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EELE 3332 – Electromagnetic II Chapter 9 Maxwell’s Equations Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik 2012 1
EELE 3332 – Electromagnetic II
Chapter 9
Maxwell’s Equations Islamic University of Gaza
Electrical Engineering Department
Dr. Talal Skaik
2012 1
2
Electrostatic Fields
produced by stationary charges.
Magnetostatic Fields
produced by steady (DC) currents or stationary magnet
materials.
Electrostatic Fields and Magnetostatic Fields do not vary
with time (time invariant).
Electromagnetic Fields
produced by time-varying currents.
Review – Electrostatics and Magnetostatics
3
Maxwell’s Equations for static fields
4
Michael Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. He demonstrated this principle of induction in 1831.
Michael Faraday (1791-1867)
9.2 Faraday’s Law
5
• In 1831, Farady discovered that a time-varying magnetic
field would produce an induced voltage (called
electromotive force or emf).
• According to Faraday’s experiments, a static magnetic field
produces no current flow.
Faraday discovered that the induced Vemf (in volts) in
any closed circuit is equal to the time rate of change of
the magnetic flux linkage by the circuit.
Faraday’s Law
6
where is the flux linkage
is the number of turns in the circuit.
is the flux through each turn.
N
N
emf
d dV N
dt dt
Faraday’s Law
7
emf
dV N
dt
Faraday’s Law
Change flux due to moving permanent magnet
8
Faraday’s Law
9
Induced EMF is in direction that opposes the change in flux that caused it
Minus Sign? Lenz’s Law
Notice that in time varying situation, both electric and
magnetic fields are present and interrelated. 10
9.3 Transformer and motional electromotive forces
emf
For a circuit with a single turn , N 1
In terms of E and B
= . .
where is the surface area of the
circuit bounded by the closed path L
emf
L S
dV
dt
dV E dl B dS
dt
S
11
The variation of flux with time may be caused in three ways:
1) By having a stationary loop in a time-varying B field.
(Transformer induction)
2) By having a time-varying loop area in static B field.
(motional induction)
3) By having a time-varying loop area in a time-varying B
field.
(general case, transformer and motional induction)
12
Application: DC Generators
13
Application: AC Generator
Water turns wheel
rotates magnet
changes flux
induces emf
drives current
14
A. Stationary loop in Time-Varying B field (Transformer EMF)
emf = . .L S
BV E dl dS
t
15
emf = . .
By applying Stoke's theorem
E . .
L S
S S
BV E dl dS
t
BdS dS
t
A. Stationary loop in Time-Varying B field (Transformer EMF)
EB
t
This is one of Maxwell’s equations for time-varying fields.
Note that (time-varying E field is non-conservative) E 0
16
B. Moving loop in static B field (Motional EMF)
Consider a conducting loo moving with uniform velocity
u, the emf induced in the loop is
emf = . ( ).m
L L
V E dl u B dl
• This type of emf is called motional emf or flux-cutting
emf (due to motion action).
17
B. Moving loop in static B field (Motional EMF)
• It is kind of emf found in electrical machines such as motors
and generators.
• Given below is in example of dc machine , where voltage is
generated as the coil rotates within the magnetic field.
18
B. Moving loop in static B field (Motional EMF)
• Another example of motional emf is illustrated below, where
a conducting bar is moving between a pair of rails.
19
m
m
emf
Recall that the force on a charge moving with uniform
velocity u in a a magnetic field B is: F u B
We define the motional electric field E as
FE u B
= . ( ).
By applying Stok
m
m
m
L L
Q
Q
V E dl u B dl
e's theorem,
. .
m
S S
m
E dS u B dS
or E u B
B. Moving loop in static B field (Motional EMF)
emf = . ( ).m
L L
V E dl u B dl
20
B. Moving loop in static B field (Motional EMF)
emf = . ( ).m
L L
V E dl u B dl
Notes:
• The integral is zero along the portion of the loop where u=0. (e.g.
dl is taken along the rod in the shown figure.
