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11
Engineering Electromagnetics
EssentialsChapter 6
Wave equation and its solution for a wave propagating through an unbounded medium
22
Objective
Topics dealt with
Background
Study of propagation of uniform plane electromagnetic waves through unbounded media such as free-space and conducting media
Representation of a wave
Establishment of wave equations in electric and magnetic fields
Wave propagation through an unbounded free-space medium
Wave propagation through a resistive medium
Skin depth, surface resistance and ac resistance
Wave propagation through sea water and choice of operating frequency vis-à-vis attenuation
Wave propagation through a medium of charge particles
Basic concepts of time-varying fields developed in Chapter 5 including Maxwell’s equations
3
A quantity that varies periodically with space and time is said to be associated with a wave. Let us see how a mathematical function can represent a propagating wave.
v
ztfztf ),(
)(condition 0 provided
)()(
),(
v
zt
v
ztf
v
zt
v
ztf
v
zzttfzzttf
Let us examine the function
t
zv
)(condition 0 dt
dz
t
z
t
Ltv
Subscript + refers to a forward wave propagating in positive z direction
Representation of a quantity associated with a wave
44
v
ztfztf ),(
)(condition 0
provided
)(),(
dt
dz
t
z
t
Ltv
v
ztfzzttf
v is identified as the velocity of wave.
v
ztfztf ),(
Thus the function
represents a forward wave propagating in positive z direction
Rewritten
55
v
ztfztf ),(
)(condition 0 provided
)()(
),(
v
zt
v
ztf
v
zt
v
ztf
v
zzttfzzttf
Similarly, now let us examine the function
t
zv
)(condition 0 dt
dz
t
z
t
Ltv
Subscript - refers to a backward wave propagating in negative z direction
v is identified as the velocity of wave.
v
ztfztf ),(
Thus the function
represents a backward wave propagating in negative z direction
66
We have thus seen that
v
ztfztf ),(
the function
represents a forward wave propagating in positive z direction
and that
v
ztfztf ),(
the function
represents a backward wave propagating in negative z direction
How to choose the functions so that they can represent the periodicity of the wave with space and tome coordinates?
Choose the functions as proportional to exp j(t-z/v) for a forward wave and to exp j(t+z/v) for a backward wave in phasor notation?
77
tjEtE
tjtEtjEE
sincos
)sin(cosexp
00
00
Phasor diagram of a time-periodic quantity, say magnitude of electric field E of amplitude E0 rotating with angular velocity
Phasor is a rotating vector
tE cospart Real 0
tE sinpart Imaginary 0
vztt /
v
ztfztf ),( (forward wave)
v
ztfztf ),( (backward wave)
Choose the functions as proportional to exp j(t-z/v) for a forward wave and to exp j(t+z/v) for a backward wave in phasor notation.
With reference to physical quantity, say, electric field for forward and backward waves, we may take the functions taking into regard their periodicity as follows:
)/(exp)exp( 00 vztjEtjEE
interpreting t in phasor notation t/ as
(forward wave)
vztt /
and
)/(expexp 00 vztjEtjEE
interpreting t in phasor notation ti/ as
(backward wave)
88
wave)(forward
)]/(sin)/([cos
)/(exp)exp(
0
00
vztjvztE
vztjEtjEE
tE cospart Real 0
tE sinpart Imaginary 0
wave)(backward
)]/(sin)/([cos
)/(exp)exp(
0
00
vztjvztE
vztjEtjEE
interpreting t as t’ and t’’ in phasor notation
wave)(backward / and wave)(forward / vzttvztt
In an alternative approach, we can choose either the real or imaginary part of the function to represent the wave. Taking the real part:
wave)(backward )/(cos
wave)(forward )/(cos
0
0
vztEE
vztEE
wavebackward theof phase
)/( )/(
waveforward theof phase
)/( )/(
ztzvtvzt
ztzvtvzt
constant n propagatio phase / v
Phase of the wave determines (i) the value of the quantity (here, typically, E- or E+) with which the wave is associated, and (ii) the state of variation of this value.
