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Transcript of 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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0variable
0constant
str
str
c
str
EE
EE
e
F
E
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Course 7 - Electromagnetic FieldTheory
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q
FE
EE
neelstr
str 0
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Course 7 - Electromagnetic FieldTheory
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Course 7 - Electromagnetic FieldTheory
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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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Course 7 - Electromagnetic FieldTheory
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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