33

Chapter 2

The macroscopic laws of electromagnetism

The subject of laws is organized in like this: firstly there are presented the

laws that rule the macroscopic theory of electromagnetism (in Chapter 2), and then a few of the conclusions and consequences derived from the laws’ system (in Chapter 3). In paragraph 3.5 there are presented the unit measures of the electric and magnetic quantities, and in Chapter 4 there is presented the use of these laws for some significant problems computation.

The laws of macroscopic theory of electromagnetism represent a set of mathematical relations coherent from the point of respecting the truth criteria, completeness and non-contradiction, relations that connect the quantities which characterize the electromagnetic field and the electromagnetic state of the bodies in variable regime, gathering in a mathematical “shape” the electromagnetic field phenomenology.

During the presentation of each law, after the statement and the comment of its mathematical formulation, we follow to mark out its physical significance and its most important consequences, firstly in variable regime, then in various particular regimes: quasi-stationary regimes (in which the variations in time of some of the quantities are sufficiently slow so they can be neglected), steady state (stationary) regimes (in which the quantities are time-invariant, but there are energy transformations) and static regimes (which are stationary regimes without energy transformations).

2.1. Electric flux law (Gauss’s law on the electric flux)

Statement: Electric flux through any closed surface is equal at any moment to the electric charge Vq form domain V bounded by the surface :

. Vq (2.1.1)

34

Figure 2.1.1

If the electric charge from domain V has the volume distribution density a v (see Figure 2.1.1), then relation (2.1.1) can be explicitly written:

V

vext dvdADdAD (2.1.2)

and it represents the integral form of the electric flux law.

Applying Gauss-Ostrogradski relation, one obtains:

,

V

vV

dvdvDdivdAD (2.1.3)

relation which, being true for any arbitrary considered domain V , imposes the equality of the integrands:

, vDdiv (2.1.4)

expression which represents the local form the electric flux law for continuity domains.

Electric flux law emphasizes one of the causes that generates electric field, namely charged bodies.

Figure 2.1.2

V

dv

v

DextdAdA

0Ddiv

0Ddiv

0 Ddiv

35

It is well-known that, for any field vector, the points from the space in which the divergence is nonzero represent end points of field lines (starting point if the divergence is positive, and, respectively, end points if the divergence is negative). It results that the lines of the electric field produced by charges are open curves, starting from the positive charged bodies and ending on the negative charged bodies (see Figure 2.1.2).

2.2. Magnetic flux law

Statement: Magnetic flux through any closed surface is zero in any moment:

.0 (2.2.1)

Replacing in equation (2.2.1) the expression of the magnetic flux, one obtains the integral form of the magnetic flux law:

.0

Vext dvBdivdABdAB (2.2.2)

Because this relation is valid for any space domain ,V it results the local form for continuity domains of the magnetic flux law:

.0 Bdiv (2.2.3)

According to relation (2.2.3), the magnetic flux density field vector is a solenoidal one (without sources). This fact underlines on one hand the non-existence of the magnetic charges similar to electric charges and, on the other hand, the inexistence of some points – extremity of magnetic field lines. Therefore the magnetic field lines are not open curves.

An immediate consequence of the magnetic flux law is that the magnetic flux through any open surface bounded by the same closed curve is the same.

In order to prove this statement one will consider an arbitrary closed curve , and two arbitrary open surfaces jS , and kS , , which rest on curve - see Figure 2.2.1.

36

Figure 2.2.1

By convention we associate the line oriented element dl with each of the area oriented elements kdA and jdA according to the right corkscrew rule, we write the relation (2.2.2) for the closed curve , reunion of the open surfaces jS , and kS , . In these conditions, the integral on will be the sum of the integral on jS , and kS , :

.0

,,,,

kS

k

jSj

kSext

jSextext dABdABdABdABdAB (2.2.4)

Therefore

S

k

kSk

jSj dABdABdAB

,,

(2.2.5)

for any open surface S which sits on the closed curve , which shows that the magnetic flux has a unique value through all open surfaces bounded by the same closed curve.

Moreover, the vector identity

0A curl div (2.2.6)

allows the introduction of a new quantity, called magnetic vector potential and denoted by A with relation:

.BA curl (2.2.7)

extj dAdA

dl

j,S

extk dAdA

k,S

37

As vector A is uniquely determined only if we know its divergence, it is common that in stationary regime to adopt the calibration condition

0Adiv (2.2.8)

such that the field vector A to be a solenoidal one too.

Applying the Stokes’ theorem, the magnetic flux which flows through an arbitrary open surface S , which sits on a closed curve , can be expressed by the line integral of the of the magnetic potential vector on the curve :

.dlAdAAcurldAB

SSS

(2.2.9)

This shows that the magnetic flux through an open surface depends only on the closed curve that bounds it.

2.3. The magnetic constitutive law (the relationship law between magnetic flux density, magnetic

field strength and magnetization)

Statement: For any point and for any time moment, between magnetic flux density vector B , magnetic field strength vector H and magnetization vector M exists the following relation:

.0 MHB (2.3.1)

We mention this law has only local form.

