Transcript VI-2

VI–2 Electromagnetic Waves
11. 8. 2003
1
Main Topics
• Properties of Electromagnetic Waves:
•
•
•
•
•
Generation of electromagnetic
waves

Relations of and B.

The speed of LightSc.
Energy Transport .
Radiation Pressure P.
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Generation of Electromagnetic
Waves
• Since changes of electric field produce
magnetic field and vice versa these fields
once generated can continue to exist and
spread into the space.
• This can be illustrated using a simple dipole
antenna and an AC generator.
• Planar waves will exist only far from the
antenna where the dipole field disappears.
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

Relations of E and B I
• All properties of electromagnetic waves can
be calculated as a general solution of
Maxwell’s equations.
• This needs understanding fairly well some
mathematical tools or it is not illustrative.
• We shall show the main properties for a
special case of planar waves and state what
can be generalized.
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

Relations of E and B II
• Let us have a polarized planar wave:
• in free space with no charges nor currents
• which moves in the
 positive x direction
• the electric field E has
 only y component
• the magnetic field B has only z component.
• We shall prove relations between time and
space derivatives of E and B which are the
result of special Maxwell’s equations.
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Maxwell’s Equations


 E  dA  0
 
d m
 E  dl   dt


 B  dA  0
 
d e
 B  dl   0 0 dt
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

Relations of E and B III
• Let’s first use the Faraday’s law:
• The line integral of the electric intensity
counterclockwise around a small rectangle
hdx in the xy plane must be equal to minus
the change of magnetic flux through this
rectangle:
E
B
x
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
t
7


Relations of E and B IV
• Now, let’s similarly use the Ampere’s law:
• The line integral of magnetic induction
counterclockwise around a small rectangle
hdx in the xz plane must be equal to the
change of electric flux through this
rectangle: B
E
x
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   0 0
t
8


Relations of E and B V
• Note the symmetry in these equations!
E
B
B
E

   0 0
x
t
x
t
• Where B decreases in time E grows in x and
where E decreases in time B grows in x.
• This is the reason why E and B must be
in-phase.
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General Harmonic Waves I
• Waves can exist in elastic environment and are
generally characterized by the transport of energy
(or information) in space but not mass.
• Deflection of a planar harmonic wave propagating
in the +x axis direction by the speed c is either in
the direction of propagation or perpendicular:
a( x, t )  a0 sin  (t  cx )
• In the point x the deflection is the same as was in the
origin before the wave has reached point x. That is
x/c = 
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General Harmonic Waves II
• Deflection is periodic both in time and in
space:
t x
a( x, t )  a0 sin  (t  )  a0 sin 2 (  ) 
T 
a0 sin( t  kx)
x
c
• We have used the definitions of the angular
frequency, the wavelength and the wave
2
 2
number

;   cT ; k  
T
c

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

Relations of E and B VI
• Now, let us suppose polarized planar
harmonic transversal waves:
E = Ey =E0sin(t - kx)
B = Bz =B0sin(t - kx)
• E and B are in phase


• Vectors c, E, B form right (hand) turning
system
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

Relations of E and B VII
E
B
• From:


x
t
E0 
kE0  B0 
 c
B0 k
• Since E and B are in-phase, generally:
E=cB
• The magnitude of the magnetic field is
c-times smaller!
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

Relations of E and B VIII
B
E
• From: x    0 0 t 
B0  0 0
kB0   0 0E0 

  0 0 c
E0
k
• Together if gives the relation of the speed of
electromagnetic waves, the permitivity and
the permeability of the free space
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1
 0 0  2
c
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The Speed of Light
• The speed can be found generally from:
E
B
B
E

   0 0
x
t
x
t
• A t-derivative of the first equation
compared to the x-derivative of the second
gives the general wave equation for B.
• Changing the derivatives we get the general
wave equation for E.
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General Properties of EMW
• The solution of ME without charges and currents
•
•
•
•
satisfies general wave equations.
Through empty space waves travel with the speed
of light c = 3108m/s.

