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Chung-Ang University Field & Wave Electromagnetics
CH 7. Time-Varying Fields and Maxwell’s Equations
Chung-Ang University Field & Wave Electromagnetics
7.1 Introduction
Fundamentl RelationsElectrostatic
ModelMagnetostatic
Model
Governing equations
Constitutive relations
(linear and isotropic media)
D
E 0
JH
B
0
In the static case, electric field vectors and
and magnetic field vectors and form separate and independent pairs.
ED BH1
E D
B H
Fundamental Relations for Electrostatic and Magnetostatic Models
In a conducting medium, static electric and magnetic fields may both exist and
form an electromagnetostatic field.
Chung-Ang University Field & Wave Electromagnetics
7.1 Introduction
A static electric field in a conducting medium causes a steady current to flow that,
In turn, gives rise to a static magnetic field.
The electric field can be completely determined from the static electric charges or
potential distributions.
The magnetic field is a consequence; it does not enter into the calculation of the
electric field.
In this chapter we will see that a changing magnetic field gives rise to an electric field,
and vice versa.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
Michael Faraday, in 1831, discovered experimentally that a current was induced in a
conducting loop when the magnetic flux linking the loop changed.
The quantitative relationship between the induced emf and the rate of change of flux
linkage, based on experimental observation, is known as Faraday’s law.
Fundamental Postulate for Electromagnetic Induction
t
BE (7 - 1)
Equation 7-1 expresses a point-function relationship; that is, it applies to every point in
space, whether it be in free space or in a material medium.
The electric field intensity in a region of time-varying magnetic flux density is therefore
nonconservative and cannot be expressed as the gradient of a scalar potential.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
Taking the surface integral of both sides of Eq.(7-1) over an open surface and applying
Stokes’s theorem, we obtain
.sdt
BldEsd
t
BE
sc
(7 - 2)
7–2.1 A STATIONARY CIRCUIT IN A TIME-VARYING MAGNETIC FIELD
Equation (7-2) is valid for any surface S with a bounding contour C, whether or not a
physical circuit exists around C.
For a stationary circuit with a contour C and surface S, Eq(7 - 2) can be written as
.sdBdt
dldE
sc (7 - 3)
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
If we define c
ldE
s
sdB
= emf induced in circuit with contour C (V) (7 – 4)
= magnetic flux crossing surface S (Wb), (7 – 5)
then Eq.(7 - 3) becomes dt
d (V). (7 – 6)
Equation (7 – 6) states that the electromotive force induced in a stationary closed circuit
is equal to the negative rate of increase of the magnetic flux linking the circuit.
This is a statement of Faraday’s law of electromagnetic induction.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
7-2.2 TRANSFORMERS
A transformer is an alternating-current (a-c) device that transforms voltage, currents,
and impendances.
(a) Schematic diagram of a transformer.
FIGURE 7-1
)(1tv )(
2tv
)(1ti )(
2ti
N 1 N 2RL
+
_
+
_
For the closed path in the magnetic circuit
In Fig.7-1(a) traced by magnetic flux ,
we have, from Eq.(6-101),
( Eq.(6-101), )
,2211
iNiN
. k
kkj
jj IN
(7 – 7)
.S
l
(7 – 8)
( core of length l, cross-sectional area S,
permeability )Substituting Eq.(7-8) in Eq.(7-7)
.2211
S
liNiN (7 – 9)
back
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
X 1 X 2
(b) An equivalent circuit.
FIGURE 7-1
Ideal transformer
R1 R2N 1 N 2i1 i2
X c Rc RLv1 v2
+
_
+
_
a) Ideal transformer.
For an ideal transformer we assume that , and
.2211
S
liNiN Eq.(7 - 9) becomes .
1
2
2
1
NN
ii
(7 – 10)
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
Eq.(7 - 10) states that the ratio of the currents in the primary and secondary
windings of an ideal transformer is equal to the inverse ratio of the numbers
of turns.
.1
2
2
1
NN
ii
dt
dNv
11,
22 dt
dNv
.2
1
2
1
NN
vv
Faraday’s law tells us that
(7 - 11) and (7 – 12)
From Eqs. (7 – 11) and (7 – 12) we have (7 – 13)
Thus, the ratio of the voltages across the primary and secondary windings of an ideal
transformer is equal to the turns ratio.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
,
2
11 )()(
2
RNN
R Leff ,
)/(
)/(1
212
221
1
1)(iNNvNN
ivR eff
RLWhen the secondary winding is terminated in a load resistance , as shown in Fig.
7-1(a) click, the effective load seen by the source connected to primary winding is
or (7 – 14a)
For a sinusoidal source and a load impedance , it is obvious that the effective
load seen by the source is , an impedance transformation.
