Electromagnetic Wave Propagation

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Electromagnetic Wave Propagation Prof. V.K.
Tripathi Department of Physics IIT Delhi October
26, 2007
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Core Issue
How does a signal Propagate from a transmitter
to a receiver?
Transmitter
Receiver
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Signal
On the path of a propagating signal if we measure
the electric field, we find a rapidly oscillating
electric field E. In the simplest form where
A is amplitude ? is frequency in
rad s-1 and The real part of RHS is
implied in Eq.(2) If we measure the magnetic
field, we get
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Signal Propagation
Z
z 0 t 0
z z t t
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Wavefront
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Oblique Propagation
If vp is the phase velocity,
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Example
Obtain the amplitude, frequency, wavelength,
phase velocity and equation of the wavefront.
Solution Compare this equation with

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Major Issues
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Maxwells Equations
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An Important Identity
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Hence.
Similarly,
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Another Important Observation
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Algebraic form of Maxwells Equations

• j k . D ?
• j k . B 0 (i.e. B is perpendicular to k)
• j k ? E j? B j?? H
• j k ? H J ? j? D ? E ? j?? E

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Effective Relative Permittivity
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Relevant Set when ? ?0
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This gives the dispersion relation
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Main Result for a plane EM Wave
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EM Wave in Vacuum
A wave propagating along z may be taken as
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In a Dielectric
Dispersion When ?r depends on ?, refractive
index depends on ? and the medium is called
dispersive, e.g. glass.
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Example An em wave propagating in a dielectric
has
Obtain ?, vp, ?r
Comparing this expression with
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Conductivity
?e
electronic charge
m electron effective mass ?
frequency of the electric field (of the wave)
? collision frequency (1012 s-1)
n0 free electron density (1022 cm-3)
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Special cases ? gtgt ?

• Phase has no dependence
• on z
• Amplitude falls off with z

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Skin depth for ? gtgt ?
No energy propagation
?
?plasmaedge
?
?
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Special case ? ltlt ?
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Applications
Generator-detector
?eff
Earth
Burried ore
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In a plasma
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For ? lt ?p no propagation For ? gt ?p
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Energy Flow
Using the identity
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Sav is called intensity
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Example
The amplitude of a laser in free space is 10 V
m-1. Estimate its intensity.
Solution
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Short Wave Communication
Plasma density n0 increases with height upto 300
km, then it falls off.
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Short waves in ionosphere
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At the turning point, ? ?/2 i.e. ?p ? cos ?i
Wave suffers reflection from a height where
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Antenna
Antenna is an exposed portion of wire through
which an ac current of frequency ? flows.
If it were a dc current
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?
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Radiated intensity ?2
sin2 ?
(? is the angle between antenna length and r )
Saveis maximum for
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Radiation Pattern
x?y plane
x?z plane
locus of the tip of Save as ? varies
locus of the tip of Save as ? varies
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Surface Wave
A wave that propagates along the interface of a
conductor and free space (or dielectric). Its
amplitude falls off away from the interface.

Boundary condition at x 0, i.e., the tangential
(Ez) component of electric field be continuous,
demands that the phase variation in both media be
same (except dependence on x)
We take phase variation of E as
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In vacuum we may write
Comparing it with
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Inside the conductor
Boundary conditions at x 0
EzI EzII and ExI ?eff ExII
A1z A2z

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High Frequency Surface Wave ? gtgt ?
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?
kz
Dispersion relation
Field amplitude falls off with x The decline
is more rapid inside conductor.
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Low Frequency Surface Wave ? ltlt ?
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Medium Wave Communication
Earth conductivity
? 1mho m-1,
Attenuation length 1000 km at 3 MHz. At high
frequency (short waves), attenuation is sever
hence SW communication is via ionospheric
reflection.
MW signals can be detected at hundreds of meters
of height.
Surface waves bend whenever there are bends or
curvature on the surface of the earth.
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Wave are trapped inside the guide by reflections
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• At x 0 and x a, Ez 0
• A1 A2 and sin kxa 0
• kx n ? /a or kx 2 ? /a

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Dispersion Relation
Every mode has a cut-off frequency.
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Optical Fibre Communication
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Conclusions

• Plane em wave propagates as

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Conclusions (Contd)

• Surface wave propagates along a conductor
  surface (X 0) as

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Conclusions (Contd)

• EM wave in waveguide propagates as
• TM modes
• 2. TE modes

for a parallel plane guide
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Conclusions (Contd)

• EM waves carry energy and momentum.
• EM waves carry information through modulation of
  their amplitude or frequency.
• EM waves travel with finite phase velocity
• EM waves suffer reflection from boundaries
  between the two media.

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Conclusions (Contd)

• EM waves suffer total internal reflection when
  angle of incidence
• A current carrying wire radiates. The radiated
  power goes as
• Antenna array emits directed beams.

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• Thank you!