Ondas Estaticas (1)
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Transcript of Ondas Estaticas (1)
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Electromagnetic Fields
Four vector quantities
E
electric field strength [Volt/meter] = [kg-m/sec3]
D
electric flux density [Coul/meter2] = [Amp-sec/m2]
H
magnetic field strength [Amp/meter] = [Amp/m]
B
magnetic flux density [Weber/meter2] or [Tesla] = [kg/Amp-sec2]
each are functions of space and timee.g. E(x,y,z,t)
J
electric current density [Amp/meter2]
v
electric charge density [Coul/meter3] = [Amp-sec/m3]
Sources generating
electromagnetic fields
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MKS units
length meter [m]mass kilogram [kg]
time second [sec]
Some common prefixes and the power of ten each represent are listed below
femto - f - 10-15
pico - p - 10-12
nano - n - 10-9
micro -
-
10-6
milli - m - 10-3
mega - M - 106
giga - G - 109
tera - T - 1012
peta - P - 1015
centi - c - 10-2
deci - d - 10-1
deka - da - 101
hecto - h - 102
kilo - k - 103
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0
v
BE
t
DH J
t
B
D
=
= +
=
=
Maxwells Equations
(time-varying, differential form)
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Maxwells Equations
James Clerk Maxwell (18311879)
James Clerk Maxwell was a Scottish mathematician andtheoretical physicist. His most significant achievement was thedevelopment of the classical electromagnetic theory, synthesizingall previous unrelated observations, experiments and equations
of electricity, magnetism and even optics into a consistent theory.His set of equationsMaxwell's equationsdemonstrated thatelectricity, magnetism and even light are all manifestations of thesame phenomenon: the electromagnetic field. From that momenton, all other classical laws or equations of these disciplines
became simplified cases of Maxwell's equations. Maxwell's workin electromagnetism has been called the "second greatunification in physics", after the first one carried out by IsaacNewton.
Maxwell demonstrated that electric and magnetic fields travel
through space in the form of waves, and at the constant speed oflight. Finally, in 1864 Maxwell wroteA Dynamical Theory of theElectromagnetic Field
where he first proposed that light was infact undulations in the same medium that is the cause of electricand magnetic phenomena. His work in producing a unified modelof electromagnetism is considered to be one of the greatest
advances in physics.
(Wikipedia)
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Maxwells Equations (cont.)
0
C S
C S S
S
v
S V
dE dr B ndS
dt
ddr J ndS D ndS
dt
B ndS
D ndS dV
=
= +
=
=
Faradays law
Amperes law
Magnetic Gauss law
Electric Gauss law
(Time-varying, integral form)
The above are the most fundamental form of Maxwells equations sinceall differential forms, all boundary forms, and all frequency domain forms
derive from them!
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Maxwells Equations (cont.)
0
v
BE
tD
H Jt
B
D
=
= +
=
=
Faradays law
Amperes law
Magnetic Gauss law
Electric Gauss law
(Time-varying, differential form)
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( )
( ) 0
D
H Jt
D
H J t
J Dt
= +
= +
= +
Law of Conservation of Electric
Charge (Continuity Equation)
vJ
t
=
Flow of electriccurrent out of volume
(per unit volume)
Rate of decrease of electriccharge (per unit volume)
[2.20]
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Continuity Equation (cont.)
vJt
=
Apply the divergence theorem:
Integrate both sides over an arbitrary volume V:
v
V V
d V d V t
=
v
S VJ n d S d Vt
= V
S
n
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Continuity Equation (cont.)
Physical interpretation:
V
S
n
v
S V
n d S d V t
=
v
o u t vV V
i d V d V t t
= =
e n c l
o u t
Q
i t
=
(This assumes that the
surface is stationary.)
e n c l
in
Q
i t
= or
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Maxwells Equations
Decouples
0
0 0
v
v
E
B DE H J B D
t t
E D H J B
= = + = =
= = = =
Time-Dependent
Time -Independent (Static s)
and is a function of and is a function ofv
H E H J
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E BH J D
B 0D v
jj
=
= +
=
=
Maxwells Equations
Time-harmonic (phasor) domain jt
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Constitutive Relations
Characteristics of media relate D
to E
and H
to B
00
0 0
( = permittivity )
( = permeability)
D E
B H
=
=
-12
0
-7
0
[F/m]8.8541878 10
= 4 10 H/m] ( )[
exact
[2.24]
[2.25]
[p. 35]
0 0
1c = c = 2.99792458
108 [m/s] (exact value that is defined)
Free Space
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Constitutive Relations (cont.)
Free space, in the phasor domain:
00
0 0
( = permittivity )
(
D
=
= E
B pe= rmeability)H
This follows from the fact that
( ) VaV t a
(where a
is a real number)
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Constitutive Relations (cont.)
In a material medium:
( = permittivity )
( = permeability
D = E
B = )H
0
0
= r
r
=
r
= relative permittivity
r
= relative permittivity
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or
Independent of Dependent on
space homogenous inhomogeneous
frequency non-dispersive dispersive
time stationary non-stationary
field strength linear non-linear
direction of isotropic anisotropicE
or H
Terminology
I i M i l
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Isotropic Materials
(or ) is a scalar quantity,which means that E
|| D
(and H
|| B
)
x xB = H
y yB = H
yH
xH
x
yx xD = E
y yD = E
yE
xE
x
y
A i t i M t i l
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(or ) is a tensor (can be written as a matrix)
This results in E
and D
being NOT parallel to each other; they are
in different directions.
0 0
0 0
0 0
x x x
y y y
x x
z z z
y y y
z z z
D E
D E
D E
D E
D E D E
=
=
=
=
Anisotropic Materials
DE