)(1tv Z L
ZNN L)/( 212
.
2
11 )()(
2
ZNNZ Leff
We have (7 – 14b)
b) Real transformer.
Referring to Eq. (7 – 9), we can write the magnetic flux linkages
of the primary and secondary windings as
.2211
S
liNiN
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
),(2211
2
111 iNNiNN l
S
).(2
2
212122 iNiNNN l
S
,212
111 dt
diLdtdiLv ,2
21
122 dtdiLdt
diLv
,2
11 NL l
S
,2
22 NL l
S
.2112 NNL l
S
(7 – 15)
(7 – 16)
Using Eqs. (7 - 15) and (7 - 16) in Eqs. (7 - 11) and (7 - 12), we obtain
(7 – 17) (7 – 18)
where (7 – 19) the self-inductance of the primary winding.
(7 – 20) the self-inductance of the secondary winding.
(7 – 21) the mutual inductance between the primary and
secondary windings.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
For an ideal transformer there is no leakage flux, and .2112 LLL
,2112 LLL kFor real transformers, k < 1 , (7 -22)
Where k is called the coefficient of coupling.
For real transformers we have the following real-life conditions.
the existence of leakage flux ( k < 1 ),
noninfinite inductances, nonzero winding resistances,
the presence of hysteresis and eddy-current losses.
The nonlinear nature of the ferromagnetic core further compounds the difficulty of
an exact analysis of real transformers.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
Eddy currents.
When time-varying magnetic flux flows in the ferromagnetic core, an induced emf will
result in accordance with Faraday’s law.
This induced emf will produce local currents in the conducting core normal to the
magnetic flux.
These currents are called eddy currents.
Eddy currents produce ohmic power loss and cause local heating.
This is the principle of induction heating.
In transformers, eddy-current power loss is undesirable and can be reduced by using
core materials that have high permeability but low conductivity (high and low ).
For low-frequency, high-power applications an economical way for eddy-current
power loss is to use laminated cores.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
7-2.3 A MOVING CONDUCTOR IN A STATIC MAGNETIC FIELD.
☉ ☉
☉
☉
☉
☉
☉
☉
☉
☉
☉
☉
2
1B
Figure 7-2 A conducting bar moving in a magnetic field.
u
u
dl
BuqF mA force will cause the
freely movable electrons in the conductor to
drift toward one end of the conductor and
leave the other end positively charged.
This separation of the positive and negative
charges creates a Coulombian force of attr-
action.
The charge-separation process continues
until the electric and magnetic forces
balance each other and a state of equilibri-
um is reached.
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
BuqF m/
ldBuc
)(
.)(2
121ldBuV
'
To an observer moving with the conductor there is no apparent motion, and the magnetic force per unit charge can be interpreted as an induced electric field acting along the conductor and producing a voltage
(7 – 23)
If the moving conductor is a part of a closed circuit C, then the emf generated around the circuit is
(V). (7 – 24)
This is referred to as a flux cutting emf or a motional emf. Obviously, only the part of the circuit that moves in a direction not parallel to the magnetic flux will contribute in Eq. (7 – 24).
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
).( BuEqF
BuEE .BuEE
7-2.4 A MOVING CIRCUIT IN A TIME-VARYING MAGNETIC FIELD.
ldBusdt
BldE
csc
)(
Lorentz’s force equation (7 – 31)
The force on q can be interpreted as caused by an electric field , whereE
(7 – 32) (7 – 33)
Hence, when a conducting circuit with contour C and surface S moves with a velocity
in a field , we use Eq. (7 – 33) in Eq. (7 – 2) to obtain
u),( BE
.sdt
BldEsd
t
BE
sc
(V). (7 – 34)
( Eq. 7-2 )
Eq. (7 – 34) is the general form of Faraday’s law for a moving circuit in a time-varying magnetic field.
or
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
C1 C2
tt Let us consider a circuit with contour that moves from at time t to at time
in a changing magnetic field .B
C
B C2
C1
S 2
S1tu
ld
3Sd
FIGURE 7-5A moving circuit in a time-varying magnetic field.
The time-rate of change of magnetic flux
through the contour is
sdBdt
d
dt
ds
.)()(1
1212
0lim
sdtB
sdttB
t sst
(7 – 35)
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
)( ttB
.,..)(
)()( TOHtt
tBtBttB
in Eq. (7 - 35) can be expanded as a Taylor’s series:
,...1
lim2 1
120
s sTOHdBdB
tsd
t
BsdB
dt
dsstss
(7 – 36)
Substitution of Eq. (7 – 36) in Eq. (7 – 35) yields
(7 – 37)
An element of the side surface is .3
tuldds (7 – 38)
Apply the divergence theorem for at time t to the region sketched in Fig. 7 – 5 :
,312
312sss d
sBd
sBd
sBdvB
V (7 – 39)
B
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
0 B
.)(12
12
ldButdsBd
sB
css Using Eq. (7 – 38) in Eq. (7 – 39) and noting that , we have
,)( ldBusdt
BsdB
dt
dcss
dt
d
sdBdt
ds
ldEc
Combining Eqs. (7 – 37) and (7 – 40), we obtain
(7 – 40)
which can be identified as the negative of the right side of Eq. (7 – 34).
