topic 6 em waves
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Oscillation and waves lecture notesTRANSCRIPT
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Topic 6
Electromagnetic Waves• Types of electromagnetic waves• Electromagnetic spectrum• Propagation of electromagnetic wave• Electric field and magnetic field• Qualitative treatment of electromagnetic waves
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
• Electromagnetic (EM) waves were first postulated by James Clerk Maxwell and subsequently confirmed by Heinrich Hertz
• Maxwell derived a wave form of the electric and magnetic equations, revealing the wave-like nature of electric and magnetic fields, and their symmetry
• Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave
• According to Maxwell’s equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa
• Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on
• These oscillating fields together form an electromagnetic wave
Introduction
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
• In the studies of electricity and magnetism, experimental physicists had determined two physical constants - the electric (o) and magnetic (o) constant in vacuum
• These two constants appeared in the EM wave equations, and Maxwell was able to calculate the velocity of the wave (i.e. the speed of light) in terms of the two constants:
• Therefore the three experimental constants, o, o and c previously thought to be independent are now related in a fixed and determined way
Speed of EM waves
m/s100.31 8
oo
c 0 = 8.8542 10-12 C2 s2/kgm3 (permittivity of vacuum)
0 = 4 10-7 kgm/A2s2 (permeability of vacuum)
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Name Differential form Integral form
Gauss's law
Gauss's law for magnetism
Maxwell–Faraday equation (Faraday's law of induction)
Ampère's circuital law(with Maxwell's correction)
Formulation in terms of free charge and current
Maxwell’s Equations
fD
t
BE
0 B
0 AdBV
)(VQAdD fV
tldE SB
S
,
t
DJH
tIldH SD
fSS
,,
zz
yy
xx
ˆˆˆ
vz
v
y
v
x
vvdiv zyx
zyx
zyx
vvv
zyx
v
ˆˆˆ
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Maxwell’s Equations
Formulation in terms of total charge and current
0
E
t
BE
0 B
t
EJB
000
0
)(
VQAdE
V
tldE SB
S
,
0 AdBV
tIldB SE
SS
,000
Differential form Integral form
Gauss's law
Gauss's law for magnetism
Maxwell–Faraday equation (Faraday's law of induction)
Ampère's circuital law(with Maxwell's correction)
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve)
line integral of the magnetic field over the closed boundary ∂S of the surface S
The electric flux (surface integral of the electric field) through the (closed) surface (the boundary of the volume V )
The magnetic flux (surface integral of the magnetic B-field) through the (closed) surface (the boundary of the volume V )
Maxwell’s Equations
ldES
ldBS
AdEV
AdBV
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UEEP1033 Oscillations and Waves
(1) Gauss’s law for the electric field
Gauss’s law is a consequence of the inverse-square nature of Coulomb’s law for the electrical force interaction between point like charges
(2) Gauss’s law for the magnetic fieldThis statement about the non existence of magnetic monopole; magnets are dipolar. Magnetic field lines form closed contours
(4) The Ampere-Maxwell law
This law is a statement that magnetic fields are caused by electric conduction currents and or by a changing electric flux (via the displacement current)
(3) Faraday’s law of electromagnetic inductionThis is a statement about how charges in magnetic flux produce (non-conservative) electric fields
Maxwell’s Equations
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Electromagnetic Spectrum
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Generating an Electromagnetic Waves
An arrangement for generating a traveling electromagnetic wave in the shortwave radio region of the spectrum: an LC oscillator produces a sinusoidal current in the antenna, which generate the wave. P is a distant point at which a detector can monitor the wave traveling past it
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Generating an Electromagnetic Waves
Variation in the electric field E and the magnetic field B at the distant point P as one wavelength of the electromagnetic wave travels past it.
The wave is traveling directly out of the page
The two fields vary sinusoidally in magnitude and direction
The electric and magnetic fields are always perpendicular to each other and to the direction of travel of the wave
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
• Close switch and current flows briefly. Sets up electric field
• Current flow sets up magnetic field as little circles around the wires
• Fields not instantaneous, but form in time
• Energy is stored in fields and cannot move infinitely fast
Generating an Electromagnetic Waves
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• Figure (a) shows first half cycle
• When current reverses in Figure (b), the fields reverse
• See the first disturbance moving outward
• These are the electromagnetic waves
Generating an Electromagnetic Waves
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• Notice that the electric and magnetic fields are at right angles to one another
• They are also perpendicular to the direction of motion of the wave
Generating an Electromagnetic Waves
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Electromagnetic Waves
• The cross product always gives the direction of travel of the wave
• Assume that the EM wave is traveling toward P in the positive direction of an x-axis, that the electric field is oscillating parallel to the y-axis, and that the magnetic filed is the oscillating parallel to the z-axis:
)sin()sin(
0
0
tkxBBtkxEE
E0 = amplitude of the electric fieldB0 = amplitude of the magnetic field = angular frequency of the wavek = angular wave number of the waveAt any specified time and place: E/B = c
cBE 00 /(speed of electromagnetic wave)
BE
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UEEP1033 Oscillations and Waves
Electromagnetic wave represents the transmission of energy
The energy density associated with the electric field in free space:
202
1EuE
The energy density associated with the magnetic field in free space:
2
0
1
2
1BuB
Electromagnetic Waves
BEBE uuuuu 22 Total energy density:
2
0
20
1BEu
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UEEP1033 Oscillations and Waves
ExampleImagine an electromagnetic plane wave in vacuum whose electric field (in SI units) is given by
0,0),109103(sin10 1462 zyx EEtzE
Determine (i) the speed, frequency, wavelength, period, initial phase and electric field amplitude and polarization, (ii) the magnetic field.
