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4-0
3.6 Polarization of a sinusoidally time-varying field
describes how the position of the tip of the field
vector at a given point in space varies with time.
Linear Polarization:
Tip of the vector
describes a line.
Circular Polarization:
Tip of the vector
describes a circle.
Elliptical Polarization:
Tip of the vector
describes an ellipse.
4-1
(i) Linear Polarization
F1  F1 cos (t   ) a x




Magnitude varies
sinusoidally with time
Direction remains
along the x axis
 Linearly polarized in the x direction.
direction
4-2
F2  F2 cos (t   ) a y



Magnitude varies

sinusoidally with time
Direction remains
along the y axis
 Linearly polarized in the y direction.
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4-3
If two (or more) component linearly polarized vectors are in
phase, (or in phase opposition), then their sum vector is also
linearly polarized, e.g.,
F  F1 cos (t   ) a x  F2 cos (t   ) a y
4-4
(ii) Circular Polarization
If two component linearly polarized vectors are
(a) equal in amplitude
(b) differ in direction by 90˚
(c) differ in phase by 90˚,
then their sum vector is circularly polarized.
4-5
Example:
F  F1 cos  t a x  F1 sin  t a y
F 
 F1
cos  t    F1 sin  t 
2
2
 F1 , constant
  tan  1
F1 sin  t
F1 cos  t
 tan  1  tan  t    t
y
F2
F

F1
x
2
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4-6
(iii) Elliptical Polarization
In the general case in which either of (i) or (ii) is not
satisfied, then the sum of the two component
linearly polarized vectors is an elliptically polarized
vector.
Example: F  F1 cos t a x  F2 sin t a y
F2
F
F1
4-7
Example: F  F0 cos t ax  F0 cos t   4  a y
y
F0
F2
F
π/4
F1 F0 x
–F0
–F0
4-8
D3.17
F1  F0 cos  2   108 t  2  z  a x
F2  F0 cos  2   108 t  3  z  a y
F1 and F2 are equal in amplitude (= F0) and differ in direction
by 90˚. The phase difference (say ) depends on z in the
manner –2z – (–3z) = z.
At (3, 4, 0),  = (0) = 0.
 F1  F2 
At (3, –2, 0.5),  = (0.5) = 0.5 .
At (–2, 1, 1),  = (1) = .
is linearly polarized.
 F1  F2  is circularly polarized.
 F1  F2 
is linearly polarized.
At (–1, –3, 0.2) =  = (0.2) = 0.2.
 F1  F2 
is elliptically polarized.
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4-9
Clockwise and Counterclockwise
Polarizations
In the case of circular and elliptical polarizations for
the field of a propagating wave, one can distinguish
between clockwise (cw) and counterclockwise (ccw)
polarizations If the field vector in a constant phase
polarizations.
plane rotates with time in the cw sense, as viewed
along the direction of propagation of the wave, it is
said to be cw- or right-circularly (or elliptically)
polarized. If it rotates in the ccw sense, it is said to
be ccw- or left- circularly (or elliptically) polarized.
4-10
For example, consider the circularly polarized electric field
of a wave propagating in the +z-direction, given by
E  E0 cos(t  z )a x  E0 sin(t  z )a y
Then, considering the time variation of the field vector in
the z = 0 plane,
plane we note that for t  0,
0 E  E0ax , and for
t   2 , E  E0a y .
Since ax × a y  a z , the polarization is cw- or right-circular.
If E  E0 cos(t   z )a x  E0 sin(t   z )a y , then the
polarization is ccw - or left - circular
4-11
Review Questions
4.16. A sinusoidally time-varying vector is expressed in
terms of its components along the x-, y-, and z- axes.
What is the polarization of each of the components?
4.17. What are the conditions for the sum of two linearly
polarized sinusoidally time-varying vectors to be
circularly
i l l polarized?
l i d?
4.18. What is the polarization for the general case of the sum
of two sinusoidally time-varying linearly polarized
vectors having arbitrary amplitudes, phase angles, and
directions?
4.19. Discuss clockwise and counterclockwise circular and
elliptical polarizations associated with sinusoidally
time-varying uniform plane waves.
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4-12
3.7 Power Flow and Energy Storage
Consider the quantity E× H . From a vector identity,
  E × H   H   × E  E   × H 
Substituting
B
t
D
D
×H  J 
 J0 
t
t
where J0 represents source current density, we have
×E  
D
B
H 
t
t
 1
 1
2
 E  J0    0 E     0 H 2      E× H 
t  2
 t  2

  E× H   E  J0  E 
4-13
Performing volume integration on both sides, and using the
divergence theorem for the last term on the right side, we get


1
  E  J  dv  t   2  E
0
V

t

2
0
V
1
 dv


  2  H
V
0
2

 dv 

 P  dS
S
where we have defined P  E × H , known as the Poynting
vector. The equation is known as the Poynting’s Theorem.
4-14
Poynting’s Theorem


1
  E  J  dv  t   2  E
0
V
Source
power density,
(power per
unit volume),
W/m3
0
V
2
 dv    1  H 2  dv  P  d S

t  2 0 

V
S
Electric stored
energy density,
J/m3

Magnetic stored
energy density,
J/m3

Power flow
out of S
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Interpretation of Poynting’s Theorem
Poynting’s Theorem says that the power delivered to the
volume V by the current source J0 is accounted for by the sum
of the time rates of increase of the energies stored in the
electric and magnetic fields in the volume, plus another term,
which we must interpret as the power carried by the
electromagnetic field out of the volume V, for conservation of
energy to be satisfied. It then follows that the Poynting vector
P has the meaning of power flow density vector associated
with the electromagnetic field. We note that the units of E x H
are volts per meter times amperes per meter, or watts per
square meter (W/m2) and do indeed represent power flow
density.
4-16
In the case of the infinite plane sheet of current, note that the
electric field adjacent to and on either side of it is directed
opposite to the current density. Hence, some work has to be
done by an external agent (source) for the current to flow, and
 E  J0  represents the power density (per unit volume)
associated with this work.


P  E × H  Power flow density W m2 associated
with the electromagnetic field
1
we   0 E 2  Energy density J m3 stored
2
in the electric field
1
wm  0 H 2  Energy density J m3 stored
2
in the magnetic field




4-17
Review Questions
4.20. What is the Poynting vector? What is the physical
interpretation of the Poynting vector over a closed
surface?
4.21. State Poynting’s theorem. How is it derived from
Maxwell’s curl equations?
4 22 Di
4.22.
Discuss th
the interpretation
i t
t ti off Poynting’s
P
ti ’ theorem.
th
4.23. What are the energy densities associated with electric
and magnetic fields?
4.24. Discuss how fields far from a physical antenna vary
inversely with distance from the antenna.
6