vector product of charge velocity and magnetic field density

iE v B

and its value by

siniE v B (6.3)

Its direction is given by right screw rule, when the vector v is turned into position of

vectorB

iE

v

iE

B

Figure 6.3 Sketch for defining the value und the direction of electric field strength


67

When the magnetic field density iE and the velocity v are perpendicular to each other,

the induced electric field strength iE has its maximal value ( 090 ).

On the length differential dl along the conductor the increase of induced electric voltage is given with the scalar product

d d cos di iU E l v B l (6.4)

When the vectors E and dl are collinear the increase of induced voltage iU has its

maximal value ( 00 ).

So in nearly all technical applications of traversed induced voltages are the angles 090 and 00 and induced voltage in a rod of length l

VipU v B l (6.5)

To define the direction of traversed induced voltage besides of vector product also the rule of the right hand is very useful:

Poleg pravila vektorskega produkta za določitev smeri prečkalne inducirane električne poljske jakosti lahko uporabimo tudi pravilo desne roke

v

BiE

Bv

Ei

Figure 6.4 The vector product rule and the right hand rule for defining the direction of traversed induced voltage.

The generalised rule of induced voltage

The traversed induced voltage is the consequence when a part of the conducting loop is crossing magnetic field (locally dependent magnetic field). Therefore we have to derive a generalised rule of induced voltage, which would include both possible causes.

If we post the right hand into magnetic field in such a way that magnetic flux density is entering into extended palm, the extended thumb is in direction of movement, then the extended fingers are in direction of induced electric field strength (figure 6.4).


68

For this purpose let examine the derivation od induced voltage in a conducting loop, where a part of the loop is moving (figure 6.5).

v

BiE

dx

dA

l

Figure 6.5 The derivation of induced voltage in a conductive loop if a part of loop is moving

The movable part of the conductive loop l is moving with velocity v

in given direction. The induced electric field strength has the direction, which can be determined by the vector product rule, by right hand rule or by Lenz's rule.

The magnetic flux through the loop increased for

d d d d d diB A B A B l x B l v t U t (6.6)

From there is the value of the induced voltage

d

diU

t V (6.7)

The direction of induced voltage is positive, if the current it caused encircled the magnetic flux by right screw. Because this is not the case, we must take

d

diU

t (6.8)

By moving the rod in opposite direction, the magnetic flux decreases, the electric current the induced voltage would encircle it by right screw. Therefore we must take

d

diU

t (6.9)

That but means just the same. The expression for the induced electric voltage has the generalised form

d

diU

t (6.10)


69

Because of

B A

The magnetic flux may change by changing the magnetic flux density or by changing its area

d d d d

d d d di

B A B AU A B A B

t t t t (6.11)

The first part includes the change of magnetic flux density and the second part the change of the loop.

The negative sign includes the substance of Lenz's rule:

The electric current caused by induced voltage has always such direction to oppose the change of primary magnetic field – action and reaction.

6.2 The Magnetic flux linkage

We already defined magnetic flux linkage as the sum of magnetic fluxes through turns of a coil. Because we have not jet mentioned induced voltages, we could not explain, why the magnetic flux linkage is so important.

N

S

N

Figure 6.6 Magnetic flux linkage with its own coil

The joint induced voltage in a coil equals the sum of induced voltages in individual turns. The magnetic fluxes through individual turns are different, so are different also the induced voltages.


70

Magnetic flux linkage represents the substitution of a coil with N turns by a single turn, where the change of magnetic flux linkage will cause an induced voltage equal the sum of induced voltages in the turns of the coil.

Magnetic flux through the turn i is

Vsi ik (6.12)

The linkage factor ki 1 tells how large part of the entire magnetic flux is linked with the turn i. The average linking factor of the coil is

1

N

iik

kN

(6.13)

The magnetic linkage is

Vsk N (6.14)

The entire induced voltage in the coil is

d

Vd

iUt

(6.15)

At induced voltages we have do distinguish between the induced voltage of own induction, where the induced voltage appears in the same coil as the current which creates magnetic field and the induced voltage of mutual induction, where the current which creates the magnetic field is in one coil and the induced voltage in another coil.