Vectors c, E, B form right turning system
The magnitude of the magnetic field is
c-times smaller than that of the electric field.
Electromagnetic waves obey the principle of
superposition.
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Energy Transport of EMW I
• The energy density of EMW at any instant
is a sum of energies of both electric and
2
2
magnetic fields:
 E
B
u  ue  u m 
o
2

2 0
• From B = E/c and c = (00)-1/2 we get:
0
u  oE 
 EB
0
0
2
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B
2
17
Energy Transport of EMW II
• We see that the energy density associated
with the magnetic field is equal to that
associated with the electric field, so each
contributes half of the total energy in spite
of the peak value difference!
• (0/0)1/2 = 0c is the impedance of the free
space = 377 .
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Energy Transport of EMW III
• The energy transported by the wave per unittime
per unit area is given by a Poynting vector S ,
which has the direction of propagating of the
wave. The units W/m2.
• The energy which passes in 1 second through
some area A is the energy density times the
volume:
U = uAct 
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1 dU
EB
2
S
 uc   0 cE 
A dt
0
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Energy Transport of EMW IV
• For general direction of the EMW a vector
definition of the Poynting vector is valid:

1  
S
( E  B)
0


• Of course, S is parallel to c .
• This is the energy transported at any instant.
We are usually interested in intensity <S>,
which is the mean (in time) value of S.
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Energy Transport of EMW V
• For a harmonic wave we can use a result we
found when dealing with AC circuits:
2
E0
2
 E 
2
• So we can express the intensity using the
peak or rms values of the field variables:
E0 B0 Erms Brms
 S 

2 0
0
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Radiation Pressure I
• If EMW carry energy, it can be expected that they
also carry linear momentum.
• If EMW strikes some surface, it can be fully or
partly absorbed or fully reflected. In either case a
force will be exerted on the surface according to

the second Newtons law: 
dp
F
dt
• The force per unit area is the radiation pressure.
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Radiation Pressure II
• It can be shown that p = U/c, where 
is a parameter between 1 for total
absorption to 2 for total reflection. So from:
F = dp/dt = /c dU/dt = <S>A/c.
we can readily get the pressure:
P = F/A = <S>/c.
• This can be significant on the atomic scale
or for ‘sailing’ in the Universe.
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The Spectrum of EMW
• Effects of very different behavior are in fact the
same EMW with ‘just’ different frequency.
•
•
•
•
•
•
•
Radio waves  > 0.1 m
Microwaves 10-1 >  > 10-3 m
Infrared 10-3 >  > 7 10-7 m
Visible 7 10-7 >  > 4 10-7 m
Ultraviolet 4 10-7 >  > 6 10-10 m
X - rays 10-8 >  > 10-12 m
Gamma rays 10-10 >  > 10-14 m
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Radio an TV
• In transmitter a wave of some carrier
frequency is either AM or FM modulated,
amplified and broadcasted.
• Receiver must use an antenna sensitive
either to electric or magnetic component of
the wave.
• Its important part is a tuning stage where
the proper frequency is selected.
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Homework
• No homework assignment today!
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Things to read and learn
• This lecture covers:
Chapter 32 – 4, 5, 6, 7, 8, 9
• Advance reading
Chapter 33 – 1, 2, 3, 4
• Try to understand the physical background
and ideas. Physics is not just inserting
numbers into formulas!
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A Rectangle in xy Plane
 
 E  dl  ( E  dE )h  Eh  hdE 
d m
dB

 hdx
dt
dt
We are trying to find an increment dE in the
+x axis direction and assume hdx is fixed.
dB
E
B
hdE   hdx


dt
x
t
^
A Rectangle in xz Plane
 
 B  dl  Bh  ( B  dB)h  hdB 
d e
dE
 0 0
  0 0 hdx
dt
dt
We are trying to find an increment dB in the
+x axis direction and assume hdx is fixed.
dE
B
E
hdB    0 0 hdx

   0 0
dt
x
t
^
Generalized Ampere’s Law
 
d e
 B  dl   0 I encl   0 0 dt
• Iencl sum of all enclosed currents taking into
account their directions and
• 0de/dt is the displacement current due to
change-in-time of the electric flux.
^
General Wave Equations I
 E
B
2E
2B
(

)

dt x
t
xt
t 2
 B
E
2B
2E
(
   0 0
)
   0 0
2
dx x
t
x
tx
• After comparing these equations we get the
wave equation for B :
2B
2B
  0 0
0
2
2
x
t
^
General Wave Equations II
 E
B
2E
2B
(

)

2
x x
t
x
tx
 B
E
2B
2E
(
   0 0
)
   0 0
t x
t
xt
t 2
• After comparing these equations we get the
wave equation for E :
2E
2E
  0 0
0
2
2
x
t
^