‫كلية العلوم‬/ ‫الجاهعة الوستنصرية‬
‫قسن الفيزياء‬
Mustansiriyah University
College of science
Physics department
Lecture (2) for Msc
Electromagnetic wave Equation
Edited by
Prof. Dr. Ali A. Al – Zuky
1
Differential wave equation :
→
Wave equation:
kx= ω t
Phase velocity:
And can be written:
Note: ( k≡β )
Group velocity
: is given by:
Refractive index:
or
So can be write group velocity as follow:
Group velocity(Vg) equal the( phase velocity) only when the medium refractive
index is constant (
, so the group velocity and phase velocity independent of
freq.
When vg & vp vary with frequency (ω) , the medium is called dispersive medium :
ω = ω (k)
Phase velocity (
Dispersive relation of the medium.
)
2
The medium refractive index (n) is a function of light frequency (f) i.e n=n(f) or
n=n( ) : (blue, green & red )
n : decreased with increased
for transparent materials in visible range:
Called normal dispersion
Example: For the following wave equation.
----------( 1)
Find the dispersion relation ω(k) for travelling wave solution of the following
forms, then Find the k(ω), graph wave vector k as a function of wave frequency
ω, then sketch phase and group velocities:
i)
ii)
iii)
.
, H.W
: H.W
Solution.
i):
,
,
So eq.1 become
√
or
3
√
 Phase velocity
:
 Group velocity:
4
EM (Electromagnetic) waves:
Wave differential equations for the electric and magnetic field of an
electromagnetic wave are as follows:
-------------- )1(
E= î E0
= î E0
= --------------)2(
B= ĵ B0
-------------)3(
or
is traveling wave in +z direction and its shape is kept
unchanged if and only if we are working in non-dispersive medium i.e:
This wave move at a constant speed , unchanged its shape in medium , have we
call this medium dispersive medium. And dispersive relation is
Now: check the reaction of angler frequency
wave:
(
Used
and the wave number (k) for the
) ------------) 4(
as a test wave function get:
√
5
Where
,
The wave loses shape
a lot of dispersion and a higher speed
dispersion
Non-dispersive
dispersive
Note: ( k≡β )
6
Transmission signal:
A transmission line is a two-port network connecting a generator circuit at the
sending end to a load at the receiving end.
If the generator voltage is cosinusoidal in time, then the voltage across the input
terminals AA' is:
Where
is the angular frequency, and if we assume that the current
flowing through the wires travels at the speed of light, c = 3 x 108 m/s, then the
voltage across the output terminals BB' will have to be delayed in time relative to
that across AA' by the travel delay-time l/c, where l is the length (l=AB).Thus,
assuming no ohmic losses in the transmission line and ignoring other transmission
line:
* (
(
)=
)+
With
Thus the time delay associated with the length of the line l manifests itself as a
constant phase shift
in the argument of the cosine. Let us compare VBB' to
VAA' at t = 0 for an ultralow-frequency electronic circuit operating at a frequency
f= l kHz. For a typical wire length l = 5cm, Eqs. (1) and (2) give VAA' = Vo and
VBB' = Vo cos(2πfl/c) = 0.999999999998Vo. Hence, for all practical purposes, the
presence of the transmission line may be ignored and terminal AA' may be treated
as identical with BB' so far as its voltage is concerned. On the other hand, had the
line been a 20-km long telephone cable carrying a 1kHz voice signal, then the
7
same calculation would have led to VBB' = 0.91Vo, a deviation of 9%. The
determining factor is the magnitude of
. The velocity of propagation of
a traveling wave is related to the oscillation frequency f and the wavelength λ. by:
( )
In the present case,
= c. Hence, the phase delays:
When is very small, transmission-line effects may be ignored, but when ≥0.01,
it may be necessary to account not only for the phase shift due to the time delay,
but also for the presence of reflected signals that may have been bounced back
by the load toward the generator.
Power loss on the line and dispersive effects may need to be considered as well. A
dispersive transmission line is one on which the wave velocity is not constant as a
function of the frequency f. This means that the shape of a rectangular pulse,
which through Fourier analysis can be decomposed into many sinusoidal waves of
different frequencies, will be distorted as it travels down the line because its
different frequency components will not propagate at the same velocity (Fig.
below). Preservation of pulse shape is very important in high speed data
transmission, not only between terminals, but also across transmission line
segments fabricated within high-speed integrated circuits. At 10 GHz, for example,
the wavelength is λ = 3cm in air but only on the order of 1 cm in a semiconductor
material. Hence, even lengths between devices on the order of millimeters become
significant, and their presence has to be accounted for in the design of the circuit.
A dispersionless line does not distort signals passing through it regardless of
its length, whereas a dispersive line distorts the shape of the input pulses
because the different frequency components propagate at different velocities.