1N1

12

N2

I1

Figure 6.7 Magnetic flux linkage with another coil

To determine the induced voltage of mutual induction the magnetic flux linkage with another coil has to be known. The linkage factor between two coils

1212

1

1k (6.16)

determines the linked magnetic flux 12 with second coil and the magnetic flux linkage

with second coil

12 12 1 2 Vsk N (6.17)


71

Meanwhile the induced voltage of own induction always opposes its cause, the direction of induced voltage of mutual induction depended besides of the direction of currents in both coils also of direction of turns of both coils.

1N1

12

N2

I1

I2

Figure 6.8 Two inductive opposite coupled coils

To take care of this possibility, we denote one end of both coils. If the current enters both coil in denoted connection (or in not denoted one), then the linked flux is added to the own one, both induced voltages have the same sign. If one current enters denoted connection and the other coil in not denoted one, then the own and linked flux subtract, both induced voltages have opposite signs. In first case we say, that the coils are coupled in the same direction, in second case they are coupled in opposite direction.

6.3 The own and the mutual inductances

The inductance could have been mentioned already at magnetic fields, but their importance can be entirely understood only after induced electric fields.

The inductance is the integral substantial-geometrical characteristic of magnetic fields which corresponds to their differential quantity – the permeability. It is defined as

LI

Vs/A=H (Henry) (6.18)

the quotient between the consequence – magnetic linkage and its cause - the electric current I.

Meanwhile the capacitance C and the conductance G are always bound on one element, in magnetic fields there are:

a) Own inductances, substantial-geometric properties, where the cause I and consequence magnetic flux linkage are bound to the same coil and


72

b) Mutual inductances, substantial-geometric properties, where cause I is in one coil and the consequence coupled magnetic linkage in another coil.

The own inductance denoted with Li, where i is he number of the coil.

Generally is the mutual inductance denoted with Lij, where i are the number of the coil with caused current and j the number of the coil with coupled flux linkage. Only in the case of only two inductive coupled coils, the mutual inductance is denoted with M

12 2112 21

1 2

M L Li i

(6.19)

The essential difference between electrostatic and magnetic field is that we distinguish between magnetic and non-magnetic substances, that is between linear and non-linear magnetic substances.

For a coil with a non-magnetic core, the induced voltage is given as

d

( ) Vd

i

iu t L

t, (6.20)

in the case of a coil with a magnetic core, because of L=L(i), as

d d( ) d( ) V

d d di

L iu t L i i L i

t i t (6.21)

6.4 Interdependence of electric currents and electric voltages on passive

elements of electric circuits

Capacitor

From the integral rule of capacitors

( ) ( )cQ t C u t

are the instantaneous values of electric current and voltage

d ( )

( )dc

C

u ti t C

t and

0

1( ) ( ) d (0)

t

C C Cu t i t t uC

(6.22)

Resistor

On resistor are the values of electric current and voltage linked with Ohm's law:

( ) ( )R Ri t G u t in ( ) ( )R Ru t i t R (6.23)


73

Coil

Using the integral rule of ideal coil

( ) ( )t L i t

We can the induced voltage represent as a driving voltage or as an inductive voltage drop.

