course_7 tce

8/12/2019 Course_7 TCE

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Course 7 - Electromagnetic FieldTheory

1

I.2. Electrokinetic fieldElectrokinetic state of conductors is evidenced by their heating and action effects, more

important than in electrostatic regime. This state is produced by ordered movement of electric

charges, volumetric distributed, in one way or another.

Electrical conductors are good conductive bodies. Classification emphasizes two types ofconductors depending on the chemical reactions that occur or not during conduction:

- First type conductors: electrokinetic state is not accompanied by chemical reactions.

Circulation of species is given only by electrons. From this category of conductors belong all

metals, few ceramics materials and semiconductors;

- Second type conductors: electrokinetic state is accompanied by chemical reactions.

Circulation of species is given by ions. From this category belong the aqueous solutions.

In both cases the circulation of species (of true electric charges) could be produced byelectric or non-electric causes. In consequence the electrokinetic state could be produced by

electric or non-electric forces. Applying an electric potential difference to a conductor, inside it

will appear an electric field moving the electrons or ions.

Electric field and the forces generated by him oppose non-electrical forces because of

local physico-chemical heterogeneities and interatomic forces. The work done by electric and

non-electric forces on electric charges that will move is:

lFFL l neel d


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It will define as electromotive force on that contour, closed or opened, the work exercised by

electric and non-electric forces to move the electronic electric charge on that contour (as a

potential theorem consequence):

l l str

neellEEel

q

FF

q

Le dd

The non-electric forces will produce a foreign electric field, called also induced. The

induced electromotive force will be always the cause of moving the electric charge for

electronic type in first and second type of conductors.

The foreign electric fields could be:

-Volumetric (acceleration, concentration, temperature);- Surface (voltaic, photovoltaic, galvanic).


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3

0variable

0constant

str

str

c

str

EE

EE

e

F

E


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4

q

FE

EE

neelstr

str 0


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5


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6

ABB

)n(A

B

)n(A str

B

)n(A

B

)n(A str

A

)m(B str

l

A

)m(B str

B

)n(A strstr

UlEelElEe

0lE

0lEE

lEElEElEEe

ddd

d

d

ddd


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7

Electric current. Electric current density

The electrokinetic state of conductor could

be evaluated by using a scalar parameter,

i, called electric current. This parameter

represents the ordered movement of a set

of particles related to a reference system.

nn SS0t t

q

t

qi

d

dlim

To define the direction in which electric charge flows, we introduce a vector parameter

that is called the current density, J, defined by:

SS

SSS0A

AJcosAJicosA

iJ

cosAA,A

i

A

i

lJ nnn

ddd

d

dd

d

dim


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Real electric charge conservation theorem under electrokinetic regime

t

qi

d

d Global form 1 of electric charge conservation law

V VV

V VuVt

i ddivd

Global form 2 of electric charge conservation law

tJ

tuJ

ut

J

VuVt

i

VJAJi

Vt

VV

VV

V VV

V

VS

divdiv

divdiv

ddivd

ddivd

Local form of electric charge

conservation law

Consider the following particular cases of local form of electric charge conservation law:

- If the conductors system is not moving then u=0, so:

tJ V

div


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- if the work regime is stationary (DC current), then:0J

div

Current density vector lines are closed on themselves. They have no beginning and no

end. The field of the current density vector has solenoid shape.

0AJAJVJ

0J

V

V

S

SV

d

dddiv

div

Global form 3 of the electric charge conservation law


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Current intensity of Hertz

If the electric charge q varies differently in time than its actual movement, then introduce the

current intensity of Hertz, that assure the continuity of conduction current in each point:

0ii,t

qi HH

d

d

If the electric charge variation in time is caused by other phenomena than the classical,

electric charge could be written by using the integral form of Maxwell postulate and the current

intensity of Hertz becomes:

