wave
Post on 04-Aug-2015
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Wave Equation
From Maxwell’s second equation
Take curl on both side
From Maxwell’s first equation
……….1
……….2
Substitute 2 in 1
The vector identity
From Maxwell's third equation
For the charge free region
……….3
……….6
Substitute 5 in 4
……….5
Substitute 6 in 3
……….4
This is the wave equation for Electric field
From Maxwell's first Equation
Take curl on both side
From Maxwell’s second equation
….. 7
…… 8
Substitute 8 in 7
The vector identity
From Maxwell’s fourth equation
………. 8
………. 9
Substitute 10 in 9
Substitute 11 in 8
……. 10
……. 11
This is the wave equation for Magnetic field
The above two equation are combinedly called as wave equations
Wave Equation in free space
The wave equations are
In free space conductivity , ,
Substitute these values in the wave equations 1 and 2
……..1
……..2
Wave Equation in Phasor form
Consider a linear, isotropic charge free medium i.e. =0The Maxwell equation in Phasor form are
Take curl on both side of equation 2
The vector identity is
Equating 4 and 5
Substitute 3 in 6
……1
……3
……2
Substitute 1 in 4
……4
……5
……6
……7
……6
The equations 6 and 7 are known as Helmholtz’s equation or Vector Wave equation
If the wave propagating along direction and it has x component only
This is scalar wave equation, a linear differential equation with solution
It has component only and component is ‘0’
To Include the time factor, multiply by
Similarly for ‘H’
……1
……2
By solving the equation 1 and 2 we get the value of and
It’s unit is nepers per meter
It’s unit is radian per meter
Plane wave in Free space
Condition
Values
Attenuation constant
Phase constant or wave number
Velocity of propagation
Wave length
Phase difference between “E” and “H”
Phase difference between “E” and “H”
Attenuation constant
Phase constant or wave number
Velocity of propagation
Wave length
Plane waves in Good conductor
Condition
Phase difference between “E” and “H”
As E or H wave travels in a medium .It’s amplitude is attenuated by a factor
The distance at which the wave amplitude is decreased to a factor is called skin depth
The skin depth is a measure of the depth to which an EM wave can penetrate the medium
Skin Depth
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