In first case it is

d ( )

( ) ( ) ( ) 0dL

L i L

i tu t u t u t L

t, (6.24)

in second case as

d ( )

( )dL

L

i tu t L

t (6.25)

The electric current through ideal coil is then

0

1( ) ( ) d (0)

t

L L Li t u t t iL

(6.26)

Serial connection of ideal elements R, L and C

In a serial connection of elements the electric current i(t) through elements is the same and the driving voltage is equal to the sum of voltage drops on elements:

0

d ( ) 1( ) ( ) ( ) ( ) ( ) ( ) d (0)

d

t

R L C C

i tu t u t u t u t i t R L i t t u

t C (6.27)

Parallel connection of ideal elements R, L and C

In a parallel connection of elements the driving voltage on each element is the same and the current of the source equals the sum of the currents through parallel elements:

0

1 d ( )( ) ( ) ( ) ( ) ( ) ( ) d (0)

d

t

R L C L

u ti t i r I t i t u t G u t t i C

L t (6.28)

All equations are valid as well as for dc circuits as for alternating circuits and other time dependent values.

6.5 The energetic considerations in magnetic fields

For the derivation of magnetic energy law the analogy with electrostatic field could be used with consideration of magnetic and non-magnetic substances.


74

The magnetism is caused by electric currents or moving electric charges. So it is understandable that the magnetic energy is analogous with kinetic energy in mechanics. To preserve the kinetic energy the friction losses have to be replaced. To preserve magnetic energy the Ohm's losses have to be replaced. In electrostatic field the electric forces are connected with the changes of electrostatic field energy. In magnetic field the magnetic forces are connected with the changes of magnetic field energy.

In the area of magnetic fields the following phenomena have to be discussed: a) The magnetic field energy in non-magnetic and magnetic substances. b) The magnetic field energy density in non-magnetic and magnetic substances. c) The magnetic forces on bordering surfaces of different magnetic substances.

It may be questioned whether the energy conversion should be also part of this chapter, or should it be treated as an application of basic laws.

6.5.1 The magnetic field energy

The inductance as substantial-geometric property of magnetic field represents the functional interdependence between the cause of magnetic phenomena (the currents or moving electric charges) and its consequence (the magnetic flux linkage). Both characteristics for non-magnetic and magnetic substances are shown on figures 6.9a and 6.9b

Wm

dWm

Ikdi

d

k

I

Wm

dWm

iki

d

k

i

( )f i

Figure 6.9 The substantial-geometric characteristic

a) of non-magnetic substances

b) of magnetic substances

Because there are no conductors or coils without resistance, for maintaining a magnetic field the Ohm's losses must be compensated. The driving voltage balanced the resistive and the inductive voltage drop.

d

dR Lu u u i R

t (6.29)


75

Multiplying this equation on both sides with di t , it becomes the equation of energetic equilibrium:

2d d du i t i R t i (6.30)

The left side of the equation designates the energy supplied to the system in time interval dt. The first expression on the right side designates the energy spent on the resistances and the second part designates the increment of magnetic field energy

d dmW i (6.31)

Because of the linear dependence in case of non-magnetic substances

d dL i

The magnetic energy stored in a coil with a non-magnetic core is

2 2

2

0

d J2 2 2

i

m

L i iW L i i

L (6.32)

The magnetic energy is in the substantial-geometric characteristic presented with the plane between with the characteristic and the ordinate.

In a magnetic substance stored magnetic energy is

0

dmW i (6.33)

Because the substantial-geometric characteristic for magnetic materials has no form of any known mathematical curve, the magnetic energy may be determined whether by a planimeter or by numerical integration.

Because the slope of substantial-geometric characteristic for non-magnetic substances is very gentle, nearly whole magnetic energy of a magnetic circuit with an aerial slot is located in the slot.

6.5.2 The magnetic field energy density

To derive equation for magnetic field energy density three approaches are possible: a) Exploiting the linkage between the differential and integral quantities. b) By using the substantial characteristics of non-magnetic and magnetic substances. c) From equation magnetic field energy.