By expressing the magnetization vector M by its two components, namely temporary component tM and permanent component pM , the magnetic constitutive law has the following form:

pt MMHB 000 (2.3.2)

and emphasizes as magnetic field generating cause – bodies that have permanent magnetization.

38

2.4. Temporary magnetization law

Statement: For any point and for any time moment, temporary magnetization tM depends on the magnetic field strength H :

.HMM tt (2.4.1)

Function of magnetic materials’ nature, relation (2.4.1) has explicit specific forms.

We remind a material is called isotropic if in any of its points there are no privileged directions (in other words, its local properties does not depend on the direction of a certain quantity) and is called linear if in any of its points there are no privileged values (in other words, its local properties does not depend on the value of a certain quantity).

Below there will be presented and discussed the various dependences between the magnetic flux density and the magnetic field strength function of magnetic materials’ nature.

♦ For isotropic linear materials, the dependence (2.4.1) has the form

,HM mt (2.4.2)

where the proportionality coefficient m is a non-dimensional material constant which is called magnetic susceptibility, and the magnetic constitutive law becomes

.1 00

000000

pm

pmpt

MH

MHHMMHB

(2.4.3)

The non-dimensional material parameter

mr 1 (2.4.4)

is called relative permeability, and the dimensional quantity

ro (2.4.5)

is called absolute permeability.

Using these, the magnetic constitutive law in magnetic field becomes

,0 pp IHMHB (2.4.6)

39

H Hμ

in which the quantity pp IM 0 is called magnetic polarization.

Figure 2.4.1, a Figure 2.4.1, b

Vectors H , H , B and pp MI 0 are placed from qualitative point of view as in Figure 2.4.1,a case for which 0pM and, respectively, as in Figure 2.4.1,b case for which .0pM

Function of their relative permeability values, the materials are being classified as follows:

• diamagnetic materials (that have a r slightly less than one); • paramagnetic (that have a r slightly more than one); • ferromagnetic materials (with very big r , touch four figures and which, from certain pair of values, they loose their linear character).

In the table below there are presented values of the relative permeability for a couple of diamagnetic and paramagnetic materials, observing that for all of them one can consider .1r

Name of diamagnetic material r

Hydrogen 1-0,063∙10-6

Copper 1-08,8∙10-6

Water 1-9∙10-6

Zinc 1-12∙10-6

pI pIHμB

Hμ H

40

Name of diamagnetic material r

Halite 1-12,6∙10-6

Silver 1-19∙10-6

Mercury 1-25∙10-6

Bismuth 1-176∙10-6

Name of the paramagnetic material r

Azoth 1+0,013∙10-6

Air 1+0,4∙10-6

Oxygen 1+1,9∙10-6

Aluminum 1+23∙10-6

Platinum 1+360∙10-6

♦ For linear anisotropic materials one notices that, if there is considered a three-orthogonal coordinate axis system, in each point in the space each scalar component of temporary magnetization vector depends in principle on all scalar components of the magnetic field strength vector. In a Cartesian coordinate system, for example, if kHjHiHH zyx ,

kMjMiMM ztytxtt and kBjBiBB zyx , then

.zzzmyzymxzxmzt

zyzmyyymxyxmyt

zxzmyxymxxxmxt

HHHM

HHHM

HHHM

(2.4.7)

Dfining the magnetic susceptibility tensor m by the attached matrix:

41

zzmzymzxm

yzmyymyxm

xzmxymxxm

m

, (2.4.8)

The law of temporary magnetization can be shortly written as

HM mt (2.4.9)

and it leads to the following form of the magnetic constitutive law

,1 0000

000000

pprpm

pmpt

IHMHMH

MHHMMHB

(2.4.10)

where mr 1 is he relative permeability tensor, and r 0 is the absolute permeability tensor.

Figure 2.4.2, a Figure 2.4.2, b

Relation (2.4.10) has the vector interpretation from Figure 2.4.2,a with the particular case in Figure 2.4.2,b corresponding in which .0pM

If we explicitly write the relation (2.4.10):

zp

yp

xp

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

I

I

I

H

HH

B

BB

, (2.4.11)

pI

Hμ

pIHμB

H

H

HμB

42

we remark the fact that the matrix corresponding to the tensor is symmetrical and positive defined. There are three orthogonal directions, called principal magnetization directions, having the unit vectors 1u , 2u , 3u , for which the scalar components of vectors 332211 uBuBuBB ,

332211 uHuHuHH and 33

22

11

uIuIuII pppp after

these three directions satisfy the matrix relation:

3

2

1

3

2

1

3

2

1

3

2

1

000000

p

p

p

I

I

I

HHH

BBB

. (2.4.12)

♦ For nonlinear materials the dependence between the magnetic flux density and magnetic field strength can be expressed (or at least approximated) analytically (see Figure 2.4.3) by a real, nonlinear function f :

HfB . (2.4.13)

Figure 2.4.3

♦ One notices there are nonlinear materials (such as for example some iron, nickel, cobalt alloys) for which the dependence of the relationship between the magnetic flux density and the magnetic field strength cannot be expressed (and not even approximated) analytically. These materials present a specific phenomena called magnetic hysteresis, which expresses the fact that a functioning point from the curve which gives the dependence HBB has coordinates dependent on the succession of states followed to arrive in that specific point, so it depends on the historical realizations of that state.