99
Wave phase velocity, wavelength and frequency and the relation between them (recalled) waveforward theof phase zt
)cos(0 ztEE
Put the phase of the wave as constant:
constant phase zt
Differentiating
dt
dz
phv (phase velocity of the wave)
velocity of the constant phase of the wave
Variation of the quantity (here, electric field magnitude) representing a wave, typically, forward wave, with space coordinate z at a fixed time (snap-shot), typically t = 0:
zEzEE cos)cos( 00
time)fixed a(at 0t
1010
)cos(0 ztEE
zEzEE cos)cos( 00
time)fixed a(at 0t (recalled)
1cos maximum Positive z
...,6,4,2,0 z (corresponding to the same phase: here, referring to positive maximum value of the quantity)
Interval of z between two consecutive same phases at a fixed time
2z
2
z Distance between two consecutive same phases at a fixed time called wavelength
2
0EE
(wavelength)
2
(wave propagation constant)
1111
)cos(0 ztEE
distance) fixed a(at 0z
tEE cos0
0EE
1cos maximum Positive t
...,6,4,2,0 t (corresponding to the same phase: here, referring to positive maximum value of the quantity)
Interval of t between two consecutive same phases at a fixed time
2t Time interval between two consecutive same phases at a fixed distance called time period T
2
t Time interval between two consecutive same phases at a fixed distance called time period T
2
T
1212
2
T
2
1
Tf (wave circular frequency or simply frequency)
f 2 (wave circular frequency or simply frequency)
f
v2
ph 2
fv ph (relation between phase velocity, frequency and wavelength)
1313
Wave equations in electric and magnetic fields
t
HE
t
EJH
t
HE
Electric and magnetic fields are coupled in the following two Maxwell’s equations:
HB
ED
In order to decouple these two equations to obtain a single equation, namely wave equation in electric field, take the curl operation on the first of these equations and read it using the second of these equations.
(Maxwell’s equations recalled)
Wave equation in electric field
1414
EEE
2)(
E
2
2
)(t
E
t
J
t
EJ
tE
2
22)(
t
E
t
JEE
2
22
t
E
t
JE
(Maxwell’s equation)
2
22
t
J
t
EE
Rearranged
(wave equation in electric field)
t
EJH
)( H
tE
(Maxwell’s equation)
t
HE
(rewritten) Partial time derivative and curl involving partial
derivatives in space coordinates are interchangeable
(vector identity)
1515
) )( t
EJ
t
EJH
t
EJH
t
HE
In order to decouple these two equations to obtain a single equation, namely wave equation in magnetic field, take the curl operation on the second of these equations and read it using the first of these equations.
(Maxwell’s equations recalled)
Wave equation in magnetic field
)( Et
JH
Partial time derivative and curl involving partial derivatives in space coordinates are interchangeable
t
HE
(Maxwell’s equation)
2
2
t
HJH
1616
2
2
t
HJH
HHH
2)(
(vector identity)
2
22 )(
t
HJHH
2
22 J
t
HH
0 H
(Maxwell’s equation)
(wave equation in magnetic field)
Wave propagation through a Free-space medium
2
22
2
22
Jt
HH
t
J
t
EE
(source-free wave equations in electric and magnetic fields)
0J
0
0
2
22
2
22
t
HH
t
EE
)0( J
1717
Consider a uniform plane wave propagating through an unbounded free-space medium.
(source-free wave equations in electric and magnetic fields)
0
0
2
22
2
22
t
HH
t
EE
)0( J
2
2
002
2
2
002
2
2
002
t
EE
t
EE
t
EE
zz
yy
xx
2
2
002
2
2
002
2
2
002
t
HH
t
HH
t
HH
zz
yy
xx
02
22
t
EE
02
22
t
HH
Formulate the problem by considering wave propagation along z in the rectangular coordinate system.
A plane wave has a plane wavefront. A wave front is an equi-phase surface over which the phase of the quantity associated with the wave remains constant. For a uniform plane wave, in addition to the phase, the amplitude of the quantity associated with the wave also remains constant over the plane wavefront.