In magnetic field there exist following pairs of integral and differential quantities

,B i H

Taking them in account the expression for magnetic field energy density becomes


76

0

dH

mw H B (6.34)

Because of the linearity for non-magnetic substances it may be remodelled into

2 2

00 3

00 0

Jd d

2 2 2 m

H H

m

H H B Bw H B H H

(6.35)

From substantial characteristics of non-magnetic and magnetic substances it is evident that the magnetic field energy density equals the area between substantial characteristic and ordinate.

wm

dwm

H Hk

wm

dwm

HkHi H

( )B f H

H

B

Bk

dB

B

Bk

dB

Figure 6.10 Substantial characteristics for

a) non-magnetic substances and

b) magnetic substances

The magnetic field energy density may also be derived from expression for the magnetic energy in homogenous magnetic field and then the result generalise. For the non-magnetic substance we get

22 2 2

mm

iW N B A i i N B H B

wV A l A l l

And for a magnetic substance

0

0

d

dB

mm

i N B AW

w H BV A l


77

6.5.3 The forces on borders of flux tubes

Faraday's statement that flux tubes behave as strained elastic bands , is valid also for magnetic fields. If the substance on both sides of the tube border is the same, the specific forces compensate, the resulting specific force is equal zero. If the substances on both sides of the tube border have different magnetic permeability, then the resulting specific force tends to shift the border into direction of the substance with lower permeability.

The equation for specific forces is the same as for magnetic field energy density

2 2

2 2 2

B H H Bf w [N/m2=J/m3] (6.36)

In technical praxis two ways of crossing on border of two magnetic substances are important:

a) A serial placing of two magnetic substances or normal crossing of magnetic field.

b) A parallel placing of two magnetic substances or tangential crossing of magnetic field.

1 2

r r2 1

B1 B2

f1 f2

f

Figure 6.11 The normal crossing of magnetic field

At normal crossing the magnetic flux density are crossing continuously. The resulting specific force is (figure 6.11):

2

1 2 30 1 2

1 1 N( )

2 mr r

Bf f f (6.37)

1

2

r r2 1

H2

H1

f1

f2 f

Figure 6.12 The tangential crossing of magnetic field


78

At tangential crossing the magnetic field strength are crossing continuously. The resulting specific force is (figure 6.12):

20

2 1 2 1 3

N( )

2 mr r

Hf f f (6.38)

6.6 The structural linking of basic magnetic field

Though the structural linking of basic quantities of magnetic field, from its cause - moving electric charges - up to its final consequence - magnetic flux linkage, has already been mentioned in the former chapter, two very important quantities – the induced voltage and the magnetic field energy have not been mentioned yet.

But now as we have them, we are able to present structural linkage of all quantities of magnetic fields. Because of non-magnetic and magnetic substances we have present them separately.

In non-magnetic substances we have the following linkage of basic quantities and laws of magnetic fields:

Li

0

B

H

m

2

B Hw

m

2

iW

i H B

Figure 6.13 Structural linkage of magnetic quantities and laws in non-magnetic substances

In magnetic substances we have the following linkage of basic quantities and laws of magnetic fields:

There are two additional basic laws of magnetic, the force on a moving electric charge

( )F I l B [N] oz. ( )F Q v B [N]


79

and

d

Vd

iUt

The laws of closed path in magnetic and induced electric field we have presented in their integral form:

d

d ( ) dd

k

l A

DH l J A

t in (6.39)

d d

d ( d )d d

i i

l A

E l U B At t

(6.40)

L i

0B H

m dw H B

W i m d

i H B

Figure 6.14 Structural linkage of magnetic quantities and laws in magnetic substances

Theirs differential equivalent are both Maxwell's law, derived with the use of Stokes's theorem

d rot dl A

v l v A ,

are,

d

rotd

k

DH J

t and (6.41)

d

rotd

i

BE

t (6.42)


80

Both laws of closed surface in magnetic and induced electric field we also presented only in their integral form:

d 0A

B A and (6.43)

d 0i

A

E A (6.44)

Theirs differential equivalent are both Maxwell's law, derived with the use of Gauss' theorem

d div dA V

v A v V

are

div 0B in (6.45)

div 0iE (6.46)

Both first two Maxwell's laws define the magnetic and the induced electric field as two curl vector fields, their sources are the time dependent current respectively magnetic field.