B

H

0

43

Below we present, from the qualitative point of view, this phenomena (see Figure 2.4.4). Starting from an initial state in which the material is not magnetized, state corresponding to point 0;0O in the plane BH; and applying a magnetic field with increasing magnetic field strength, one notices first a nonlinear increase of the magnetic flux density followed by a saturation plateau until point maxmax1 ;BHA . On this plateau, an increase of the magnetic field strength does not lead to significant increases of the magnetic flux density. 1OA curve is called the first magnetization curve.

Figure 2.4.4

When the magnetic field strength is decreased, the values of the magnetic flux density do not coincide to those corresponding to the first magnetization curve for the same values of H . The decrease of the magnetic flux density is realized by applying an increasing magnetic field but opposite as direction to the initial one. To magnetic field cancellation it corresponds a nonzero remanent magnetic flux density – the functioning point rBA ;02 . The magnetic field that cancels the magnetic flux density is called coercive magnetic field – functioning point .0;3 cHA

44

The continuation of decreasing the magnetic field strength until value maxH is reached, brings the functioning point maxmax4 ; BHA in a point

1A symmetric with respect to the origin of the coordinate system axis. Then, if the magnetic field is increased, the functioning points are being placed on the curve 1654 AAAA which is symmetric to curve 4321 AAAA with respect to the origin. In this way the hysteresis cycle closes.

Ferromagnetic materials magnetize following irreversible evolutions, which for a given value of H corresponding in principle to three possible values for B , corresponding to the first magnetization curve 1OA , to the descending part 4321 AAAA or to ascending part 1654 AAAA of the hysteresis cycle.

We also mention that, once we arrive in point A , for example BH ; coordinate, we decrease the value of the magnetic field by value H , after which we come back to the starting value point, describing in this way a secondary cycle. This cycle is very flat, it can be approximated by its median and, the transformation that took place can be considered as being quasi-reversible.

In point P there are defined the following three relative permeabilities (see Figure 2.4.4):

• static relative permeability sr, , equal to the to the trigonometric tangent of the angle between AO chord and OH axis:

tgH

B

Asr

0, ; (2.4.14)

• dynamic relative permeability dr , , equal to the to the trigonometric tangent of the angle between the trigonometric tangent in point A and the OH axis:

;lim00

0,

tgH

B

AHHdr

(2.4.15)

• reversible relative permeability revr , , equal to the to the trigonometric tangent between the median of the secondary cycle and the OH axis:

,lim00

0,

tgH

B

AHHrevr

(2.4.16)

with .; Max

45

According to the width of the hysteresis cycle, the ferromagnetic materials can be divided into two big categories:

o soft magnetic materials, characterized by narrow cycles, respectively small coercive fields (see dependence (a) from Figure 2.4.5);

o hard magnetic materials, characterized by wide cycles, respectively big coercive fields (see dependence (b) from Figure 2.4.5).

Figure 2.4.5

The curves from Figure 2.4.5 gives us only a qualitative information, that do not make out the real magnitude order of the ratio between the widths of the hysteresis cycles. For comparison, we present below the values of the remanent flux density rB and of the coercive field cH for a few of the soft ferromagnetic materials, respectively for hard magnetic materials.

Name of the soft ferromagnetic material rB

T

cH

m/A

Super Malloy (79% Ni; 15% Fe; 5% Mo; 0,5% Mn) 0,6 0,4

Perm alloy (78,5% Ni; 21,5% Fe) 0,6 4

Pure Iron 1,4 4

Nickel – Zinc Ferrite 0,13 10

B

H 0

46

Name of the hard ferromagnetic material rB

T

cH

m/A

Electrotechnical Steel (with 4% Si) 1,8 40

Manganese – Zinc Ferrite 0,15 20

Steel (cu 1% C) 0,7 5∙103

Chrome Steel, Wolfram Steel 1,1 5∙103

Alnico (12% Al; 20% Ni; 5% Co; 63% Fe) 0,73 34∙103

Oerstit (20% Ni; 30% Co; 20% Ti; 30% Fe) 0,55 65∙103

Cobalt Ferrite 0,16 90∙103

Barium Ferrite 0,35 200∙103

Cobalt – Platinum Alloy (77% Pt; 23% Co) 0,45 260∙103

One can notice that, for both categories, the magnitude order of the remanent flux densities is the same, while the values of coercive fields are slightly different. For example, taking into consideration two materials with the same value of the remanent flux density, manganese –zinc ferrite from the soft materials category and cobalt ferrite from the hard materials category, one can notice a ratio between the widths of the hysteresis cycles, expressed by the ratio of their coercive fields, of 1/4.500.

Finally, it’s important to mention that the completion of a hysteresis cycle is accompanied by energy transfer from the magnetic field to the ferromagnetic body, the volume density of this energy being proportional to the cycle’s area (as Warburg theorem proves, theorem that will be presented in a following chapter of this book).

We also remember that the properties of the ferromagnetic material are greatly influenced by temperature, these properties even disappearing for certain limit values of temperature, proper for each material, called Curie points. Curie point for cobalt is C. 01371 , for iron is C0753 , and for nickel is C0376 . Increasing the temperature over the Curie point value leads to the transformation of ferromagnetic materials into paramagnetic materials.