The degree of distortion is proportional to the length of the dispersive line
8
Transverse electromagnetic (TEM) transmission lines: Waves propagating along
these lines are characterized by electric and magnetic fields that are entirely
transverse to the direction of propagation. Such an orthogonal configuration is
called a TEM mode.
Transmission line parameters. These are:
 R: The combined resistance of both conductors per unit length, in Ω/m,
 L: The combined inductance of both conductors per unit length, in H/m.
 G: The conductance of the insulation medium between the two conductors
per unit length, in S/m, and
 C: The capacitance of the two conductors per unit length, in F/m.
The resistance R and conductance G of a conductor of uniform cross section,
therefore, can be computed as
And
Where l is the length of the conductor, measured in meters (m), A is the
cross-sectional the electrical conductivity measured in siemens per meter
(S·m−1), the electrical resistivity of the material, measured in ohm-metres
(Ω·m).
In general, a transmission line can support two traveling waves, an incident
wave [with voltage and current amplitudes (V0+, I0+)] traveling along the
+z-direction (towards the load) and a reflected wave [with (V0-, I0-)] traveling
along the -z-direction (towards the source)
9
The two wave equations, one for V(z) and another for I(z). The wave equation for
V (z) is:
Or
Where
√
Also, can be find the wave equation for I(z) as follow:
Where
Here γ is called the complex propagation constant of the transmission line. As
such, y consists of a real part , called the attenuation constant of the line with
units of Np/m, and an imaginary part , called the phase constant of the line with
units of rad/m.
The solution of V(z) and I(z) wave equations as follows:
The characteristic impedance Z0 given by:
where
√
10
It should be noted that
is equal to the ratio of the voltage amplitude to the
current amplitude for each of the traveling waves individually (with an
additional minus sign in the case of the -z propagating wave), but it is not equal
to the ratio of the total voltage V(z) to the total current I(z), unless one of the two
waves is absent.
The voltage is related to the electric field E and the current is related to the
magnetic field H. Current can be written in terms of voltages and vice versa
In general, each will be a complex quantity characterized by a magnitude and a
phase angle:
| |
|
|
Both waves propagate with a phase velocity
…(21)
given by:
Maxwell’s Equations:
The general differential form of Maxwell’s equations can be used only from
medium and fields are continuous in the space and space derivatives exist:
𝛻⃗ ⃗
or
𝛻⃗ ∙ ⃗
Gauss’ Law
𝛻⃗ ∙ ⃗
Gauss’ Law
𝛻⃗ × ⃗
𝛻⃗ × ⃗
𝜇
Faraday’s Law
𝜖
⃗
or
Ampere’s & Maxwell’s Law
Wave equations:
and
11
(C⋅m−2) Electric flux (electric displacement field)
𝜇
(T) Magnetic flux
(A/m2) current density
Speed of light in vacuum
√
ε
medium permittivity(F/m)the lowest possible value for vacuum
=8.85*
F/m. Permittivity is the measure of medium ability to store an
electric field in the polarization of the medium:
𝜇: permeability of the medium represents of the measure of ability of the medium
of support the formation of a magnetic field with itself. 𝜇
×
H/m, for
vacuum
𝜇
̂
,
̂
: In vacuum (free space): = 0 & J = 0
𝛻⃗ ∙ ⃗
, 𝛻⃗ ∙ ⃗
⃗
, 𝛻⃗ × ⃗
and we get
and 𝛻⃗ × ⃗
𝜇 𝜖
⃗
When in the last two equation see the change in magnetic field generate as electric
field and changing electric field product a magnetic field
To solve Maxwell's equations, in free space need to use the identity
⃗𝜵
⃗ × (𝜵
⃗⃗ × ⃗⃗ )
⃗𝜵
⃗ (𝜵
⃗⃗ ⃗⃗ )
⃗⃗ ⃗𝜵
⃗ )⃗⃗
(𝜵
⃗ ⃗𝜵
⃗
and ⃗𝜵
12
⃗𝜵
⃗
is the Laplace operator
Then can find the wave function of an electric and magnetic fields as follows:
⃗⃗ × 𝑬
⃗⃗ )
𝛻⃗ × (𝜵
𝛻⃗ (𝛻⃗ ⃗ )
⃗⃗ 𝑬
⃗⃗
𝜵
𝛻⃗
⃗⃗
𝛻⃗ ×
⃗⃗ × ⃗⃗ )
(𝜵
⃗𝜵
⃗ ⃗𝑬
⃗
𝛻⃗ ×
⃗
𝝁 𝝐
⃗𝑬
⃗𝑬
𝝁 𝝐
Where :
⃗⃗
)𝑬
(
𝝁 𝝐
⃗𝑬
⃗
These equations changed the world, and Maxwell was the first to create them, as
he deduced the wave equations for electric and magnetic fields and found that the
velocity of electric and magnetic fields wave is constant and can be calculated
from:
×
√𝝐
Also can be find wave equation for magnetic field:
⃗⃗
𝝁 𝝐
It is very important that the associated magnetic field also satisfies the wave
equation. So from Maxwell’s equations E create B and B create E.