The second two Maxwell's laws state that those two fields have no scalar sources

The jointly electric field may be the sum of electrostatic and induced electric field

s iE E E , (6.47)

div div div 0s iE E E (6.48)

d d

rot rot rot 0d d

s i

B BE E E

t t (6.49)

The electric field may have a scalar or a vector source or both of them, but the magnetic field has only a vector source.

The force acting on a moving electric charge in a joint electric field is known as a Lorentz force:

( ) NsF Q E Q E v B

7 MAXWELL'S FIELD EQUATIONS

Mathematics is for the technician a wonderful tool and weapon, but he has to master it and know where and how it to employ.

Zorič Tine

Contents

7.1 The physical background of Maxwell's field equations 7.1.1 The theorems which are the basis for Maxwell's field equations

7.2 The application of Gauss' and the Stokes' theorem in the derivation of Maxwell's field equations


82

7.1 The physical background of Maxwell's field equations

To master an area of physical or technical sciences the derivation and application of basic laws is a necessary preposition. The derivation of basic laws is based on already known facts or additional experiments. In the former chapters we have shown how to derive the basic laws of electricity.

The space in them the physical phenomenon is taken place is whether a scalar or a vector field. The scalar field is defined as a space where a physical phenomena is in every point of space described with a scalar quantity. The vector field is in the same way defined as a space where a physical phenomena is in every point of space described with a vector quantity. Therefore is a very good knowledge of both types of fields a preliminary condition to get the whole set of basic laws for an area of physical or technical sciences. Maxwell derived those after him named laws on hand of theorems defined by mathematicians Gauss and Stokes.

Maxwell complemented the already known integral laws of electric and magnetic fields with their differential equivalents. Our main interest lies in the physical importance of the Maxwell's field equations, therefore we will present them only in Cartesian co-ordinates, though they are universally valid in every orthogonal system of co-ordinates. We will in point T(x,y,z) present a scalar quantity as u(x,y,z) and a vector quantity as

( , , )v x y z .

7.1.1 The theorems which are the basis for Maxwell's field equations

The Maxwell's field equation are based on three theorems

a) The theorem of Gauss

b) The theorem of Stokes

c) The gradient of a scalar field

The Gauss' theorem1

Let to be V a volume partially bordered with continuous surfaces in them a vector function ( , , )v x y z exists. The Gauss' theorem has the form

div d dV A

v V v A (7.1)

1 Vector function ( , , )v x y z is in the space a continuous function with continuous first derivatives.

Maxwell's field equation

83

In mathematical sense enables the Gauss' theorem the transformation of a volume integral into a surface integral or vice versa. In a physical sense has the Gauss' theorem the following meaning:

Has the scalar quantity divv in a point T(x,y,z) of a volume V(x,y,z) a from zero different

value, then there exists is this point of space a scalar source of a vector field v .

The definition of divv is evident from the Gauss' theorem

1

div lim dV

A

v v AV

(7.2)

In a system of Cartesian co-ordinates it has the form

31 21 2 3div ( ) (

vv vv v i j k v i v j v k

x y z x y z (7.3)

But in the technical sciences the expression divv is not found from equation (7.3)

Instead of this we have for the given vector field to find an equation in the form of equation (7.2). The expression for divv is from it evident.

The Stokes theorem

Let be ( , , )A x y z a continuous surface bordered by a piecemeal continuous curved line

( , , )l x y z . The vector function ( , , )v x y z crosses this area. The Stokes' theorem has the

form

rot d dA l

v A v l (7.4)

Mathematically enables the Stokes' theorem the transformation of a volume integral into a surface integral and vice versa. But the physical meaning of the Stokes' theorem is the

next: If a vector quantity rotv has in a point of space T(x,y,z) a from zero different value,

then there exists in this point a source of the vector field v

.