47

2.5. The electric constitutive law (the relationship between electric displacement, electric field

strength and electric polarization)

Statement: For any point and for any time moment, between electric displacement vector, D electric field strength vector E and polarization vector P exists the following relation:

.0 PED (2.5.1)

As for the magnetic constitutive law, this law has only local form.

Expressing the electric polarization vector P by its two components, namely the temporary tP and the permanent component pP , the electric constitutive law has the following form:

pt PPED 0 (2.5.2)

and emphasizes as electric field generating cause – bodies that have permanent electric polarization.

2.6. The law of temporary polarization

Statement: For any point and for any time moment, temporary electric polarization tP depends on electric field strength E :

.EPP tt (2.6.1)

Depending on the electric material nature, relation (2.6.1) has explicit particular forms.

As follows we present and comment various types of dependences between electric displacement and electric field strength as a function of magnetic materials nature.

♦ For linear and isotropic materials, the dependence (2.6.1) has the form

,0 EP et (2.6.2)

48

where the proportionality coefficient e is a material numeric (non-dimensional) constant called electric susceptibility, and the electric constitutive law becomes

.1 0000 pepept PEPEEPPED (2.6.3) The numeric quantity

er 1 (2.6.4) is called relative permittivity, and the numeric quantity

r 0 (2.6.5) is called the absolute permittivity or electric constant.

Using these notations, the electric constitutive law has the form

.pPED (2.6.6)

Figure 2.6.1, a Figure 2.6.1, b

Vectors E , E , D and pP are qualitative placed as in Figure 2.6.1,a for the case in which 0pP and, respectively, as in Figure 2.6.1,b for the case in which .0pP

Depending on their relative permittivities, materials can be classified as follows:

diaelectric materials (which have r closed to 1); paraelectric materials (which have unit order r ); ferroelectric materials (which have very big r , thousands order and which, if bigger than certain pair of values, they loose their linear characteristics).

In the table below e present the values of the relative permeabilities for some of the diaelectric and paraelectric materials.

pP

E

pPED

E ED E

49

Name of the material Aggregation state r

H2 1,0003

Air 1,0006

O2 1,0006

CO 1,0007

CO2 1,001

CH4 1,001

C2H6

Gaseous

1,0015

Air ( C0191 ) 1,43

Transformer oil ( C020 ) 2,2

Acetone ( C020 ) 21,2

Ethyl alcohol ( C015 ) 26

Methyl alcohol ( C015 ) 32,5

Nitrobenzene ( C018 ) 36

Distilled water ( C020 ) 81,1

Hydrocyanic acid ( C015 )

Liquid

95

Paraffin 2,2

Polyethylene 2,3

Polyamide 2,4

Insulant paper 2,4

Bakelite

Solid

2,8

Plexiglas 3,0÷3,6

Pressboard

Solid

3,4÷4,3

50

Name of the material Aggregation state r

Ebonite 2,5÷5,0

Rubber 3,0÷6,0

Quartz glass 4,0÷4,2

Porcelain 5,0÷6,5

ClNa 5,5

Mica 5,0÷7,0

Glass 5,5÷8,0

SO4K2 8,35

Diamond

Solid

16,5

♦ For linear and anisotropic materials, one can notice that, if a three-orthogonal coordinate system is taken into consideration, in each point from the space, each scalar component of temporary electric polarization vector depends, in principle on all scalar components of electric field strength vector. In a Cartesian coordinate system, for example, if kEjEiEE zyx ,

kPjPiPP ztytxtt and

kDjDiDD zyx , then

.000

000

000

zzzeyzyexzxezt

zyzeyyyexyxeyt

zxzeyxyexxxext

EEEP

EEEP

EEEP

(2.6.7)

Defining the electric susceptibility e by the attached matrix:

zzezyezxe

yzeyyeyxe

xzexyexxe

e

, (2.6.8)

the temporary electric polarization law can be shortly written

51

EP et 0 (2.6.9)

and it leads to the following magnetic constitutive

,1 00

000

pprpe

pept

PEPEPE

PEEPPED

(2.6.10)

where er 1 is the relative permittivity tensor, and r 0 is the absolute permittivity tensor.

Figure 2.6.2, a Figure 2.6.2, b

Relation (2.6.10) has the qualitative vector interpretation from Figure 2.6.2,a with the particularization from Figure 2.6.2,b corresponding to situation for which .0pP

If relation (2.6.10) is written explicitly:

zp

yp

xp

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

P

P

P

E

EE

D

DD

, (2.6.11)

and it can be noticed that the matrix corresponding got tensor is symmetrical and positive defined. So, there are three orthogonal directions (called electrization principal directions), having the unit vectors 1u , 2u , 3u for which the scalar components of the vectors 332211 uDuDuDD ,

332211 uEuEuEE and 33

22

11

uPuPuPP pppp with

respect to these three directions, satisfy the matrix relation:

pP

E

pPED

E

ED E

52

zp

yp

xp

z

y

x

z

y

x

P

P

P

E

EE

D

DD

3

2

1

000000

. (2.6.12)

2.7. Electromagnetic induction law (Faraday’s law)

Statement: The electromotive force (emf) u along a closed curve is equal to the rate of decrease (in time) of the magnetic flux S across any surface S bordered by the closed curve :

.dt

du S

(2.7.1)

Considering the definition relations of emf and of the magnetic flux it results the integral form of the law:

.