Solution(i) The wave function has the form: )(sin),( 0 vtzkEtzE xx
)]103(103sin[10Here, 862 tzEx
1816 ms103,m103 vk
Hz105.4,nm7.6662 14
vf
k
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Solution (continued)
Period , and the initial phase = 0s102.2/1 15 fT
Electric field amplitude V/m1020 xE
The wave is linearly polarized in the x-direction and propagates along the z-axis
(ii) The wave is propagating in the z-direction whereas the electric field oscillates along the x-axis, i.e. resides in the xz-plane.
Now, is normal to both and z-axis, so it resides in the yz-plane. Thus,
E
B
E
),(ˆand,0,0 tzBjBBB yzx
Since, cBE
T)109103(sin1033.0),( 1466 tztzBy
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refer to the fields of a wave at a particular point in space and
indicates the Poynting vector at that point
Energy Transport and the Poynting Vector S
• Like any form of wave, an EM wave can transport from one location to another, e.g. light from a bulb and radiant heat from a fire
• The energy flow in an EM is measured in terms of the rate of energy flow per unit area
• The magnitude and direction of the energy flow is described in terms of a vector called the Poynting vector: S
BES
0
1
B ,E
S
is perpendicular to the plane formed by , the direction is determined by the right-hand rule.
S
B E
and
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Energy Transport and the Poynting Vector S
Because are perpendicular to each other in an EM wave, the magnitude of is:
B E
andS
EBS0
1
2E
cS
0
1
E/cB Instantaneous
energy flow rate
Intensity I of the wave = time average of S, taken over one or more cycles of the wave
)(sin11 22
00
tkxEc
Ec
SI m2
rmsrmsrms BEEc
SI0
2
0
1
2
1
In terms of rms :
rmsm EE 2mmm BEE
cI
0
2
0 2
1
2
1
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UEEP1033 Oscillations and Waves
Example[source: Halliday, Resnick, Walker, Fundamentals of Physics 6 th Edition, Sample Problem 34-1
An observer is 1.8 m from a light source whose power Ps is 250 W. Calculate the rms values of the electric and magnetic fields due to the source at the position of the observer.
Energy Transport and the Poynting Vector S
0
2
24
c
E
r
PI rms
V/m48)m8.1(4
H/m)10m/s)(4π10(250W)(3
4 2
78
20
r
PcErms
T106.1m/s103
V/m48 78
c
EB rms
rms
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Polarization of Electromagnetic Wave
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Polarization of Electromagnetic WaveThe transverse EM wave is said to be polarized (more specifically, plane polarized) if the electric field vectors are parallel to a particular direction for all points in the wave
direction of the electric field vector E = direction of polarization
xtkzEE ˆ)sin(0
Example, consider an electric field propagating in the positive z-direction and polarized in the x-direction
ytkzEc
B ˆ)sin(1
0
ztkzEcS ˆ)sin(200
BES
0
1
oo
1
c
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UEEP1033 Oscillations and Waves
ExampleA plane electromagnetic harmonic wave of frequency 6001012 Hz, propagating in the positive x-direction in vacuum, has an electric field amplitude of 42.42 V/m. The wave is linearly polarized such that the plane of vibration of the electric field is at 45o to the xz-plane. Obtain the vector BE
and
Solution
:bygiven isvectorelectricThe E
here 2/120
200,0 zyx EEEE
812
0103
106002sinx
tEE
102
100 Vm30 EEE zy
x
y
z
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UEEP1033 Oscillations and Waves
Solution (continued)
So
812
103106002sin30,0
xtEEE zyx
8127
103106002sin10,0
xtBBB yzx
cBE
)ˆˆ(ˆˆ kjEkEjEE yzy
)ˆˆ(ˆˆ kjBkBjBB yzy
BEBE
tonormalis,0Then
required.as,ˆ2)ˆˆ(and
2
ic
EiiBEBES y
zy
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UEEP1033 Oscillations and Waves
Harmonic Waves
)](sin[ txkAy v
)sin( tkxAy
A = amplitude k = 2/ (propagation constant)
)](cos[ txkAy vor
v = f = f (2/k) k v = 2f = (angular frequency)
)cos( tkxAy or
Phase : = k(x + vt) = kx + t moving in the – x-direction = k(x - vt) = kx - t moving in the + x-
direction
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UEEP1033 Oscillations and Waves
Harmonic Waves
)sin( 0 tkxAy
In general, to accommodate any arbitrary initial displacement, some angle 0 must be added to the phase, e.g.
