electromagnetic power density introduction – …eie.polyu.edu.hk/~em/em06pdf/poynting vector.pdf1...
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Electromagnetic Power Density
Introduction– Power is transmitted by an electromagnetic wave in the
direction of propagation.
– Poynting vector
• not only applied to uniform plane waves• can be used for any electromagnetic waves
)/(),,,(),,,(),,,( 2mWtxyxtzyxtzyx HES ×=
mV / mA /
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Poynting Vector from Maxwell’s Equations
It can be derived from Maxwell’s equation in the time domain that:
dvHEt
dvdVVS ∫∫∫
+
∂∂
+⋅=⋅×−22
22 µεJESHE
power loss
the rate of increase of energies stored in the electric and magnetic fields
V
Snet power supplied to the enclosed surface
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Poynting Vector
Net power supplied to the volume:Power density of the wave:
Therefore, Poynting vector P (in W/m2) represents the direction and density of power flow at a point.
Example: If a plane wave propagating in a direction that makes an angle θ with the normal vector of an aperture
HEP
SHE
×=
⋅×= ∫Sout dW
θcos
ˆ
SA
dAnPS
=
⋅= ∫S
k̂n̂
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Example
Find the Poynting vector on the surface of a long straight conducting wire (of radius b and conductivity σ) that carries a direct current I.
Solution:Since we have a d-c situation, the current in the wire is uniformly distributed over its cross-sectional area.
L2ˆbIz
π=J
2ˆbIz
σπσ==
JE
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Example
On the surface of the wire,
Integrating the Poynting vector over the wall of the wire
bIπ
φ2
ˆ=H
32
2
32
2
2ˆ
2ˆˆ
bIr
bIz
σπσπφ −=×=×= HEP
RIbLI
bLb
Idsrdss
22
2
32
2
22
ˆ
=
=
=×=⋅−=⋅− ∫∫
σπ
πσπ
HEPsP
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Average Power Density
In time-varying fields, it is more important to find the average power. We define the average Poynting vector for periodic signals as Pavg
It can be shown that the average power density can be expressed as
∫=Tdt
T 0
1 PPavg
( )*),,(),,(Re21 zyxzyx HEPavg ×=
Complex conjugate
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Example: Plane Wave in a lossless Medium
The general expression for the phasor electric field of a uniform plane wave with arbitrary polarization, traveling in the +z direction, is given by
The magnetic field is
( ) zjyoxo
yx
eEyEx
zEyzExzβ−+=
+=
ˆˆ
)(ˆ)(ˆ)(E
( ) zjyoxo eExEy
zzz
β
η
η
−−−=
×=
ˆˆ1
)(ˆ1)( EH
( )22
21ˆ yoxoav EEzP +=∴η
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Example: Solar Power
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Example: Solar Power
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Example: Solar Power
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Example: Solar Power
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Example: Plane wave in a lossy medium
The general expression for the phasor electric field of a uniform plane wave with arbitrary polarization, traveling in the +z direction, is given by
and magnetic field is
( ) zjzyoxo
yx
eeEyEx
zEyzExzβα −−+=
+=
ˆˆ
)(ˆ)(ˆ)(E
( ) zjzyoxo eeExEy
zzz
βα
η
η
−−−−=
×=
ˆˆ1
)(ˆ1)( EH
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Example: Plane wave in a lossy medium
The average power density is
Demonstration: D7.6
( )
( )( ) ηθ
ηα
α
ηηθη
η
jzyoxo
zyoxo
eeEEz
eEEz
zyxzyx
=+=
+=
×=
−
−
ifcos21ˆ
1Re21ˆ
*),,(),,(Re21
222
*222
HEPavg
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Reflection of Plane Waves at Normal Incidence
x
z
Medium 2 (σ2,ε2,µ2)Medium 1(σ1,ε1,µ1)
(incident wave)
(reflected wave)
(transmitted wave)
Ei
ak
ak
ak
Et
Er
Hi
Hr
Ht
ak – propagation direction
y direction
-y direction
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Incident wave:
yzio
yz
ioi
xz
ioi
eEeH
eE
aaH
aE
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1
1
γγ
γ
η−−
−
==
=
Reflected wave:
yzro
yz
ror
xz
ror
eEeH
eE
aaH
aE
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1
1
)( γγ
γ
η−=−=
=
Transmitted wave:
yzto
yz
tot
xz
tot
eEeH
eE
aaH
aE
22
2
2
γγ
γ
η−−
−
==
=
Reflection of Plane Waves at Normal Incidence
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Boundary conditions at the interface (z=0):
Tangential components of E and H are continuous in the absence of current sources at the interface, so that
( )21
1ηηto
roio
toroio
toroio
EEE
HHHEEE
=−
=+=+
Reflection coefficient12
12
ηηηη
+−
==Γio
ro
EE
Transmission coefficient21
22ηη
ητ+
==io
to
EE
Reflection of Plane Waves at Normal Incidence
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Note that Γ and τ may be complex, and
101
≤Γ≤
=Γ+ τ
Similar expressions may be derived for the magnetic field.
In medium 1, a standing wave is formed due to the superimposition of the incident and reflected waves. Standing wave ratio can be defined as in transmission lines.
Reflection of Plane Waves at Normal Incidence
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Example: Normal Incidence on a good conductor
For good conductor,
12
12
ηηηη
+−
=Γ21
22ηη
ητ+
=
∞→σ
02 →+
=ωεσ
ωµηj
j
1−=Γ
0=τ