Scalar fields: every position in the space gets a number
Vector Files: Instead of a number or scalar every point gets a vector
̂
̂
̂
13
Divergence: using the definition for above vector:
⃗ ⃗⃗
⃗ ⃗⃗
Divergence of vector ⃗⃗
The divergence is a measure of how much the vector spreads out (diverges) from a
point.
Curl : (rotate)
̂
̂
̂
⃗ ×
[
⃗ ×
̂(
)
]
̂(
)
̂
What exactly does Curl mean: Curl is mean measure of how much the vector
(Curl around) a point.
14
Solving problems
Problem1: Given the dispersion relation ω = ak2, calculate: (a) phase velocity and
(b)group velocity.
Solution
ω = ak2
a)
b)
Problem 2: Show that the group velocity
where
is the phase velocity and
Solution:
ω=
k
(
)
15
can be expressed as
.
Problem3: Show that if the phase velocity varies inversely with the wavelength
then the group velocity is twice the phase velocity.
Solution:
(by problem)
Problem4: Prove that the usual expression for the group velocity of a light wave in
a medium can be rearranged as
, where c is the phase velocity of the
waves in free space, f is the frequency and n is the refractive index of the medium.
Solution:
But:
Problem5: A plane EM-wave E = 100 cos (6×108 t – 4z) (V/m) propagates in a
medium. What is the refractive index of this medium.
Solution:
×
The standard wave equation given by:
16
×
×
×
×
, so
×
Problem6: Check if a plane wave satisfying wave equations:
⃗
⃗
⃗
̂
⃗
then
Solution:
𝜇 𝜖
𝜇 𝜖
√𝜖
Then we can fined ⃗
⃗
(⃗ × ⃗ )
̂
S
[
̂
̂
]
(
̂)
̂
17
̂
So eq6 become
⃗
̂
⃗
⃗
⃗
̂
∫
̂
̂
Here form eq.2 and eq.9 can be concluding
1- The ⃗ field must come with ⃗ field, the two fields are perpendicular and
they are in phase. If ̂ is the direction of propagation then:
⃗
̂× ⃗
The amplitude of the magnetic field is equal to the amplitude of electric field
divided by the speed of light.
2- The EMW in free space is non-dispersive wave that mean the speed of light
(c) is independent of the wave number (k).
√𝝐
3- The direction of propagation of EM is in same direction of (⃗ × ⃗ ).
⃗⃗
̂×𝑬
⃗
̂
⃗
18
Problem7: a) Show that for a magnetic field the wave equation has the form
⃗
𝜇 𝜖
in vacuum, and b) Then prove the resulted electromagnetic waves
propagate at the speed of light:
Solution:
a) Maxwell's equation in vacuum:
𝛻⃗ ∙ ⃗
, 𝛻⃗ ∙ ⃗
⃗
, 𝛻⃗ × ⃗
and𝛻⃗ × ⃗
𝜇 𝜖
⃗
By using
⃗𝜵
⃗ × (𝜵
⃗⃗ × ⃗⃗ )
⃗⃗⃗⃗ ⃗⃗⃗
⃗⃗ ⃗𝜵
⃗ )⃗⃗ = 𝜵
(𝜵
𝜵 ⃗⃗
⃗𝜵
⃗ (𝜵
⃗⃗ ⃗⃗ )
𝛻⃗ × [𝛻⃗ × ⃗
⃗𝜵
⃗ × (𝜇 𝜖
⃗
𝜇 𝜖
⃗
(
𝜇 𝜖
⃗
𝜇 𝜖
⃗
⃗𝜵
⃗ ⃗⃗
]
⃗𝜵
⃗ ⃗⃗
)
⃗𝜵
⃗ ×⃗
𝜇 𝜖
⃗𝜵
⃗ ⃗⃗
⃗𝜵
⃗ ⃗⃗
⃗𝜵
⃗ ⃗⃗
)
=
⃗
𝜇 𝜖
b) the resulted free space wave equation for a medium without absorption is:
⃗
𝜇 𝜖
Compare with standard wave equation:
19
𝝁 𝝐
𝟖
√𝝁 𝝐
𝟖×
√𝟒
×
× 𝟖 𝟖𝟓𝟒 ×
References
1. Fawwaz T. Ulaby,Eric Michielssen,Umberto Ravaioli'sFundamentals of
Applied Electromagnetics (6th Edition) [Hardcover], Prentice Hall, 2010.
2. Jackson, John D. (1998). Classical Electrodynamics (3rd ed.),Wiley.
3. 1000 Solved Problems in Classical Physics. Ahmad A. Kamal . 2011.
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