Download - EE 307 Chapter 9 - Maxwells Eqns
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Chapter 9: Maxwells Equations
Topics Covered Faradays Law
Transformer and Motional
Electromotive Forces
Displacement Current
Magnetization in Materials Maxwells Equations in Final
Form
Time Varying Potentials
(Optional)
Time Harmonic Fields (Optional)
Homework: 3, 7, 9, 12, 13, 16,
18, 21, 22, 30, 33
All figures taken from primary textbook unless otherwise cited.
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Faradays Law (1)
We have introduced several methods of examining magnetic fields in terms of forces,energy, and inductances.
Magnetic fields appear to be a direct result of charge moving through a system and
demonstrate extremely similar field solutions for multipoles, and boundary condition
problems.
So is it not logical to attempt to model a magnetic field in terms of an electric one? This is
the question asked by Michael Faraday and Joseph Henry in 1831. The result is Faradays
Law for induced emf
Induced electromotive force (emf) (in volts) in any closed circuit is equal to the time rate of
change of magnetic flux by the circuit
where, as before, is the flux linkage, is the magnetic flux, N is the number of turns in the
inductor, and t represents a time interval. The negative sign shows that the induced voltageacts to oppose the flux producing it.
The statement in blue above is known as Lenzs Law: the induced voltage acts to oppose the
flux producing it.
Examples of emf generated electric fields: electric generators, batteries, thermocouples, fuel
cells, photovoltaic cells, transformers.
dt
dN
dt
dVemf
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Faradays Law (2)
To elaborate on emf, lets consider a battery circuit. The electrochemical action within the battery results and in emf produced electric field, Ef
Acuminated charges at the terminals provide an electrostatic field Eethat also exist that
counteracts the emf generated potential
The total emf generated in the between the two open terminals in the battery is therefore
Note the following important facts An electrostatic field cannot maintain a steady current in a close circuit since
An emf-produced field is nonconservative
Except in electrostatics, voltage and potential differences are usually not equivalent
IRldEldEV
P
N
e
P
N
femf
L
e IRldE 0
P
Nf
Lf
L
ef
ldEldEldE
EEE
0
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Transformer and Motional
Electromotive Forces (1)
For a single circuit of 1 turn
The variation of flux with time may be caused by three ways
1. Having a stationary loop in a time-varyingB field
2. Having a time-varying loop in a static Bfield
3. Having a time-varying loop in a time-varying Bfield
A stationary loop in a time-varyingB field
SL
emf
emf
SdBdt
dldEV
dt
d
dt
dV
dt
BdE
SdBdt
dSdEldEV
SSL
emf
One of Maxwells for time varying fields
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A time-varying loop in a static Bfield
BuE
ldBuSdE
TheoremsStokesby
uBlV
IlBF
BIlF
ldBuldEV
BuQ
FE
fieldEmotionalain
BuQF
m
LL
m
emf
m
m
LLemf
m
m
_'_
____
Some care must be used when applying this equation
1. The integral of presented is zero in the portion of the
loop where u=0. Thus dl is taken along the portion
of the lop that is cutting the field where u is not
equal to zero
2. The direction of the induced field is the same as that
of Em. The limits of the integral are selected in the
direction opposite of the induced current, thereby
satisfying Lenzs Law
Transformer and Motional
Electromotive Forces (2)
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A time-varying loop in a time-varying Bfield
Budt
BdE
ldBuSddt
BdldEV
m
LSL
emf
One of Maxwells for time varying fields
Transformer and Motional
Electromotive Forces (3)
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Conducting element is stationary and themagnetic field varies with time
Assume the bar is held stationary at y =0.08 m
and B= 4cos(106t)azmWb/m2
Assume the length between the two conducting
rails the bar slides along is 0.06 m
Transformer and Motional
Electromotive Forces: Example1
dt
BdE
Sddt
BdV
m
S
emf
Vt
t
txy
dxdyt
SdatdtdSd
dtBdV
S
S
z
S
emf
)10sin(2.19)10sin()10)(10)(4(06.008.0
)10sin()10)(10)(4(
)10sin()10)(10)(4(
))10cos(004.0(
6
663
663
663
6
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Conductor moves at a velocity u= 20aym/s inconstant magnetic field B=4azmWb/m2
Assume the length between the two
conducting rails the bar slides along is 0.06 m
mV
xdx
adxaaV
BuE
ldBuldEV
x
L
zyemf
m
LL
emf
8.406.008.0
08.008.0
004.020
Transformer and Motional
Electromotive Forces: Example 2
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Conductor moves at a velocity u= 20aym/s in timevarying magnetic field B=4cos(106t-y)azmWb/m2
Assume the length between the two conducting
rails the bar slides along is 0.06 m
Transformer and Motional
Electromotive Forces: Example 3
L
xzy
z
S
zemf
LSL
emf
adxayta
adxdyaytdt
dV
ldBuSddt
Bd
ldEV
)10cos()4)(10(20
)10cos()4)(10(
63
63
)10cos(240)10cos(240
)10cos(4000)10cos(4000
)10cos()4(10)10cos()10(8)4)(10(
)10cos()4)(10(20
)10cos()4(10)10cos()4)(10(
)10cos()4)(10(20
)10cos()4(10)10cos()4)(10(
)10cos()4)(10(20
)10sin(10)4)(10(
66
66
63623
63
6363
63
6363
63
663
tytV
txytxV
xtxytV
xyt
xtxytV
dxyt
xtxytV
dxyt
adxdyaytV
emf
emf
emf
emf
emf
z
S
zemf
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Lets now examine time dependent fields from the perspective on Amperes Law.