The definition of curl rotv is evident from the Stokes' theorem

1

rot lim dA

A

v v lA

(7.5)

In the Cartesian co-ordinate system the curl rot v has the form

3 32 1 2 1

1 2 3

rot ( ) ( ) ( )

i j k

v vv v v vv v i j k

x y z y z z x x y

v v v

(7.6)


84

But in the technical sciences the expression rotv is not found from equation (7.6)

Instead of this we for the given vector field found an equation in the form of equation (7.5). The expression for rotv is from it evident.

The Gradient of a scalar field

A scalar function ( , , )V x y z denotes in electricity the electric potential. The surfaces

( , , )V x y z konst are therefore called equipotential surfaces. The direction and the value

of the greatest increase of the scalar function ( , , )V x y z are defined

max

dgrad

d

V V V VV V i j k

l x y z (7.7)

The gradient is orthogonal to the equipotential surface. It defines a vector field that belongs to a scalar field.

7.2 The application of Gauss' and the Stokes' theorem in the derivation of

Maxwell's field equations

The electrostatic field

The electrostatic field exists in a space without movable electric charges, which is because of other fixed electric charges brought into an electric strained state. We will give a short account of derivation of basic laws and then complement them with the derivation of belonging Maxwell field equations.

For a lone spherical, cylindrical or flat electric charge, the electric flux density was defined as

Q

DA

(7.8)

The electrostatic field strength is

D

E (7.9)

This equation is also called the constitutional equation of electrostatic fields and in the set of Maxwell field equations also as one of the subsidiary Maxwell's equations.

The work necessary to bring a positive unity of electric charge from the chosen point of departure T0 to the point T1 is defined as electric potential V1 and the work necessary to bring the positive unity of electric charge from point T1 to point T2 as difference of electric potentials or as electric voltage between those points of electrostatic field


85

1 2

1 12 1 2

0 1

d , in dV E l U E l V V (7.10)

The constitutional equation of electrostatic fields in integral form is

Q

UC

(7.11)

All phenomena in an electrostatic field are reversible. The electric energy density is also the specific force on the border of two dielectric substances

2

el el

D Ew f (7.12)

The electric energy accumulated in space is

2

el

Q UW (7.13)

Besides of them we formulated also both laws of closed path and closed surface in electrostatic field in capacitive circuits called also Kirchhoff's equations but in the set of Maxwell field equations also as Maxwell field equations of electrostatic field in the integral form.

Choosing in the electrostatic field a closed surface ( , , )A x y z , the surface integral of

electric flux density has two possible values

0d Q

A

D A (7.14)

The first result denoted the closed surface encircles an electric charge Q or an equal sum of electric charges. The second result denotes that the encircled electric charge or the sum of electric charges is equal zero.

The electric charge inside the surface can also be written in the form

d dA V

Q D A V (7.15)

Comparing it with the Gauss' theorem (ΔV→0):

divD or

divE (7.16)

at the first possibility, and as

div 0D (7.17)

at the second possibility.

The equation 7.16 is called also the forth Maxwell' field equation, but is also only the differential equivalent of the law of closed surface in the electrostatic field.

The line integral of electric field strength sE along a closed path l is equal


86

d 0s

l

E l (7.18)

Its starting and the end point are the same.

From the comparison with Stokes' theorem

rot d dA l

v A v l ,

the zero value is evident

rot 0sE (7.19)

The electrostatic field is a curl free field. The equation is only the differential equivalent of the law of closed path in an electrostatic field.


The electric current field is a consequence of an electric field in substances with movable electric charges. The methodology of deriving basic laws of electric current fields was as a whole presented in chapter 4. At this place we will give only a short account of basic laws and complement them with the belonging Maxwell field equations.

