Electromagnetic Fields
Electromagnetic Waves in Material Media
In a material medium free charges may be present, which generate a current under the influence of the wave electric field. The current
J c
is related to the electric field E through the conductivity σ as
J c
=
E
The material may also have specific relative values of dielectric permittivity and magnetic permeability
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Electromagnetic Fields
Maxwell’s equations become j H
H E j E j ( j
σ
ω
)E
In phasor notation, it is as if the material conductivity introduces an imaginary part for the dielectric constant ε . The wave equation for the phasor electric field is given by
E E
2
E j H j (J c j E)
⇒ ∇ 2
E j ( j )E
We have assumed that the net charge density is zero, even if a conductivity is present, so that the electric field divergence is zero.
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Electromagnetic Fields
In 1-D the wave equation is simply
∂ 2 j
∂
E z
2 x
( j )E x
= γ with general solution
2
E x z = A exp( z ) B γ z y
= − j
1 ∂ E
ωµ ∂ z x = j ωµ
(
A exp( −γ − B γ z
)
=
1
η
(
A exp( −γ − B γ z
)
These resemble the voltage and current solutions in lossy transmission lines .
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Electromagnetic Fields
The intrinsic impedance of the medium is defined as
η = η e j τ
= j ωµ
For the propagation constant , one can obtain the real and imaginary parts as
γ = j ( j ) j
α =
β =
ω µε


2
ω µε


2
1 +
 σ 
2
 ωε 
1 +
 σ 
 ωε 
2
− 1



1 / 2
+ 1



1 / 2
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Electromagnetic Fields
Phase velocity and wavelength are now functions of frequency v p
=
ω
=
β
2
µε


1 +
( )
2
ωε
+ 1


− 1 / 2
λ =
2 π
=
β f
2
µε


1 +
( )
ωε
2
+ 1



− 1 / 2

The intrinsic impedance of the medium is complex as long as the conductivity is not zero. The phase angle of the intrinsic impedance indicates that electric field and magnetic field are out of phase. Considering only the forward wave solutions z = A exp( −γ z ) = A exp( −α z ) exp( − β ) z =
1
η
A exp( z j
1
η
A exp( −α z ) exp( j z j )
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Electromagnetic Fields
In time-dependent form x
=
{
A θ
= A exp( −α z ) cos( exp( −α z ) exp( t z ) y
=
1
η
Re
{
A θ exp( −α z ) exp(
) exp( ω )
} j z j ) exp( ω )
}
=
1
A exp( −α z ) cos( t z )
η where the integration constant has been assumed to be in general a complex quantity as
A = A θ
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Electromagnetic Fields
Classification of materials
Perfect dielectrics - For these materials
σ
= 0
Propagation constant
β = ω ε ε µ µ
α = 0
Medium Impedance
η = j j
ωµ
=
µ µ
ωε ε ε
Phase velocity v p
λ =
2
β
π
= v p
= f f
1
β µ µ ε ε
Wavelength
1
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Electromagnetic Fields
Imperfect dielectrics – For these materials
σ ≠ 0 but (
σ / ωε )<< 1
γ = j ( j ) j 1 − j
σ
ωε
≈
σ µ
2 ε
α ≈
σ µ
2 ε
β
…
≈ ω µ ε v p
β
1
µε
λ =
2
β
π
≈ f
1
µε
η = j ωµ
= j ωµ j ω ε 
1 − j
σ
ωε 
− 1
2
≈
µ
ε
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Electromagnetic Fields
If (
σ / ωε )<< 1 , the errors made in the approximations for
α , β , v p and
λ are very small, since only terms of order (
σ / ωε ) 2 or higher appear in the expansions. The error is slightly higher fo the medium impedance
η
since the expansion contains a term of order
(
σ / ωε ).
The simple rule of thumb is that approximations for imperfect dielectric can be applied when
σ
ωε
≤ 0.1
When the condition above is verified, the imperfect dielectric behaves in all respects like a perfect dielectric, except for an attenuation term in the fields.
The quantity
σ / ωε is called Loss Tangent .
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Electromagnetic Fields
Good conductors – For these materials
σ ≠ 0 but (
σ / ωε )>> 1
γ = j ( j ) j ωµσ = ωµσ
= ωµσ exp( j
π
4
) = ωµσ


1
2
+ j
1
2

 j
f
v p
=
ω
β
≈
4 π
µσ f
λ =
2
β
π
≈
4 π f µσ
= j ωµ
≈ j ωµ
σ
=
ωµ
σ exp( j
π
4
)
=
ωµ 
σ 
1
2
+ j
1
2


=
(1
j )
(1 j )
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Electromagnetic Fields
The simple rule of thumb is that approximations for good conductor can be applied when
10
Note that for a good conductor the attenuation constant
α and the propagation constant
β are approximately equal.
The medium impedance
η
has nearly equal real and imaginary parts, therefore its phase angle is approximately 45 ° .
This means that in a good conductor the electric and magnetic fields have always a phase difference
τ
= 45 ° =
π / 4.
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Electromagnetic Fields
Also, in a good conductor the fields attenuate very rapidly. The distance over which fields are attenuated by a factor exp( − 1.0) is
1
α
= δ =
1
=
A typical good conductor is copper , which has the following parameters:
σ =
ε ≈ ε o
µ ≈ µ o
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Electromagnetic Fields
Copper remains a good conductor at extremely high frequencies.
Another good conductor example is sea water at relatively low frequencies
σ ≈ 4.0 [S/m]
At a frequency of 25 kHz
ε ≈ 80 ε o
µ ≈ µ o
σ
ωε
≈ 36, 000
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Electromagnetic Fields
Perfect conductor - For this ideal material
σ →
For this material, the attenuation is also infinite and the skin depth goes to zero. This means that the electromagnetic field must go to zero below the perfect conductor surface.
General medium When a material is not covered by one of the limit cases, the complete formulation must be used. We can classify a material for which the conditions (
σ / ωε )<< 1 or (
σ / ωε )>> 10 are invalid as a general medium .
The simple rule of thumb for general medium is
10 >
σ
>
ωε
0.1
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