If we designate
emf induced in circuit C measured in the moving frame, (7 – 42)
(7 – 41)
Eq. (7 – 34) can be written simply as
(V),
(7 – 43)
Chung-Ang University Field & Wave Electromagnetics
7.2 Faraday’s Law of Electromagnetic Induction
dt
d
Eq. (7 – 43) is of the same form as Eq. (7 – 6).
If a circuit is not in motion, reduces to , and Eqs. (7 – 43) and (7 – 6) are exactly the same.
Faraday’s law that the emf induced in a closed circuit equals the negative time-rate of increase of the magnetic flux linking a circuit applies to a stationary circuit as well as a moving one.
Chung-Ang University Field & Wave Electromagnetics
7.3 Maxwell’s Equations
The fundamental postulate for electromagnetic induction assures us that a time-varying magnetic field gives rise to an electric field.
Time-varying case: 0 E .t
BE
,JH
, D .0 B
The revised set of two curl and two divergence equations from Table 7 – 1:
,t
BE
(7 – 47a) (7 – 47b)
(7 – 47c) (7 – 47d)
The mathematical expression of charge conservation is the equation of continuity :
.t
J
(7 – 48)
Divergence of Eq. (7 – 47b) : ,0)( JH (7 – 49) (null identity)
since Eq. (7 – 48) asserts that does not vanish in a time-varying situation,
Eq. (7 – 49) is, in general, not true.
J
Chung-Ang University Field & Wave Electromagnetics
7.3 Maxwell’s Equations
How should Eqs. (7 – 47a, b, c, d) be modified so that they are consistent with
Eq. (7 – 48)?
First of all, a term must be added to the right side of Eq. (7 – 49) :
Using Eq. (7 – 47c) in Eq. (7 – 50), we have
which implies that
Eq. (7 – 52) indicates that a time-varying electric field will give rise to a magnetic field, even in the absence of a current flow.
The additional term is necessary to make Eq. (7 – 52) consistent with the principle of conservation of charge.
t /
.0)(t
JH
),()(t
DJH
.t
DJH
tD /
(7 – 50)
(7 – 51)
(7 – 52)
The term is called displacement current density.tD /
Chung-Ang University Field & Wave Electromagnetics
7.3 Maxwell’s Equations
In order to be consistent with the equation of continuity in a time varying situation, both of the curl equations in Table 7 – 1 must be generalized.
The set of four consistent equations to replace the inconsistent equations,
Eqs. (7 – 47a, b, c, d), are
,t
DJH
, D
.0 B
,t
BE
(7 – 53a)
(7 – 53b)
(7 – 53c)
(7 – 53d)
They are known as Maxwell’s equations.
Chung-Ang University Field & Wave Electromagnetics
7.3 Maxwell’s Equations
7-3.1 INTEGRAL FORM OF MAXWELL’S EQUATIONS.
The four Maxwell’s equations in (7 – 53a, b, c, d) are differential equations that are valid at every point in space.
In explaining electromagnetic phenomena in a physical environment we must deal with finite objects of specified shapes and boundaries.
It is convenient to convert the differential forms into their integral-form equivalents.
We take the surface integral of both sides of the curl equations in Eqs. (7 – 53a, b) over an open surface S with contour C and apply Stokes’s theorem to obtain
sdt
BldE
sc
.)( sdt
DJldH
sc
(7 – 54a)
(7 – 54b)
Chung-Ang University Field & Wave Electromagnetics
7.3 Maxwell’s Equations
Taking the volume integral of both sides of the divergence equations in Eqs. (7 – 53c, d) over a volume V with a closed surface S and using divergence theorem, we have
dvsdDvs
.0 sdBs
(7 – 54c)
(7 – 54d)
The set of four equations in (7 – 54a, b, c, d) are the integral form of Maxwell’s
equations.
Chung-Ang University Field & Wave Electromagnetics
7.3 Maxwell’s Equations
t
DJH
D
0 B
t
BE
dt
dldE
c
sdt
DIldH
sc
QsdDs
0 sdBs
Maxwell’s Equations
Differential Form Integral Form Significance
Faraday’s law
Ampere’s circuital law
Gauss’s law
No isolated magnetic charge