The electric current density is defined as

d

d

iJ

A (7.20)

The vector of electric current density J is proportional to the vector of electric field

strength E and the substantial property of the conductor – the conductivity

J E (7.21)

This equation is also called the constitutional equation of electric current fields and in the set of Maxwell's field equations also as one of the belonging subsidiary Maxwell's equations. The belonging integral form is

I U G , (7.22)

In the expression

dl

U E l , (7.23)

U is the voltage of the voltage source and G the conductance of the conductor.

The power transformed into heat in a unity of volume is

p E J , (7.24)

The power transformed into heat in the whole volume

P U I (7.25)


87

The set of basic laws is complemented with both Kirchhoff's laws, the law of the closed path

gi j ji j

U i R (7.26)

In an electric resistor circuit and the law of closed surface

0jj

i (7.27)

Maxwell has this equation shaped in a different way. Out a closed surface an electric current is able to leave only if the electric charge inside the closed surface is reduced

d ( ) d

A V

QJ A V

t t (7.28)

Comparing this equation with the Gauss' theorem it is evident:

div Jt

(7.29)

Both equations are also called the continuity equation or the equation of preservation of electric charge.


As we already stated can all basic laws of magnetic fields be derived on three different ways: a) From electric current as the cause of all magnetic phenomena. b) From the force on a current carrying conductor in magnetic field. c) From the magnetic flux (Kalantarow-Neumann)

We explained in Chapter 5 why we decided for the first choice. We defined the electric current in a conductor or a coil as magnetic voltage:

, jj

i i N ali i (7.30)

The Ampere' rule states that the line integral of magnetic field strength along the closed path equals the algebraic sum of currents encircled by the path. The magnetic voltage is

dl

H l (7.31)

The chapters 5.1 and 5.2 show haw by known magnetic voltages and given substantial-geometric configuration the belonging magnetic field strength may be calculated.

With known magnetic field strength H and given the substantial property the

magnetic flux density is

B H (7.32)


88

This is the constitutions equation of magnetic field in differential form or one of the belonging subsidiary Maxwell equations.

The magnetic flux caused by magnetic voltage is

dA

B A (7.33)

The magnetic flux coupled with a coil with N turns is named magnetic flux linkage

kN (7.34)

The constitutions equation of magnetic field in integral form is

L i (7.35)

L is the own inductance of a coil or a closed path.

The magnetic field energy density is for a non-magnetic substance

2

m

B Hw (7.36)

and for a magnetic substance

dm

B

w H B (7.37)

The equation for a magnetic flux density again equals the expression for the specific force on border of two substances.

The whole in the space accumulated magnetic energy is

2

m

iW oz.

dmW i (7.38)

The integral form of the Ampere' law (the law of the closed path) may be also written in the form

d ( ) d

l S

DH l J A

t (7.39)

From a comparison with the Stokes' theorem the corresponding differential form is evident

rotD

H Jt

(7.40)

The curl of the magnetic field strength (the source of magnetic field) is caused in a point of space by the sum of current densities.

The rule of closed surface (the integral form) has in magnetic field the form.

d 0A

B A (7.41)


89

From a comparison with the Gauss' theorem the corresponding differential equation is evident

div 0B (7.42)

In a physical sense both differential Maxwell equation have the following meaning: a) The source of magnetic field in a point of space are electric current densities. b) The magnetic field is a curl field. .

The induced electric field

The time dependent magnetic field has as a consequence phenomenon an induced electric field. In a unique chosen conducting loop an induced electric voltage may be realised

d di i

l A

BU E l A

t (7.43)

This is one of the Maxwell' equation in the integral form or the Faraday's law of induced electric voltage.

From the comparison with the Gauss' theorem it is evident

rot i

BE

t (7.44)

In a point of space the electric field strength may have a static or a dynamic component

s iE E E (7.45)

The static component of the electric field strength sE has a scalar source

div div div 0sE E E

(7.46)

The dynamic component has a vector source.

rot rot rot 0s i

B BE E E

t dt

(7.47)

In the Lorentz' force

( ) NsF Q E v B ,

both components of the electric field strength are included.