• The direction of the induced current is the same that of Em or uxB.
The limits of integration are selected in the direction opposite to
the direction of u x B to satisfy Lenz’s law. (e.g. induced current
flows in the rod along ay, the integration over L is along -ay).
21
C. Moving loop in Time Varying Field
Both Transformer emf and motional emf are present.
Note that eq. can always be applied in place of the
equations in cases A, B, and C.
emf = . l . S ( ). lL S L
BV E d d u B d
t
B
E= u Bort
emf
dV
dt
Motional Transformer
22
a conducting bar can slide freely over two conducting rails as
shown in the figure. Calculate the induced voltage in the bar
6 2
z
2
z
6
z
( ) If the bar is staioned at y=8 cm and 4 cos 10 a mWb/m
( ) If the bar slides at a velocity u 20 m/s and B=4a mWb/m
( ) If the bar slided at a velocity u 20 m/s and
B=4 cos 10 a
y
y
a B t
b a
c a
t y
2 mWb/m
Example 9.1
23
0.08 0.06
3 6 6
0 0
3 6
6
( ) we have transformer emf
Vemf=- . 4(10 )(10 )sin10
=4(10 )(0.08)(0.06)sin10
=19.2 sin 10 V
y x
a
BdS t dx dy
t
t
t
0
3
( ) This is motional emf
Vemf= . .
= - 20(4.10 )(0.06)
=- 4.8 mV
y z x
x l
b
u B dl ua Ba dx a
ubl
24
0.06
3 6 6
0 0
( ) Both transformer emf and motional emf are present in this case.
this problem can be solved in two ways:
Method 1:
Vemf= - . ( ).
4.(10 )(10 )sin(10 )
y
x
c
BdS u B dl
t
t y dy dx
0
3 6
x
0.06
6 3 6
0
6 6 3 6
6 6
+ 20 4.10 cos(10 ) . a
240cos 10 80(10 )(0.06)cos(10 )
240cos 10 240cos10 4.8(10 )cos 10
240cos 10 240cos10
y z
y
a t y a dx
t y t y
t y t t y
t y t
25
0.06
6
0 0
6
0
Method 2:
Vemf=- , where B. S
= 4cos(10 )
=-40(0.06)sin(10 )
y
y x
y
y
dt
t y dx dy
t y
6 6
6 6
= -0.24sin(10 )+0.24 sin10 t mWb
20
,
0.24sin(10 20 ) 0.24sin10 mWb
Vem
t y
But
dyu y ut t
dt
Hence
t t t
6 6 6 6
6 6
f=- 0.24(10 20)cos(10 20 ) 0.24(10 )cos10 mV
240cos 10 240cos10
t t tt
t y t
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The loop shown in the figure is inside a uniform magnetic field B=50
ax mW/m2. If side DC of the loop cuts the flux lines at the frequency
of 50 Hz and the loop lies in the yz-plane at the time t=0, find
(a) The induced emf at at t=1 ms.
(b) The induced current at t=3 ms.
Example 9.2
27
0 x 0
The B field is time invariant, the induced emf is motional
Vemf= ( ). , where , u=
4 cm, 2 100
Transform B into cylindrical corrdinates:
B=B a B cos sin , wher
z zu B dl dl d a a at
AD f
a a
0
0
0 0
0
0.03
0
e B =0.05
u×B 0 0 B cos
B cos B sin 0
u×B . B cos 0.04(100 ) 0.05 cos
= 0.2 cos
0.2 cos 6 cos mV
z
z
z
a a a
a
dl dz dz
dz
Vemf dz
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0 0
0
To determine recall that
= ( is the integration constant)
At t=0, / 2 beacause the loop is in the yz-plane at that time,
/ 2. Hence, / 2
6 cos / 2 =6 sin(100 ) mV
A
dt C C
dt
C t
Vemf t t
t t=1 ms, Vemf=6 sin(0.1 ) =5.825 mV
(b) The current induced is
60 sin(100 ) mA
At t=3ms, i=60 sin(0.3 ) mA=0.1525 A
-------------------------------------------------
Vemfi t
R
---------------
Note that for sides AD and BC u×B . 0 since . 0zdl a a
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9.4 Displacement current
Here we will consider Maxwell's curl equation for magnetic fields
(Ampere's Law) for time-varying conditions.
