Lecture 6: Dynamic Fields
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Faraday’s Law
Stationery Loop in a Time-Varying Magnetic Field
Ideal Transformer
Moving Conductor in a Static Magnetic Field
Electromagnetic Generator
Moving Conductor in a Time-Varying Field
Displacement Current
Boundary Conditions for Electromagnetics
Charge-Current Continuity Relation
Electromagnetic Potentials
Prof Joshua Le-Wei Li, EM Research Group
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Faraday’s Law
• Michael Faraday’s Hypothesis
– If I o H, then H o I or I l H.
• Verified by Michael Faraday & Joseph Henry
• Final Observation
– The magnetic field can
produce an electric current
in a closed loop, but only
if the magnetic flux linking
the surface area of the loop
changes with time.
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EE2011: Engineering Electromagnetics
Faraday’s Law
•
Mathematical Representation
)
³³ B ˜ ds, so that electromotive force Vemf
S
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N
d)
dt
N
d
B ˜ ds, (V).
dt ³³
S
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Faraday’s Law
• EMF Produced in 3 Conditions:
– Transformer emf: A time-varying magnetic field
tr
linking a stationery loop, Vemf
;
– Motional emf: A moving loop with a time-varying
area (relative to the normal component of B) in a
m
V
static field B, emf;
– Total emf: Sum of the transformer emf and the
motional emf when a moving loop in a time
tr
m
Vemf
.
varying field B, Vemf Vemf
)
³³ B ˜ ds, so that electromotive force Vemf
S
Prof Joshua Le-Wei Li, EM Research Group
5
N
d)
dt
N
d
B ˜ ds, (V).
dt ³³
S
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Stationery Loop in a TimeVarying Magnetic Field
• Lenz’ Law
– The current in the loop is always in such a direction as to opposite the
change of magnetic flux )(t) that produced it.
³ E˜dl ³³
C
wB
N ³³
˜ ds
wt
S
tr
emf
V
Stokes's theorem
S
wB
˜ ds
wt
N ³ E ˜ dl
C
N ³³ ’ u E ˜ ds.
S
– In other words, the induced magnetic field by the current is always
opposite to the change (wB/wt) of external field.
’uE
wB
, Faraday's law . I
wt
Prof Joshua Le-Wei Li, EM Research Group
6
tr
Vemf
R Ri
EE2011: Engineering Electromagnetics
Stationery Loop in a TimeVarying Magnetic Field
• Example 1: Question
Question: An inductor is formed by winding N turns of a thin conducting wire
into a circular loop of radius a. The inductor loop is in the x-y plane with
its center at the origin, and it is connected to a resistor R, as shown. In the
presence of a magnetic field given by B B0 yˆ 2 zˆ 6 sin Zt. Find
a)
the magnetic flux linking a single turn of the inductor;
b) the transformer emf, given that N = 10, B0 = 0.2 T, a = 10 cm, and Z = 103
rad/s,
c)
tr
the polarity of Vemf
at t = 0, and
d) the induced current in the circuit for
R = 1 k: (assume the wire resistance
to be negligibly small).
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Stationery Loop in a TimeVarying Magnetic Field
• Example 1: Solution
Solution:
a) The magnetic flux linking each turn of the inductor is
)
³³ B ˜ ds
S
tr
emf
³³ >B0 yˆ 2 zˆ 6sin Z t @˜ zˆ ds
6Sa 2 B0 sin Z t .
S
b) To find V we can apply the general expression defined earlier. The
latter approach gives
d)
d
tr
Vemf
N
N
6SNa 2 B0 sin Zt 6SNZa 2 B0 cosZt .
dt
dt
For N = 10, a = 0.1 m, Z = 103 rad/s, and B0 = 0.2 T,
tr
Vemf
377 cos 103 t .
>
@
c)
tr
At t = 0, d)/dt > 0, and thus Vemf
d) The current I is given by
V2 V1 377
3
I
cos
10
t
3
R
10
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V1 V2
377, (V).
0.38 cos 103 t .
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Questions & Answers
• Explain Faraday’s law and the function of
Lenz’s law?
• Solution
– Faraday’s law-Changing magnetic field could
produce induced current?
– Lenz’s law-The current direction is always
opposite to the change of magnetic flux.
• Under what circumstances is the net voltage
around a closed loop equal to zero?
• Solution: Magnetostatic case where B is static
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The Ideal Transformer
• Ideal Transformer: V
– Transformer is used to transform
currents, voltages, and impedances
between its primary and secondary
circuits.
V1
V1
V2
N1
d)
, and V2
dt
N1
, and P1
N2
N2
d)
.
dt
P2 .
Coupling ignored!!!
In a transformer, the directions of I1 and I2 are such that the flux ) generated by one of
them is opposite that generated by the other. The direction of the secondary winding in
(b) is opposite that in (a), and so are the direction of I2 and the polarity of V2.
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The Ideal Transformer
• Ideal Transformer: I & Z
P1
I1V1 , and P2
I1
I2
N2
N1
V2
I 2 RL , and V1
Rin
V1
I1
V2 § N1 ·
¨¨
¸¸
I2 © N2 ¹
I 2V2 ;
Equivalent circuit for the primary
side of the transformer
I1 Rin .
2
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§ N1 ·
¨¨
¸¸ RL .
© N2 ¹
11
2
Z in
§ N1 ·
¨¨
¸¸ Z L .