90

Conclusion

In those chapters derived basic laws of the electricity present the minimal set of basic laws valid in all its branches. Without all of them our knowledge of basic electricity is incomplete. The Maxwell' field equation offer a deeper physical insight, therefore they belong to that minimal set of basic laws.

8 CONCLUSION

Good teachers are expensive, but the bad one even more, if we take into account all damages they may cause.

Talbert Bob

This paper has no intention to be a textbook Basic Electricity. It is meant only as a guidance for the teachers and the same time an attempt to define the minimal set of basic laws, which every electrician on a university level must master.

But that does not mean, the lectures may not exceed those borders. Something like this we did too by deriving a set of special laws. We did this on purpose, to show, how to employ the teaching methods at data obtained at experiments.

The next step will be the application of basic laws in different branches of electricity. Because the fields of our experience are besides of Basic Electricity and Theory of Electricity only Electric Power Systems, we invite the teacher of other technical branches to elaborate on their area.

Technical sciences mean the application of physics in real, most not ideal circumstances. There oft even special laws do not enough for a successful mastery of problems. We have to employ special methods, to get the solutions for a narrow special problem. Some of them we will try provide ourselves.

Engineering pedagogy is the branch of science engaged in solution of problems in the field of engineering. Besides of guides for successful presentation of learning media, it also includes for a successful teaching needed sociological, psychological and ecological knowledge. The Labor didactic, the use of computer methods for modelling or simulation of industrial processes are becoming unavoidable parts of teaching processes.


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LITERATURE

A. Ingenieurpedagogik

A.1 K. GEIGER

Methodik der Lehre der Wechselstromtechnik

VEB Verlag Technik Berlin, 1956

A.2 A. HAUG, H. G. BRUCHMUELLER

Labordidaktik für Hochschulen

Leuchtturm Verlag – LTV, Alsbach, 2001

A.4 A. MELEZINEK

Ingenieurpädagogik – Praxis der Vermittlung technischen Wissens

Springer Verlag, Wien New York, 1999

B. Basic Electricity

B.1 G. BOSSE

Grundlagen der Elektrotechnik I

(Bibliographisches Institut Mannheim/Zürich 1969

B.2 G. BOSSE

Grundlagen der Elektrotechnik II

Bibliographisches Institut Mannheim/Zürich 1969

B.3 G. BOSSE

Grundlagen der Elektrotechnik III

Bibliographisches Institut Mannheim/Zürich 1969

B.4 A. FÜHRER, K. HEIDEMANN, W. NERRETER

Grundgebiete der Elektrotechnik, Band 1:Stationäre Vorgänge

Carl Hanser Verlag München Wien 1989

B.5 A. FÜHRER, K. HEIDEMANN, W. NERRETER

Grundgebiete der Elektrotechnik, Band 2:Zeitabhängige Vorgänge

Literature

93

Carl Hanser Verlag München Wien 1989

B.6. I. TIČAR, T. ZORIČ

Osnove elektrotehnike 1. zvezek: Električna in tokovna polja

(Basic Eletricity , 1. Part: Electrostatic and electric current fields)

FERI UM, Maribor 2000

B.7. P. Kitak, T. ZORIČ

Osnove elektrotehnike 2. zvezek: Magnetna in inducirana električna polja

(Basic Eletricity , 2. Part: Magnetic and induced electric fields)

FERI UM, Maribor 2011.

B.8 I.TIČAR, T. ZORIČ

Osnove elektrotehnike 3. zvezek: Izmenični tokokrogi in prehodni pojavi

(Basic Eletricity , 3. Part: Alternating electric circuits and Transients)

FERI UM, Maribor 2001

B.9 T. ZORIČ

Zbirka rešenih nalog iz Osnov elektrotehnike

(A collection of solved problems in Basic Electricity)

Published by the author, Maribor 2008

methodology of teaching basic electricity the magnetic flux linkage..... 69 6.3 the own and the...

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