1818
2
2
002
2
2
002
2
2
002
t
EE
t
EE
t
EE
zz
yy
xx
2
2
002
2
2
002
2
2
002
t
HH
t
HH
t
HH
zz
yy
xx
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
t
E
z
E
y
E
x
E
t
E
z
E
y
E
x
E
t
E
z
E
y
E
x
E
zzzz
yyyy
xxxx
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
t
H
z
H
y
H
x
H
t
H
z
H
y
H
x
H
t
H
z
H
y
H
x
H
zzzz
yyyy
xxxx
1919
0
yx
For a uniform plane wave propagating along z
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
t
E
z
E
y
E
x
E
t
E
z
E
y
E
x
E
t
E
z
E
y
E
x
E
zzzz
yyyy
xxxx
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
t
H
z
H
y
H
x
H
t
H
z
H
y
H
x
H
t
H
z
H
y
H
x
H
zzzz
yyyy
xxxx
2
2
002
2
2
2
002
2
2
2
002
2
t
E
z
E
t
E
z
E
t
E
z
E
zz
yy
xx
2
2
002
2
2
2
002
2
2
2
002
2
t
H
z
H
t
H
z
H
t
H
z
H
zz
yy
xx
(wave equation in electric field for a uniform plane wave in free space)
(wave equation in magnetic field for a uniform plane wave in free space)
2020
E
(Maxwell’s equation) 0 H
(Maxwell’s equation)
medium space-free afor
0
0 E
0
z
E
y
E
x
E zyx 0
z
H
y
H
x
H zyx0
yx
(for a uniform plane wave propagating along z)
0z
Ez 0z
H z
zwith
constant each are and zz HE
02
2
z
Ez 02
2
z
H z2
2
002
2
t
E
z
E zz
2
2
002
2
t
H
z
H zz
02
2
t
Ez 02
2
t
H z
2121
02
2
t
Ez 02
2
t
H zIntegrating
At
Ez
BAtEz
Ct
H z
DCtH z
Integrating
tDCBA with constants are ,,,
Ez and Hz are each constant with z and that they are linearly dependent on t.
(linearly dependent on t) (linearly dependent on t)
0
0
z
z
H
E
A uniform plane wave propagating through an unbounded free-space medium is essentially a transverse electromagnetic wave (TEM) with no electric and magnetic field components in the direction of propagation (along z)
Contradicts with the dependence of these quantities as exp j(t-z/v) (for a forward wave in free space considered) with non-zero values of the constants A, B, C and D.
Avoidance of contradiction
0 DCBABAtEz DCtH z
2222
)(0 yyxxyx
xy aHaH
ta
z
Ea
z
E
t
EJH
t
HE
(Maxwell’s equations recalled)
space) free(for 0J
0
0
z
z
H
Et
EH
t
HE
)(0 yyxxyx
xy aEaE
ta
z
Ha
z
H
Equating the x-components and y-components of the left- and right-hand sides
t
H
z
E
t
H
z
E
yx
xy
0
0
t
E
z
H
t
E
z
H
yx
xy
0
0
Ex
Hy
Z
A uniform plane wave propagating through an unbounded free-space medium is essentially a transverse electromagnetic wave (TEM) with no electric and magnetic field components in the direction of propagation (along z)
2323
t
H
z
E
t
H
z
E
yx
xy
0
0
t
E
z
H
t
E
z
H
yx
xy
0
0
)(exp
as vary quantities Field
ztj
jz
jt
yx
xy
HE
HE
0
0
yx
xy
EH
EH
0
0
0
x
y
y
x
H
E
H
E
0
x
y
y
x
H
E
H
E0 yyxx HEHE
0HE
0
0
Electric and magnetic fields are at right angle to each other.
0022 cv
00
ph
1
2424
Unbounded free-space medium supports transverse electromagnetic (TEM) wave, such that the directions of the electric field, magnetic field and propagation are mutually perpendicular to one another, propagating with phase velocity equal to the speed of light c:
0HE cv
00
ph
1
m/s 1031 8
00
cH/m 104 70
F/m 19854.810)36/(1 1290
Intrinsic impedance 0 of the free-space medium is
0
x
y
y
x
H
E
H
E2/122
2/122
0)(
)(
yx
yx
HH
EE
H
E
02/122
2/1
222
0
2/122
2/122
0)(
)(
)(
)(
yx
yx
yx
yx
HH
HH
HH
EE
10
2525
)(rewritten 00
0022
00
0
00
H/m 104 70
F/m 19854.810)36/(1 1290
120
0
00
(intrinsic impedance of a free-space medium)
2626
Wave propagation through a conducting medium
2
22
t
J
t
EE
(wave equation in electric field)
(recalled)
2
22
t
E
t
EE
medium) g(conductin 0
EJ
Let us recall wave equation in electric field:
22
2
))((
jjt
jt
EjjE
)(2 )( jj
EE
22
2727
EE
22 Expanding Laplacian 2
0
yx
(for a uniform plane wave propagating along z)
yyy
xxx
EjjEz
E
EjjEz
E
)(
)(
22
2
22
2
)( jj
is called the propagation constant of the wave
)exp(
)exp(
0
0
zEE
zEE
yy
xx
(solution for a forward wave) (subscript 0 referring to field amplitudes)
jjj )(
)exp()exp(])(exp[
)exp()exp(])(exp[
00
00
zjzEzjEE
zjzEzjEE
yyy
xxx
wave theofconstant n propagatio phase wave theofconstant n attenuatio
2828
Skin depth
)exp()exp(])(exp[
)exp()exp(])(exp[
00
00
zjzEzjEE
zjzEzjEE
yyy
xxx
wave theofconstnat n propagatio phase
wave theofconstant n attenuatio
jjj )( (recalled)
Invoking time dependence exp(jt)
)exp()exp()exp(])(exp[
)exp()exp()exp(])(exp[
00
00
tjzjzEzjEE
tjzjzEzjEE
yyy
xxx
Amplitude exponentially attenuates as )exp( z
For a good conductor:
jj
2929
Separate real and imaginary parts Squaring
jj (for a good conductor)
22)()( 222220 jjjjj
Equating imaginary partEquating real part
022 20
20
)(
)( (for a good conductor)
3030
20
)( (for a good conductor)
)exp()exp()exp(])(exp[
)exp()exp()exp(])(exp[
00
00
tjzjzEzjEE
tjzjzEzjEE
yyy
xxx
Amplitude exponentially attenuates as )exp( z
Taking = 1/, the exponential attenuating factor of the field amplitude may be put as
)/exp()exp( zz
Putting z = , we appreciate that the field amplitude at the surface (skin) of a good conductor at z = 0 attenuates by a factor of exp (-1) = 1/e at a depth equal to z = called the skin depth of the conductor.