S

dABdtddlE (2.7.2)

Figure 2.7.1

The electromagnetic induction law has the above form only with the condition that the reference direction of the closed curve (the reference direction of the oriented line element dl ) and the direction of the normal to the surface S (the oriented area element dA) are associated according to right corkscrew rule (see Figure 2.7.1).

dl

S

dA B

E

53

For moving media, the integration domains follows the bodies in their movement, and the derivative with respect to time of the magnetic flux is a substantial derivative and it is computed using the following relation:

,dAwBcurlB divw

tBdAB

dtd

SS

(2.7.3)

where w is the local speed vector of the medium.

Using Stokes relation, it results that

S

dAE curldlE (2.7.4)

and

.

dlwBdAwBrotS

(2.7.5)

Taking into account the local form (2.2.3) of the magnetic flux law, one obtains a new integral form of the electromagnetic induction law:

.

dlwBdAtBdlEu

S (2.7.6)

Relation (2.7.6) emphasizes the physical significance of the law: the time variable magnetic field produces (induces) an electric field by the electromagnetic induction phenomena. Therefore, the electromagnetic induction is a physical phenomena, unlike electric displacement D and magnetic induction B which are physical quantities.

Moreover, relation (2.7.6) allows the decomposition of the emf into two components:

,mt uuu (2.7.7)

with

SSt dA

tBdA

tBu (2.7.8)

54

called emf induced by transformation and, respectively,

dlBwdlwBum (2.7.9)

called emf induced by movement.

The two components correspond to the two ways of electromagnetic induction phenomena production: the time variation of the magnetic induction B at rest (no movement) ( tu is on zero in this case), respectively, the movement of at least a portion of a closed curve in a magnetic field in such a way that the field lines are cut away by the field (only in this case

mu is non zero).

We remark that the sum from relation (2.7.7) does not depend on the reference system chosen for the movement, while the separation into two components depends in general o the adopted reference system.

The reference system of the emf can be arbitrary chosen. To determine the real sense of the induced emf, Lenz formulated his famous rule: the real direction of emf is such that its effects oppose the generating causes.

Therefore, the minus sign from relation (2.7.1) does not have to be assimilated from mathematical point of view with Lenz’s law, this coming from the adopted convention when we associate the direction for dl with the direction of dA according to the corkscrew rule.

For continuity domains the equality:

SS

dABwcurltBdAE curl (2.7.10)

is valid for any surface S and it imposes the equality of the integrands:

.BwcurltBE curl

(2.7.11)

Relation (2.7.11) represents the local form for continuity domains of the electromagnetic induction law.

For media at rest 0w it results the local form

55

tBEcurl

(2.7.12)

and respectively, the integral form

S

dAtBu . (2.7.13)

Relation (2.7.12) shows that E is a rotational field and it underlines that the lines of the electric field produced by electromagnetic induction phenomena are closed curves which surrounds the lines of the time varying magnetic field which generated them.

In stationary regime, the electromagnetic induction law becomes the theorem of the stationary electric potential, which ha the local form

0E curl (2.7.14)

and, respectively, the integral form

.0

dlEu (2.7.15)

The local form (2.7.14) of electromagnetic induction law in stationary regime shows that, in this regime, the vector field E is non-rotational and, according to the vector identity which states that the curl of the gradient for any scalar field is zero, it can be introduced the scalar quantity V , called electric potential, with relation:

. VgradE (2.7.16)

Figure 2.7.2

56

The integral form of the electromagnetic induction law in stationary regime allows the demonstration of the following theorem: the voltage drop between two points kM and jM from space (arbitrary point) does not depend on the paths (integration curve) between them.

Indeed (see Figure 2.7.2), let consider two paths (arbitrary) between kM and jM , represented by the open curves 1C and 2C , whose reunion is the closed curve .

According to relation (2.7.15) it results that

0

2121

jM

CkM

jM

CkM

kM

CjM

jM

CkM

dlEdlEdlEdlEdlEu (2.7.17)

that is

.

21

jMkM

jM

CkM

jM

CkM

udlEdlE (2.7.18)

The electric voltage is therefore a scalar physical quantity referring to an ordered pair consisting of two points from the space, while the electric potential is a physical scalar quantity associated to each point in the space.

As relation (2.7.16) is a differential type relation, it means that electric potential is defined up to an arbitrary additive constant. This constant represents the value 00 MVMV , arbitrary, of potential 0M from the space, arbitrary, considered to be reference value for all potentials. Then, the potential of any point from the space, for example jM , is

.0

0 jM

MMjMj dlEVVMV (2.7.19)

This relation is immediately computed if it is computed the electric voltage between points jM and .0M For easiness, we use Cartesian coordinate system:

57

,VVVdVdzzVdy

yVdx

xV

kdzjdyidxkzVj

yVi

xV

dlVgraddlEu

MjMjM

M

M

jM

M

jM

M

jM

M

jM

M

jMPjP

00

00

0

00

0

(2.7.20)

which represents exactly relation (2.7.19).

