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Electromagnetic Power Density
Introduction
– Power is transmitted by an electromagnetic wave in the
direction of propagation.
– Poynting vector
S ( x, y , z , t ) = E( x, y , z , t ) × H ( x, y , x, t )
V /m
(W / m 2 )
A/ m
• not only applied to uniform plane waves
• can be used for any electromagnetic waves
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Poynting Vector from Maxwell’s Equations
It can be derived from Maxwell’s equation in the time
domain that:
∂
− ∫ E × H ⋅ dS = ∫ E ⋅ Jdv +
S
V
∂t
 ε E 2 µH 2 
∫V  2 + 2 dv
power loss
the rate of increase of
energies stored in the electric
and magnetic fields
S
net power supplied
to the enclosed surface
V
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Poynting Vector
Net power supplied to the volume: Wout = ∫SE × H ⋅ dS
Power density of the wave: P = E × H
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 k̂
that makes an angle θ with the normal vector n̂ of an
aperture
P = ∫ S ⋅ nˆ dA
S
= SA cos θ
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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.
I
J = zˆ 2
πb
L
J
I
E = = zˆ
2
σ
σπb
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Example
On the surface of the wire,
I
ˆ
H =φ
2πb
P = E × H = zˆ × φˆ
I2
2σπ b
2 3
= −rˆ
I2
2σπ 2b 3
Integrating the Poynting vector over the wall of the wire
 I2 
2πbL
− ∫ P ⋅ ds = − ∫ P ⋅ rˆds =E × H = 
2
3
s
s
 2σπ b 
L 
2
2
=I 
=
I
R
2 
 σπb 
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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
Pavg
1
=
T
∫
T
0
Pdt
It can be shown that the average power density can be
expressed as
Pavg
1
= Re(E( x, y, z ) × H ( x, y, z ) *)
2
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
E( z ) = xˆE x ( z ) + yˆ E y ( z )
= (xˆE xo + yˆ E yo )e − jβz
The magnetic field is
1
H ( z ) = zˆ × E( z )
η
1
= − ( yˆ E xo − xˆE yo )e − jβz
η
(
2
1
2
∴ Pav = zˆ
E xo + E yo
2η
)
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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
E( z ) = xˆE x ( z ) + yˆ E y ( z )
= (xˆE xo + yˆ E yo )e −αz e − jβz
and magnetic field is
1
H ( z ) = zˆ × E( z )
η
1
= − ( yˆ E xo − xˆE yo )e −αz e − jβz
η
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Example: Plane wave in a lossy medium
The average power density is
Pavg
1
= Re(E( x, y, z ) × H ( x, y, z ) *)
2
2 − 2αz
 1 
1
2
Re * 
= zˆ E xo + E yo e
2
η 
)
(
(
)
2 − 2αz
1
2
= zˆ
E xo + E yo e
cos θη
2η
if η = η e
jθη
Demonstration: D7.6
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Reflection of Plane Waves at Normal Incidence
Medium 1(σ1,ε1,µ1)
x
Medium 2 (σ2,ε2,µ2)
Ei
(incident wave)
Er
ak
Et
ak
Hi
Ht
ak
(transmitted wave)
z
Hr
(reflected wave)
y direction
-y direction
ak – propagation direction
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Reflection of Plane Waves at Normal Incidence
Incident wave:
Ei = Eio e
−γ 1 z
H i = H io e
Reflected wave:
ax
−γ 1 z
ay =
Eio
η1
e
−γ 1 z
ay
γ 1z
E r = Ero e a x
γ 1z
H r = H ro e (−a y ) = −
Transmitted wave:
Et = Eto e
−γ 2 z
H t = H to e
Ero
η1
γ 1z
e ay
ax
−γ 2 z
ay =
Eto
η2
e
−γ 2 z
ay
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Reflection of Plane Waves at Normal Incidence
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
Eio + Ero = Eto
H io + H ro = H to
1
η1
Reflection coefficient
(Eio − Ero ) =
Eto
η2
Ero η 2 − η1
Γ=
=
Eio η 2 + η1
Eto
2η 2
=
Transmission coefficient τ =
Eio η1 + η 2
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Reflection of Plane Waves at Normal Incidence
Note that Γ and τ may be complex, and
1+ Γ = τ
0 ≤ Γ ≤1
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.
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Example: Normal Incidence on a good conductor
η 2 − η1
Γ=
η 2 + η1
2η 2
τ=
η1 + η 2
For good conductor,σ → ∞
jωµ
η2 =
→0
σ + jωε
Γ = −1
τ =0
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