t
DJH
t
DJ
t
DD
ttJJ
JJH
JJH
tJ
JH
JH
d
v
d
d
d
v
0
0
0
Another of Maxwells for time varying fields
This one relates Magnetic Field Intensity to conduction
and displacement current densities
Displacement Current (1)
This vector identity for the cross product is mathematically
valid. However, it requires that the continuity eqn. equals
zero, which is not valid from an electrostatics standpoint!
Thus, lets add an additional current density termto balance the electrostatic field requirement
We can now define the displacement current density as
the time derivative of the displacement vector
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Using our understanding of conduction and displacement current density. Lets test thistheory on the simple case of a capacitive element in a simple electronic circuit.
Idt
dQSdD
tSdJldH
ISdJldH
IISdJldH
Sdt
DSdJI
t
DJH
SS
d
L
enc
SL
enc
SL
d
22
2
1
0
If J =0 on the second surface then Jd must be
generated on the second surface to create a time
displaced current equal to current on surface 1
Displacement Current (2)
Amperes circuit law to a closed path provides the following eqn.
for current on the first side of the capacitive element
However surface 2 is the opposite side of the capacitor and has no
conduction current allowing for no enclosed current at surface 2
Based on the equation for displacement current density, we can
define the displacement current in a circuit as shown
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Show that Ienc on surface 1 and dQ/dt on surface 2of the capacitor are both equal to C(dV/dt)
dt
dVC
dt
dV
d
S
dt
dES
dt
dDS
dt
dS
dt
dQI
surfacefrom
dt
dVCdt
dV
d
SSJI
dt
dV
dt
DJ
d
VED
sc
dd
d
1__
Displacement Current (3)
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It was James Clark Maxwell that put all of this together and reduced electromagnetic field
theory to 4 simple equations. It was only through this clarification that the discovery of
electromagnetic waves were discovered and the theory of light was developed.
The equations Maxwell is credited with to completely describe any electromagnetic field
(either statically or dynamically) are written as:
Maxwells Time Dependent Equations
Differential Form Integral Form Remarks
Gausss Law
Nonexistence of the
Magnetic Monopole
Faradays Law
Amperes Circuit Law
t
DJH
t
BE
B
D v
0
0 SdBS
Sdt
DJldH
SL
SL
SdBt
ldE
dvSdDv
S
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A few other key equations that are routinely used are listed over the next couple of slides
Maxwells Time Dependent Equations (2)
0
0
12
21
21
21
n
sn
n
n
aBB
aDD
KaHH
aEE
Continuity Equation
Compatibility Equations
Boundary Conditions for Perfect Conductor
Boundary Conditions
Equilibrium Equations
0E
m
m
Jt
BE
B
t
DJH
Dv
m= free magnetic density
0J
0H
0nB
0tE
tJ
v
Lorentz Force Law
BuEQF
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Time Varying Potentials
Jt
A
A
t
VV
ApotentialsforConditionLorentzApply
t
VA
gchoobyconditionsfieldvectortheLimit
t
A
t
VJAA
yields
AAA
identityvectortheApplying
t
A
t
VJA
t
AV
tJ
t
EJA
dtDdJABH
LawCircuitsAmpereApplying
v
2
22
2
22
2
22
2
2
2
0:____
:sin______
:
:___
11
:__'_
At
VE
t
AVE
Vt
AE
t
AE
Att
BE
LawsFaradayApplying
AB
AfromBofDefinition
R
dvJA
RdvV
potentialsField
v
v
v
v
2
0
:_'_
:____
4
4
:_
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Wave Equation
00
00
2
22
2
22
2
22
2
22
1
1
___
u
cn
c
u
t
BB
t
E
E
yieldsspacefreeIn
Jt
AA
t
V
V v
Refractive index
Speed of the wave in a medium
Speed of light in a vacuum