Computing electric voltage between two points kM and jM from the space following a path 3C , arbitrary, but crossing the point 0M (see Figure 2.7.2), one obtains:

,0

00

0

000

0

3

jMkM

jM

MM

kM

MM

MM

jM

M

M

kM

jM

CkM

jMkM

VVdlEVdlEV

VVdlEdlEdlEu

(2.7.21)

relation which represents the expression of the voltage between any two points function of their potentials.

If one considers 00 MV , it is said that point 0M is „grounded” or „earthed” and in an electric circuit it has a specific symbol (see Figure 2.7.2).

In these conditions

.0

M

jMjMj dlEVMV (2.7.22)

58

2.8. The magnetic circuit law (Ampère Law)

Statement: The magnetomotive force (mmf) mu along any closed curve is equal to the sum between the conduction electric current Si through an open surface S , arbitrary, bordered by the closed curve and the time derivative of the electric flux over the same surface S :

.dt

diu SSm

(2.8.1)

The term Si is also called solenaţie and it is denoted by S .

By rewriting relation (2.8.1) one obtains the integral form of the law:

SS

dADdtddAJdlH (2.8.2)

which has the above presented form only if the association between the reference direction of the orientation the oriented line element dl and the oriented area element dA is according to the corkscrew rule (as for the electromagnetic induction law).

For moving media, the integration domains follows the bodies in their movement, following a similar path as the one described for the electromagnetic induction law, and the integral developed form of magnetic circuit law is obtained:

,dAwDcurldADdivw

dAtD dAJ

dAwDcurlD divwtD

dAJdAHcurldlH

SS

SS

S

SS

(2.8.3)

59

respectively,

. SRSvSdSm iiiiu (2.8.4)

On the right side of equality (2.8.4) we have four terms:

-the conduction current:

;

S

S dAJi (2.8.5)

- the displacement current:

,

S

Sd dAtDi (2.8.6)

where

dJtD

(2.8.7)

is the displacement current density; - the convection current:

,

Sv

SSv dAwdADdivwi (2.8.8)

where

vv Jw (2.8.9)

Is the convection current density; - Roentgen current

.dlwDdAwDcurliS

SR

(2.8.10)

The magnetic circuit law emphasizes two causes that can generate the magnetic field: conducting bodies transited by conduction currents and/or time variable electric fields (by displacement currents, convection and Roentgen currents).

60

For continuity domains, the equality (2.8.3), true for any surface S , leads to:

,wDcurlwtDJH curl v

(2.8.11)

relation that represents the local form of the magnetic circuit law.

For stationary media 0w the local form becomes:

tDJHrot

(2.8.12)

and it shows that the closed lines of the magnetic field surrounds the conductors transited by conduction currents, respectively the lines of the time variable electric field that generate them.

In steady state regime the magnetic circuit law becomes the Ampère theorem, and it has the local form:

JH curl (2.8.13)

and, respectively, the global form

. sm iu (2.8.14)

In the particular cases of the point from the space for which the vector field H is non-rotational ( 0H curl ), according to the vector identity which states that the curl of the gradient of any scalar field is zero, the scalar quantity

mV , can be introduced, called scalar magnetic potential, using the relation:

. mVgradH (2.8.15)

2.9. The electric charge conservation law

Statement: The conduction current i which exists from a closed surface , arbitrary, is equal to the rate of decrease (in time) of the electric charge Vq contained in a domain V bordered by the closed surface :

61

.dt

dqi V (2.9.1)

By rewriting the relation (2.9.1) one obtains the integral form of the law:

V

vext dvdtddAJdAJ (2.9.2)

According to the adopted convention referring to the orientation of the unit normal to a closed surface (towards its exterior as in Figure 2.9.1), it results that the reference direction of the current i is towards the exterior of the space domain .V

Figure 2.9.1

For moving media, the time derivative from relation (2.9.2) is a substantial derivative, which is computed using the relation:

V

vv

Vv dvwdiv

tdv

dtd ,

(2.9.3)

such that the developed integral form of the electric charge conservation law becomes:

V

vv dvwdivt

dAJ ,

(2.9.4)

or, using Gauss-Ostrogradski relation:

V

dv

v

J extdAdA

62

,

V

vv

V

dvwdivt

dvJdiv

(2.9.5)

in which

vv Jw (2.9.6)

is the convection current density.

As relation (2.9.5) is true for any domain V , from the integrands equality one obtains the local form of the electric charge conservation law for continuity domains:

vv Jdivt

Jdiv

(2.9.7)

or

. t

JJdiv vv

(2.9.8)

For stationary media 0w the local form of the law is:

. t

Jdiv v

(2.9.9)

In steady state regime the local form becomes:

0Jdiv , (2.9.10)

And the global form has the expression

0Σi (2.9.11)

and it shows that, in this regime, for any closed surface the electric conduction current is preserved.

The electric charge conservation law underlines from macroscopic point of view that the electric convection current is generated by moving bodies.

