32
11. 8. 2003 1 VI–2 Electromagnetic Waves
11. 8. 2003 1
VI–2 Electromagnetic Waves
11. 8. 2003 2
Main Topics
• Properties of Electromagnetic Waves:• Generation of electromagnetic waves
• Relations of and .
• The speed of Light c.
• Energy Transport .
• Radiation Pressure P.
SB
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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 and 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.
B
E
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Relations of and 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 has only y component• the magnetic field 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.
B
E
E
B
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Maxwell’s Equations
dt
dldB
AdB
dt
dldE
AdE
e
m
00
0
0
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Relations of and 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:
B
E
t
B
x
E
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Relations of and 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
t
E
x
B
00
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Relations of and V
• Note the symmetry in these equations!
• 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.
B
E
t
E
x
B
00t
B
x
E
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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:
• 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 =
)(sin),( 0 cxtatxa
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General Harmonic Waves II
• Deflection is periodic both in time and in space:
• We have used the definitions of the angular frequency, the wavelength and the wave number
)sin(
)(2sin)(sin),(
0
00
kxta
x
T
tatatxa c
x
2
;;2
c
kcTT
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Relations of and 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 , , form right (hand) turning system
B
E
c
E
B
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Relations of and VII
• From:
• Since E and B are in-phase, generally:
E = c B
• The magnitude of the magnetic field is
c-times smaller!
B
E
t
B
x
E
ckB
EBkE
0
000
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Relations of and VIII
• From:
• Together if gives the relation of the speed of electromagnetic waves, the permitivity and the permeability of the free space
B
E
t
E
x
B00
ckE
BEkB 00
00
0
00000
200
1
c
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The Speed of Light
• The speed can be found generally from:
• 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.
t
B
x
E
t
E
x
B
00
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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 , , 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.
c
E
B
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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 magnetic fields:
• From B = E/c and c = (00)-1/2 we get:
0
22
22 BE
uuu ome
0
0
0
22
EB
BEu o
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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 , 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
S
0
20
1
EB
cEucdt
dU
AS
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Energy Transport of EMW IV
• For general direction of the EMW a vector definition of the Poynting vector is valid:
• Of course, is parallel to .• This is the energy transported at any instant.
We are usually interested in intensity <S>, which is the mean (in time) value of S.
)(1
0
BES
S
c
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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:
• So we can express the intensity using the peak or rms values of the field variables:
2
202 E
E
00
00
2 rmsrms BEBE
S
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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:
• The force per unit area is the radiation pressure.dt
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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!
A Rectangle in xy Plane
^
dt
dBhdx
dt
d
hdEEhhdEEldE
m
)(
We are trying to find an increment dE in the +x axis direction and assume hdx is fixed.
t
B
x
E
dt
dBhdxhdE
A Rectangle in xz Plane
^
dt
dEhdx
dt
d
hdBhdBBBhldB
e0000
)(
We are trying to find an increment dB in the +x axis direction and assume hdx is fixed.
t
E
x
B
dt
dEhdxhdB
0000
Generalized Ampere’s Law
^
dt
dIldB e
encl
000
• Iencl sum of all enclosed currents taking into account their directions and
0de/dt is the displacement current due to change-in-time of the electric flux.
General Wave Equations I
^
xt
E
x
B
t
E
x
B
dx
t
B
tx
E
t
B
x
E
dt
2
002
2
00
2
22
)(
)(
• After comparing these equations we get the wave equation for B :
02
2
002
2
t
B
x
B
General Wave Equations II
^
2
2
00
2
00
2
2
2
)(
)(
t
E
tx
B
t
E
x
B
t
xt
B
x
E
t
B
x
E
x
• After comparing these equations we get the wave equation for E :
02
2
002
2
t
E
x
E