For static EM fields, recall that
H J
(1)
But the divergence of the curl is zero:
0 (2)
However, the equation of continuity requires that
V
H J
Jt
(3)
Thus, equations (2) and (3) are incompatible for time
varying conditions !!
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d
Here We must modify equation (1) to consider time-varying situation:
To do this, add a term to eq. (1):
J J (4)
again, taking the divergence
dJ
H
d
d
d d
, we have:
0 J J (5)
Since J , then, J
Since D
D DJ J = (6)
Substituting eq (6) into
V V
V
H
t t
Dt t t
eq (4) results in,
D J (7)H
t
Displacement current
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DJH
t
Displacement current This is Maxwell's equation ( based on
Ampere's circuit law) for a time varying field.
The term is known as displacement current density.
This is the third type of current density we have met:
•Conduction current density:
(motion of charge in a conductor)
•Convection Current Density:
(doesn’t involve conductors, current flows through an insulating
medium, such as liquid, or vacuum).
•Displacement Current Density:
(is a result of time-varying electric field).
DJ =d
t
J= E
VJ= u
DJ =d
t
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The insertion of Jd into Ampere’s equation was one of the major
contributions of Maxwell.
Without the term Jd, the propagation of electromagnetic waves
(e.g., radio or TV) would be impossible.
Displacement current is the mechanism which allows
electromagnetic waves to propagate in a non-conducting medium.
The displacement current is defined as:
DI = J . S . Sd d d d
t
Displacement current
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Displacement current
A typical example of displacement current is the current through a
capacitor when alternating voltage source is applied to its plates.
The total current density is J+Jd.
Take Amperian path as shown. Consider 2 surfaces bounded by path L.
If surface S1 is chosen: Jd=0
If surface S2 is chosen: J=0
1
encH. = J. S= = L S
dl d I I
2 2
dH. = J . S= D. S = L S S
d dQdl d d I
dt dt
So we obtain the same current for either surface.
A time-varying electric field induces magnetic field inside the capacitor.
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Conduction to Displacement Current Ratio
The conduction current density is given by The displacement current density is given by Assume that the electric field is a sinusoidal function of time: Then, We have Therefore
Note: In free space (or other perfect dielectric), the conduction current is zero and only displacement current can exist.
0 cosE E t
0 0cos , sinc dJ E t J E t
cJ E
d
EJ
t
0 0max max , c dJ E J E
max
max
c
d
J
J
Good Conductor ( neglible)
Good Insulator ( neglible)
d
c
I
I
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A parallel plate capacitor with plate area of 5 cm2 and plate
separation of 3 mm has a voltage 50 sin 103 t V applied to its plates.
Calculate the displacement current assuming ε=2 ε0.
Example 9.4
9 43 3
3
3
DJ . , J =
D J =
( same as conduction current ).
10 5 102. . .10 50cos(10 )
36 3 10
147.4cos(10 ) nA
d d d
d
d c
d
d
I St
V dVbut D E
d t d dt
S dV dVI C I
d dt dt
I t
I t
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9.5 Maxwell’s Equations in Final Forms
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The concepts of linearity, isotropy, and homogeneity of a material
medium still apply for time varying fields.
In a linear, homogeneous, and isotropic medium characterized by
σ, ε, and μ, the following relations hold for time varying fields:
Consequently, the boundary conditions remain valid for time
varying fields.
0
0
D E E P
B H (H M)
J E uV
1 2 1 2
1 2 1 2
0, -
, - 0
t t n n S
t t n n
E E D D
H H K B B
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Electromagnetic flow diagrams
Figure 9.11 Electromagnetic flow diagrams showing the relationship between the potentials and vector fields: (a) electrostatic system, (b) magnetostatic system, (c)
electromagnetic system.