63

2.10. The constitutive law of electric conduction

Statement: In any point and at any moment of time, the electric

conduction current density J depends on the electric field strength E :

.EJJ (2.10.1)

Function of the nature of the various conducting materials, relation (2.10.1) has explicit forms. As follows, there will be studied only the conducting linear and isotropic materials. In this case the dependence (2.10.1) becomes:

,iEEJ (2.10.2)

in which is a dimensional material quantity called electric conductivity, and iE represents impressed electric field strength.

Despite it has the attribute electric, the impressed electric field in not an electric field in the accepted notion as it was presented up to now.

For it understanding it is very useful to call the microscopic configuration of the conductors, noticing that upon the free charge carriers, in certain conditions (if, for example, the conductors are heterogeneous, or they are accelerated, or they are not isotherms), beside the electric nature forces, there are also acting non-electric forces.

Admitting as a simplified model of the microscopic structure of a homogenous, not-accelerated and isotherm conductor, a ionic network positive charged, placed in an electronic „fluid” negatively charged and without ordered component for its constitutive particles’ movement, the volume electric charge density of the conductors is zero.

Supposing that upon this conductor there is applied a non electric perturbation, it has a new regime: due to non electric forces, the electrons will migrate and they will generate in this way a separation of opposite signs charges, showing an electric field strength E , oriented from the region with excess positive charges towards the region with excess of negative charges. Upon each electron (having the charge 0q ) act two types of forces: an electric nature one:

EqF el 0 (2.10.3)

64

opposite direction to the electric field strength E (because 00 q ) and a non electric nature one neelF having the direction of the field. The electron will move under the action of the non zero resulting force

.00

00

qFEqFEqFF neel

neelneelel (2.10.4)

The term

ineel Eq

F

0 (2.10.5)

has the dimension of an electric field strength and it is called impressed electric field strength. Impressed electric field is a quantity that capture from the electric point of view the non electric nature actions upon the charge carriers from the conductors, in heterogeneous areas of the chemical-physical properties.

Function on the cause nature that generates them, the impressed electric fields can be of mechanical, thermal, chemical, etc nature, and function of space repartition of heterogeneities, the impressed electric field can be surface or interface (contact) ones.

Relation (2.10.2) can be also written under the form:

,JEE i (2.10.6)

in which is emphasized the dimensional material quantity called electric resistivity

.1 (2.10.7)

The temperature dependence of the resistivity is well approximated by the linear relation:

,1 00 (2.10.8)

where 0 is a reference temperature, and represents the resistivity temperature variation coefficient.

For the majority of the metals 0 (resistivity increases with temperature), but there are other conducting materials such as coal for example, for which 0 (their resistivity decreases while the temperature increases).

65

In the table below there are presented the values of the resistivity and of the resistivity temperature variation coefficient for a few of conducting materials.

Name of the material at C020

][ 2 m/mm

310 between

C00 and C0100

Silver 0,0161 4

Technical Cooper 0,0175 4,45

Electrolytic Cooper 0,0175 4,4

Gold 0,0237 3,77

Aluminum 0,0278 4,23

Yellow Cooper 0,08 1,5

Platinum 0,0866 2,47

Iron 0,0918 6,25

Nickel 0,138 6,21

Plumb 0,221 4,11

Cooper Nickel 0,4 0,2

Manganin 0,43 0,6

Constantan 0,45 0,4

Retort carbon 7,25 -0,3

Carbon 40100 -0,2

Uranium Oxide 501000 -1,5

Copper Oxide 0,051000 -2,6

Function of the values for resistivity, respectively of conductivity, the materials can be classified into three big categories, namely:

• insulators, having big values for resistivity (respectively vary small values for conductivity); a perfect insulator is the material for which 0 ;

66

semiconductors, with average values for resistivity (respectively for conductivity) and with exponential shape for their temperature function variation;

conductors, with very small values for resistivity (respectively very big values for conductivity); a perfect conductor is the material for which

0 .

The physical significance of the electric conduction law is to underline as a cause for the apparition of the electric field the bodies that possess impressed electric field.

For all constitutive laws, the local forms are those that give the consistent information. The constitutive law of electric conduction is not an exception from this rule. The integral form of this law is very useful for electric circuits study and it will be presented for the particular case of the conductors having transverse sections sufficiently small in order to consider that the repartitions of the conduction electric currents through the conductors’ sections are uniform. These conductors are called filiform.

Figure 2.10.1, a Figure 2.10.1, b

Let consider a section from such a filiform conductor, having the area of the section constant A (see Figure 2.10.1, a), through which flows a current i . The module of the current density J will be then

.AiJ (2.10.9)

1M

2M

e

R

i

fu bu

67

Integrating the local form of the constitutive law of the electric conduction along the axis C of the conducting section, between the extremities 1M and

2M of this section, one obtains:

2

1

2

1

2

1

2

1

2

1

2

1

2

1

M

CM

M

CM

M

CM

M

CM

M

CM

i

M

CM

M

CM

i

Adlidl

AidlJ

dlJdlEdlEdlEE

(2.10.10)

because the vectors J and dl are omo-parallel, and the electric conduction current is along the conductor. In this relation appear the following quantities:

electric voltage along the line:

;2

1

M

CM

f dlEu (2.10.11)

impressed electromotive force

;2

1

M

CM

iiie dlEeeu (2.10.12)

electric resistance of the conducting section

.2

1

M

CM A

dlR (2.10.13)

So, the integral form of the constitutive law of electric conduction for filiform conductors is:

iReu f (2.10.14)

and it has the following remarks.

