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CA 4042: Campos
Electromagnticos
Instructor: Ing. Hctor C. Vergara V.
Profesor de Facultad de Ingeniera Mecnica
Centro Regional de Azuero
Universidad Tecnolgica de Panam
Mvil: (507) 6677-5920, email: [email protected]
Libro de Texto:M.N.O. Sadiku,Elementos de Electromagnetismo 5th ed. Oxford University Press, 2009.
Lectura Auxiliar:
W.Hayt, J.Buck, Teora Electromagntica, 8va ed. McGrawHill, 2012.
Todas las figuras son tomadas del libro de texto principal a menos que se diga lo contrario
Cap. 9:Ecuaciones de Maxwell
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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 dt
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d d NVemf
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Faradays Law (2)
The total emf generated in the between the two open terminals in the battery is therefore
Note the following important factsAn electrostatic field cannot maintain a steady current in a close circuit since
Ee dl 0IRL
An emf-produced field is nonconservative
Except in electrostatics, voltage and potential differences are usually not equivalent
Ef dl Ee dl IRVemfP P
N N
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 fieldEe that also exist that
counteracts the emf generated potential
EEf Ee
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Edl
E
fdl 0
E
f dl
P
L L N
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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.
2.
3.
Having a stationary loop in a time-varyingB field
Having a time-varying loop in a static B field Having
a time-varying loop in a time-varying B field
A stationary loop in a time-varyingB field
E dl dtB dS
d ddt dt
L S
Vemf
emf
d
V
V Edl E dS d B dS
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E dB
dt
dtSSLemf
One of Maxwells for time varying fields
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A time-varying loop in a static B field
Qu B
u B dl
E dS u Bdl
E u B
by_Stokes's_Theorem
Fm IlB
uBlVemf
Fm IlB
V Edl
E Fm u BQ
fieldin a motional E
F
m
L L
m
LLemf
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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Transformer and Motional
Electromotive Forces (3)
A time-varying loop in a time-varyingB field
u B dl
dB u Bdt
E
E dl dB dSdt
V
m
LL S
emf
One of Maxwells for time varying fields
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Electromotive Forces: Example1
Conducting element is stationary and themagnetic field varies with time
Assume the bar is held stationary at y =0.08 m
andB = 4cos(106t)az mWb/m2
Assume the length between the two conducting
rails the bar slides along is 0.06 m
Transformer and Motional
dt dB
E
V dB dSdt
m
S
emf
(4)(103)(106)sin(106t)dxdyS
xy(4)(103)(106)sin(106t)
0.080.06(4)(103
)(106
)sin(106
t) 19.2sin(106t)V
d (0.004cos(106t))a dSS dt
V dB dS dt
z
S
emf
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Transformer and Motional
Electromotive Forces: Example
2
Conductor moves at a velocity u = 20ay m/s in
constant magnetic field B=4az mWb/m2
Assume the length between the two
conducting rails the bar slides along is 0.06 m
u B dl
E u B
V 20a 0.004a dxaL
0.08dx 0.08x 0.080.064.8mV
V E dl
xy zemf
m
L L
emf
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Conductor moves at a velocity u = 20a
ym/s in time
varying magnetic field B=4cos(106t-y)az mWb/m2
Assume the length between the two conducting
rails the bar slides along is 0.06 m
Transformer and Motional
Electromotive Forces: Example 3
Edl dS u Bdld
V (103)(4)cos(106ty)a dxdya
20a (103)(4)cos(106ty)a dxaL
y z x
z
S
emf z
L S LVemf
dt
dt
dB
V (103)(4)106 sin(106ty)a dxdyaS
20(103)(4)cos(106ty)dxV (103)(4)cos(106ty)x 103(4)cos(106t)xemf
20(103
)(4)cos(106
ty)dxV (103)(4)cos(106ty)x 103(4)cos(106t)xemf 20(103)(4)cos(106 ty)xV (103)(4) 8(102 )cos(106ty)x 103 (4)cos(106 t)x
V 240cos(106 ty) 240cos(106t)
4000xcos(106ty) 4000xcos(106t)Vemf
emf
emf
zzemf
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Lets now examine time dependent fields from the perspective onAmperes Law.
H0 J
H0 J J J J v D D
t HJD
tDJ
t
We can now define the displacement current density as
the time derivative of the displacement vector
t t
HJJd
t J v 0
HJ
d
d
d
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
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Using our understanding of conduction and displacement current density. Lets test this
theory on the simple case of a capacitive element in a simple electronic circuit.
H dl J dS D dS dQ IdttS
H dl J dSIenc 0
H dl J dSIenc I
DI J dS dSt
tHJD
S
dL
S2L
S1
L
d
22
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)
Based on the equation for displacement current density, we can
define the displacement current in a circuit as shown
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
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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)
dQ Sds SdD SdES dV CdVdt dt dt dt d dt dt
I
d dt dt
from surface 1
SS dV CdVI Jt d dt
D dVJ
D E Vd
c
d d
d
Displacement Current (3)
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Maxwells Time Dependent Equations
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:
Differential Form Integral Form Remarks
Gausss Law
Nonexistence of the
Magnetic Monopole
Faradays Law
Amperes Circuit Law
t
HJD
tE B
B0
D v
B dS0S
Hdl J t dSD
SL
Edl B dStL S
D dS vdv
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)
E E a 0H H a KD D a B B a 02 1 n
2 n s1
2 n1
1 2 n
Boundary Conditions
Compatibility Equations
Boundary Conditions for Perfect Conductor
Equilibrium Equations
E 0
Jm
B m
Bt
E
HJD
t
Dv
m = free magneticdensity
H 0 J0
Bn 0
Et 0
Lorentz Force Law Continuity Equation
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t J v
F QE u B
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Maxwells Time Dependent Equations:
Identity Map
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Time Varying Potentials
A A 2Ayields:
A 2A J
t2
2
2V V v
Apply Lorentz Condition for potentials: A 02
t A V
by choo sing:conditionsvector field Limit the
t2V At
identity:Applying the vector
t2V At
A J
t A JE J VA
tt
dt H1 B 1 A J dD
Applying_Ampere's Circuit Law :
2
2
B t
At
E v 2V
At
E V
VAt E
t A E
At
E
Applying _Faraday 's_Law :
Definition
B A
of B from A :
Jdv4R
A
v4R
dvV
Field _potentials :
v
v
0
2
A A
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Wave Equation
0 0
0 0
t2
t2
2B 2B
2E 2E
t2
t2
2A 2A
2V
2
V
1
1
J
n c
u
c
u
y ie ld sI n fr ee s p a ce
v
Refractive index
Speed of the wave in a medium
Speed of light in a vacuum
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