Suppose the initial boundary conditions are such that y = y0 when x = 0 and t = 0 , then
y = A sin 0 = y0
0 = sin-1 (y0/A)
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UEEP1033 Oscillations and Waves
Plane WavesThe wave “displacement” or disturbance y at spatial coordinates (x, y, z): )sin( tkxAy
Traveling wave moving along the +x-direction
At fixed time, let take at t = 0: kxAy sin
When x = constant, the phase = kx = constant
the surface of constant phase are a family of planes perpendicular to the x-axis
these surfaces of constant phase are called the wavefronts
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UEEP1033 Oscillations and Waves
Plane Waves
Plane wave along x-axis. The waves penetrate the planes x = a, x = b, and , x = c at the points shown
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UEEP1033 Oscillations and Waves
Plane Waves
Generalization of the plane wave to an arbitrary direction. The wave direction is given by the vector k along the x-axis in (a) and an arbitrary direction in (b)
x= r cos
)cossin( krAy
)sin( tAy rk
zyx xkxkxk rk
)( zyx k,k,k
are the components of the propagation direction
)( tiAey rk
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UEEP1033 Oscillations and Waves
Spherical & Cylindrical Waves
Spherical Waves:
Cylindrical Waves:
)( tkrier
Ay
)( tkieA
y
r = radial distance from the point source to a given point on the waveform
A/ r = amplitude
= perpendicular distance from the line of symmetry to a point on the waveform
e.g. of the z-axis is the line of symmetry, then 22 yx
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UEEP1033 Oscillations and Waves
Mathematical Representation of Polarized Light
yExEE yx ˆˆ
Consider an EM wave propagating along the z-direction of the coordinate system shown in figure.
The electric field of this wave at the origin of the axis system is given by:
z
x
y
E
xE
yE
0
Propagation direction
Complex field components for waves traveling in the +z-direction
with amplitude E0x and E0y and phases x and y :
)(0
~xtkzi
xx eEE )(0
~xtkzi
yy eEE
xx EE~
Re yy EE~
Re
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UEEP1033 Oscillations and Waves
yxE ˆˆ~ )(
0)(
0yx
tkziy
tkzix eEeE
)(0
)(00
~~ˆˆ[
~
tkzi
tkziiy
ix
e
eeEeE yx
EE
]yxE
]ˆˆ[~
000 yxE yxi
yi
x eEeE = complex amplitude vector for the polarized wave
Since the state of polarization of the light is completely determined by the relative amplitudes and phases of these components, we just concentrate only on the complex amplitude, written as a two-element matrix – called Jones vector:
y
x
iy
ix
y
x
eE
eE
E
E
0
0
0
00 ~
~~E
Mathematical Representation of Polarized Light
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UEEP1033 Oscillations and Waves
Linear PolarizationFigures representation of -vectors of linearly polarized light with various special orientations. The direction of the light is along the z-axis
oscillations along the y-axis between +A and A
Vertically polarized Horizontally polarized Linearly polarized
+A
A
linear polarization along y
1
00~
0
00 A
AeE
eEy
x
iy
ix
E
E
E
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Linear Polarization
1
0 = Jones vector for vertically linearly polarized light
b
a = vector expression in normalized from for 1
22 ba
In general:
Topic 3 Electromagnetic Waves
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UEEP1033 Oscillations and Waves
Linear Polarization
AEE xxy 00 ,0,0
0
1
0~
0
00 A
A
eE
eEy
x
iy
ixE
linear polarization along x
Horizontally polarized
+A-A
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UEEP1033 Oscillations and Waves
Linear Polarization
0
cos,sin 00
yx
yx AEAE
sin
cos
sin
cos~
0
00 A
A
A
eE
eEy
x
iy
ixE
linear polarization at
oscillations along the a line making angle with respect to the x-axis
E
Linearly polarized
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UEEP1033 Oscillations and Waves
3
1
2
1
2/3
2/1
60sin
60cos~0E
Linear Polarization
For example = 60o :
b
a0
~EGiven a vector a, b = real numbers
the inclination of the corresponding linearly polarized light is given by
ox
oy
E
E
a
b 11 tantan
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UEEP1033 Oscillations and Waves
Suppose = negative angle
E0y = negative number
Since the sine is an odd function, thus E0x remain positive
The negative sign ensures that the two vibrations are out of phase, as needed to produce linearly polarized light with -vectors lying in the second and fourth quadrants
E
The resultant vibration takes places place along a line with negative slope
b
aJones vector with both a and b real numbers, not both zero,
represents linearly polarized light at inclination angle
a
b1tan
Linear Polarization
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• In determining the resultant vibration due to two perpendicular components, we are in fact determining the appropriate Lissajous figure
• If other than 0 or , the resultant E-vector traces out an ellipse
Lissajous Figures
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UEEP1033 Oscillations and Waves
Lissajous figures as a function of relative phase for orthogonal vibrations of unequal amplitude. An angle lead greater than 180o may also be represented as an angle lag of less that 180o . For all figures we have adopted the phase lag convention =
y-
x
Lissajous Figures
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Linear Polarization ( = m)
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Circular Polarization ( = /2)
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Elliptical Polarization