68

▪ The reference directions for global quantities fu ,e and i that appear

in relation (2.10.14) are associated using the convention depicted in the graphical representation from Figure 2.10.1,b. Because in general the reference directions are arbitrary chosen, they can be optionally chosen; if one uses other reference directions that the ones presented in Figure 2.10.1,b, the integral form of the constitutive law of electric conduction changes by changing the corresponding signs of the terms with changed reference directions.

▪ In the particular case of the steady state regime, one proved that the voltage does not depend on the path and, as a consequence, the voltage along the path

fu is equal to the voltage bu computed along any open curve bC having as

extremities the point 1M and 2M (called terminals in the electric circuit theory); the electric voltage

2

1

M

bCM

b dlEu (2.10.15)

is called the terminals’ voltage, and the integral form of the constitutive law of electric conduction becomes

.iReub (2.10.16)

2.11. The law of power transfer associated to electric conduction

Statement: The power volume density p of the power transferred from electromagnetic field to substance in the electric conduction process is equal, at any moment of time, to the dot product between the electric field strength E and the conduction electric current density J :

.JEp (2.11.1)

We remark that the sign of the dot product shows the real direction of the transferred power: from the field towards the conducting body if 0p , respectively from the conducting body towards the field if 0p .

69

From relation (2.10.6) and (2.11.1) one can obtain a new expression of the local form of the law, for linear and isotropic conductors:

.2

JEJJEJp ii (2.11.2)

The two terms from the right side of equation (2.11.2) have the following interpretation:

Term 2J is positive and it represents the power volume density irreversibly transferred by the electromagnetic field of the conductor.

Term JE i has the direction imposed by the orientation of the two vectors iE and J ; if is the smallest angle between the two vectors, then we can

have the following situations: 2;0 , so 0 JE i and the power is transferred from the impressed electric field sources (of the conductor) to the electromagnetic field or ;2 , so 0 JE i and the power is

transferred from the electromagnetic field to impressed electric field sources (of the conductor) or 2 and there is no power transfer between the electromagnetic field and the conductor.

The integral form of the law is obtained by first integrating the local form (2.11.1) on domain V in which the electric conduction process takes place. The total power transferred by the electromagnetic field to the carrying current conductors from V

,

VV

dvJEdvpP (2.11.3)

and the corresponding energy to this phenomena between two time moments 1t and 2t is

.2

1

2

1

dtdvJEdtPWt

t V

t

t

(2.11.4)

The measure units S.I. for the quantities p , P and W are, respectively, watt per cubic meter (W / 3m ), watt (W ) and watt multiplied by second ( sW ).

For a filiform conductor in steady state regime (see Figure 2.10.1,a), taking into account that dlAdv , it results that the integral relation (2.11.3) takes the particular form

70

,bbf

C

VV V

PiuiudlEi

dlAJEdvJEdvJEP

(2.11.5)

that shows that the power transferred by the electromagnetic field to the conducting section in the electric conduction process is equal to the power bP called received power at its terminals.

Regarding the physical significance, the law characterizes the thermal effect of the electromagnetic field, referring to heat producing phenomena (Joule-Lenz effect) which is associated to the electric conduction current flow through the conductors, that corresponds to the transformation of electromagnetic energy into caloric energy.

2.12. The electrolysis law

Statement: The substance mass m deposited at one of the electrodes of an electrolytic bath in a time interval 21;tt depends on the electric current ti flowing through the bath as in relation:

,1 2

10 t

t

dttivA

Fm (2.12.1)

where 0F is a universal constant called Faraday’s constant, that has the value

,.

6,484.960 gramechCF (2.12.2)

A is the mol’s mass of the deposited substance, and v is the valence of a ion from the deposited substance.

The ratio v/A is called the chemical equivalent, and its value are presented in the table below.

71

Substance Atomic mass

gA

Valence

v Chemical

equivalent v/A g

Aluminum 1 1 1

Oxygen 16 2 8

Aluminum 27 3 9

Magnesium 24 2 12

Iron 3 55,9 3 18,6

Nickel 58,6 2 29,3

Cooper 2 63,2 2 31,6

Zinc 68,8 2 34,4

Potassium 39 1 39

tin 117,4 2 58,7

Cooper 1 63,2 1 63,2

Gold 196,1 3 65,4

Platinum 194,4 2 97,2

Mercury 199,8 2 99,9

Plumb 206,4 2 103,2

Silver 107,7 1 107,7

The physical significance of the electrolysis law is that it shows the mass transport phenomena associated to electric current flow through certain substances.

The electrolysis process has many practical applications such that electro-metallurgy (metal extraction from ore, metal refining, substance preparation etc.) and galvanotechnics (galvanostegy – the coating of some object by nickel plating, chrome plating, cadmium plating a.s.o and galvanoplasty – object shape reproduction).