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# Electromagnetic waves

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• 1. Electromagnetic Waves
Presented by :
Anup Kr Bordoloi
ECE Department ,Tezpur University
11/11/2008
• 2. Electromagnetic Waves in homogeneous medium:
The following field equation must be satisfied for solution of electromagnetic problem
there are three constitutional relation which determines
characteristic of the medium in which the fields exist.
in particular case of e.m. phenomena in free space or in a perfect dielectric containing no
charge an no conduction current
Differentiating 1st
• 3. Also since and are independent of time
Now the 1st equation becomes on differentiating it
Taking curl of 2nd equation
(But )
this is the law that E must obey
lly for H
these are wave equation so E and H satisfy wave equation.
• 4. For charge free region
for uniform plane wave
There is no component in X direction be either zero, constant in
time or increasing uniformly with time .similar analysis holds for H
Uniform plane electromagnetic waves are transverse and have components in E and H only in the direction perpendicular to direction of propagation
Relation between E and H in a uniform plane wave:
For a plane uniform wave travelling in x direction
a)E and H are both independent of y and z
b)E and H have no x component
From Maxwell’s 1st equation
From Maxwell’s 2nd equation
• 5. Comparing y and z terms from the above equations
on solving finally we get
lly
Since
The ratio has the dimension of impedance or ohms , called characteristic impedance or intrinsic impedance
of the (non conducting) medium. For space
• 6. The relativeorientation of E and H may be determined by taking their dot product and using above relation
In a uniform plane wave ,E and H are at right angles to each other.
electric field vector crossed into the magnetic field vector gives the direction in which the wave travels.
• 7. The wave equation for conducting medium:
From Maxwell’s equation if the medium has conductivity
Taking curl of 2nd eq. ( )
For any homogeneous medium in which is constant
But there is no net charge within a conductor
Hence wave equation for E.
lly , wave equation for H.
Sinusoidal time variations:
where is the frequency of variation.
time factor may be suppressed through the use phasor
notation.
Time varying field may be expressed in terms of corresponding phasor quan
-tity
• 8. as
Phasor is defined by
real
Phase is determined by of the
complex number ,time varying
field quantity may be expressed as
Maxwell’s equation in phasor form:
for sinusoidal steady state we may substitute the phasor
relation as
Imaginary axis
Real axis
• 9. which is the differential equation in phasor form.
Observation point:
Time varying quantity is replaced by phasor quantity
Time derivative is replaced with a factor
Maxwell’s equation becomes
The above equations contain the equation of continuity
The constitutive relation retain their forms
For sinusoidal time variations the wave equation for electric field in lossless medium
• 10. becomes
In a conducting medium the wave equation becomes
Wave propagation in lossless medium:
For uniform plane wave there is no variation w.r.t. Y or Z.
For Ey component solution may be written as
The time varying field is
real
• 11. When c1 and c2 are real,
if c1 = c2 the two travelling waves combine to form standing wave which does not
progress.
Wave velocity: if velocity is given by
or
phase –shift constant.
From fig.
Again
• 12. Thank you