© N2 ¹
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Moving Conductor in a Static
Magnetic Field
• Magnetic force
Fm
qu u B • Motional electric field
Em
Fm
q
u u B,
• Motional emf
m
Vemf
m
emf
V
V12
V12
1
1
³ Em ˜ dl
³ u u B ˜ dl,
2
2
Conducting wire moving in a static
magnetic field
uB0l. This means that V1 V2 is negative or V2 is higher.
Only those segments of the circuit that cross
magnetic field lines contribute to induced emf.
Prof Joshua Le-Wei Li, EM Research Group
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m
Vemf
³ u u B ˜ dl.
C
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Moving Conductor in a Static
Magnetic Field
• Example 2: Moving Loop
Given that B( y )
zˆ 0.2e 0.1 y .
UsingȱtheȱLenz’sȱlaw,ȱtheȱcurrent
directionȱshouldȱbeȱclockwise.ȱ
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Electromagnetic Generator
• AC Motor and AC Generator
Fm
I ³ dl u B.
m
Vemf
C
³ u u B ˜ dl.
C
Principles of the a-c motor and the a-c generator. (a) The magnetic torque on
the wire causes the loop to rotate; (b) The rotating loop generates an emf.
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EE2011: Engineering Electromagnetics
Electromagnetic Generator
• Motional emf
– Method 1
nˆ u zˆ
m
Vemf
B
zˆ B0 , u nˆ ZA nˆ Z
xˆ sin D , so we have
1
V14
w
;
2
3
³ u u B ˜ dl ³ u u B ˜ dl
2
4
l / 2
ª§
º
w·
ˆ
ˆ
ˆ
u
n
z
Z
B
¸
0 » ˜ xdx
Ǭ
³
2¹
¼
l / 2 ©
l / 2
ª§
º
w·
ˆ
ˆ
ˆ
u
n
z
Z
B
¸
0 » ˜ xdx
Ǭ
³
2¹
¼
l / 2 ©
wlZB0 sin D AZB0 sin Zt C0 .
– Method 2
m
Vemf
d)
dt
d
>AB0 cosZt C0 @ AZB0 sin Zt C0 .
dt
d
B ˜ ds
dt ³³
S
Prof Joshua Le-Wei Li, EM Research Group
d
zˆ B0 ˜ nˆ ds ³³
dt S
15
A loop rotating in a magnetic
field induces an emf
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Questions & Answers
• Contrast the operation of an a-c motor with that of
an a-c generator.
• The rotating loop considered earlier has a single
turn. What would be the emf generated by a loop
with 10 turns?
• The magnetic flux linking the loop shown earlier is a
maximum when D=0, (loop in the x-y plane), and
yet according to Eq. (6.34), the induced emf is zero
when D=0. Conversely, when D=90o, the flux linking
the loop is zero, but the emf is at a maximum. Is this
consistent with your expectations? Why?
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EE2011: Engineering Electromagnetics
Moving Conductor in a TimeVarying Field
• Induced emf
– Sum of Transformer Component and
motional component
tr
m
Vemf
Vemf
Vemf
³ E ˜ dl
C
wB
³³
˜ ds ³ u u B ˜ dl.
wt
S
C
– Faraday’s Law
d)
dt
Vemf
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d
³³ B ˜ ds.
dt S
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Displacement Current
• Ampère’s Law
’uH
The displacement current I2d in the insulating
material of the capacitor is equal to the
conducting current I1c in the wire.
wD
J
.
wt
³³ ’ u H ˜ ds
Stokes's Theorem
S
³ H ˜ dl
C
• Displacement
Current
Ic=JA=VEA & Id=JdA=HAwE/wt.
wD
³³S J ˜ ds ³³S wt ˜ ds
³ H ˜ dl
C
³³ J
c
S
˜ ds ³³ J d ˜ ds
wD
³³S J ˜ ds ³³S wt ˜ ds
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Ic Id .
S
Ic Id
18
I.
•Ic and Id have a 90o phase difference;
•Ic is very much larger (a109) than Id.
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Boundary Conditions for
Electromagnetics
• Boundary Conditions for Electromagnetics
– The same as those for electrostatic and
magnetostatic cases;
– But only two of the four conditions are usually
used instead of the four conditions.
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Charge-Current Continuity
Relation
• Charge Continuity Relation
³³ J ˜ ds
S
³³ J ˜ ds
I
dQ
dt
divergence theorem
S
’˜J
d
³³³ U v dv
dt v
dU v
³³³
dv
dt
v
³³³ ’ ˜ Jdv
v
dU v J VE
. Ÿ V’ ˜ E
dt
’˜E
Uv
H
dU v
.
dt
V
Uv
H
• Kirchhoff’s Current Law
’˜J
0 or
³³ J ˜ ds
0.
S
¦ Ii
i
0.
U v t U v 0 e V / H t
U v 0 e t /W .
r
where Wr= H/V is relaxation time constant.
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Electromagnetic Potentials
• Faraday’s Law & Magnetic Vector Potential
’˜B
’uE
˜’u A { 0
0 (no magnetic charge) ’
o B ’ u A.
wB B
wt
’u A
wA
w
’ u A ’ u
wt
wt
’V
’V
• Electrical Scalar Potential & Field Expressions
wA
wA ·
§
’u’V { 0
’V .
’u¨E ¸ 0,  o E wt
wt ¹
©
wA
E ’V and B ’ u A.
wt
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