SSHS

At

DAD

tiADq d

d

dd

d

dd

uDDu

t

D

t

Dt),t(z),t(y),t(xDD rotdiv

d

d


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AuDDut

Di

SH drotdiv

uDut

DJ

D

AJiVH

V

S HH rot

div

d

Basically three types of current densities may occur. There are observed in previous relation.

a) Displacement current density - will be prevalent in areas where there are dielectrics (areas

occupied with capacitors)

t

P

t

EJ

PED

t

DJ

0D

0

D

b) Convection current density, Jc

d dd d d

d dV V V V

c V c V

q l i

q V A l i A ut t A

J u J u


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c) Rontgen type current density, JR

uPuEJ

PED

uDJ

0R

0

R

rotrot

rot


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Electrical conduction law (Ohms law)

It is a law of material that can be demonstrated. For demonstration are used someassumptions which have physical correspondent. It takes into account thediscontinuous structure of electric current. He is basically a stream of electrons whichhave huge spaces between them. Due to naturally internal agitation on conductors,

the group of electrons moving from one metal atom to another can be treated as anelectronic gas. In this electronic gas the interaction between the electrons isnegligible. The only factor to be taken into consideration is that the electrons collidewith the metal ions after attending the mean free path. The speed of the electronscan be calculated with classic energy balance equation. Due to natural internalagitation, electrons have different speeds on different directions and senses. It canbe considered an average speed of their group, vm. Because of real conditionsmentioned above it may be associated to electronic gas a monoatomic type model of

ideal gas. In this case the average speed of the group is:

m

Tkvm

3

where k = 1,3810-23 J/K is Boltzmanns constant, T absolute

temperature and mphysical mass group.


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When apply to the conductor an electric field from outside, some electrons of the conductor

are plucked by electric field forces. They ordered move in one direction and a sense with

speed u. Reported to conductor, the conduction current density can be considered as one of

convection, on a section of copper conductor:

uenuJJ VVC

where nVis the number of electrons, ethe electric charge of one electron,

uspeed of electrons imposed by total force having electric and non-electric component.

neel

neel

u a F Fu

mF F m a

m

neelV

m

neelm

vm

FFenJ

vm

FFuv


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strstrstrm

2V

m

str

Vstrneel

EEJEEEEvm

enJ

vm

EEe

enJEEeFF

is called electrical conductivity and is a material constant for a given temperature and

under certain conditions. The result of the expressions above is the material second equation

of electromagnetic field and this form is called local electrical conduction law.

Electric conductivity depends on frequency only when it exceeds 1014Hz (1/). At industrial

frequency the conductivity is independent of it. If the conductor is heated, the average speed

vm increases, the mean free path, , will decrease and therefore will decrease the

conductivity, .

It defines the electrical resistivity: = 1/


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Global form of Ohmslaw: refer to portions of conductor in electrokinetic regime and is

demonstrated by integrating local form 2 to a field line of current density vector.

2 2

1 1

2 2 2 2 2 2

1 1 1 1 1 1

2 2 2

1 1 1

1

d d

dd d d d d

dd d

str str str

str str

str

J E E E E J E E l J l

i lE l E l l E l E l i

A A

lE l u , E l e , i i R u e R i

AR is ohmic resistance which is defined by a material constant, the electricalresistivity, ,and by geometrical dimensions of the conductor.

The last expression represents the global form of Ohmslaw and is available in any

working regime (stationary or passive elements). An important consequence of this

global form of Ohmslaw is the second theorem of Kirchhoff.


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p

1jl j

j

jj

p

1jl jjj

p

1jl j

strj

p

1jl jjjl

p

1j l jstrj

l str

lll str

l

ll strstr

jjjj

j

lA

ilJlElJlJ

,lElE,0lE,lJlElE

lJlEEJEE

dddd

dddddd

dd

p

1j

jjp

1j

j

p

1j

p

1j

p

1j

jjlj

jjjl j

j

jj

p

1j

j

p

1jl j

jstr

iRe

iRA

lil

A

i,elE

jjj

ddd

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