/1
/12
0
0
2 (skin depth)
31
0
2 (skin depth) f 2
0
1
f Skin depth decreases with the increase of the conductivity of
the medium and operating frequency
/2
)(2
0
2
1 Skin depth is directly proportional to the wavelength in
the conducting medium.
Typically, for copper ( = 5.8x107 mho/m), at operating frequency 50 Hz, the value of the skin depth is ~ 9.4 mm. The skin depths are ~ 2.1 mm, ~ 66.1 m, and ~ 2.1 m, for frequencies 1 kHz, 1 MHz, and 1 GHz, respectively. For 10 GHz frequency, we find the skin depth as 6.61x10-7 m, which happens to be in the range of wavelength of visible light.
32
Surface resistance and ac resistance
(recalled) )(exp0 zjEE yy
(RF time dependence jt understood)
Let us recall the expression for electric field intensity at depth z from the surface of the conductor:
0yE Electric field intensity amplitude at the surface of the conductor
)exp()(exp0 tjzjEE yy
33
Current density along y at depth z from the surface of the conductor
]][)(exp[ 0 WdzzjEy
zjEy )(exp0
Current through the strip of infinitesimal thickness dz and width W
Current through the entire conductor of width W
j
WE
j
zjWE
zdzjWE
yy
y
0
0
0
0
0
)(
)(exp
)(exp
Current through the entire conductor of unit width (W = 1)
j
Ey
0
Ey0 = E0
34
Surface impedance Zs is defined as
widthunit'' ofconductor entire rough theCurrent th
conductor theof surface at thelength unit''per drop PotentialsZ
ssy
y jXRjjj
j
E
E
11
0
0
/1
Rs = Surface resistance ( or /)
Xs = Surface reactance ( or /)
1
ss XR
0
1
f
0f
XR SS
j
Ey
0
0yE
Comparing the real and imaginary parts
35
Expression for the dc resistance Rdc of a wire of length l and radius a is well known:
ac resistance of a straight conducting wire:
We can find the ac resistance of a straight round wire (that is, of circular cross section) made of a good conductor at high frequencies and compare it with its dc resistance to show that the ac-to-DC resistance ratio of the wire is a/2, where a is the radius of the wire and is the skin depth of the conductor making the wire.
2dc
1
a
lR
Let us next proceed to find an expression for the ac resistance
Rac of the wire using the same approach as used to find the surface resistance.
36
For a round wire of radius a large compared to the skin depth , a point inside the wire where the electric field is significant will not ‘see’ the curvature of the round wire and therefore we can take the surface of the wire as a planar surface.
Current through the wire of width W = 2a
)2(0 aj
Ey
Therefore, following exactly the same approach as used to find the surface resistance of a planar conductor we can find the resistance of the round wire interpreting the width as the circumference of the wire (W = 2a).
Ey0 = E0
Potential drop across length l at the surface of the wire
lEy0
aW
lZ
2 width of wirerough theCurrent th
wire theof surface at the length across drop Potential wire theof Impedance
ac
acac0
0ac )2()2()2(
)(
)2(jXR
a
lj
a
l
a
lj
aj
E
lEZ
y
y
Separated in the real part Rac, which is the ac resistance of the wire, and the imaginary part Xac, which is the ac reactance of the wire
37
2dc
1
a
lR
21
21
2
ac a
ala
l
R
R
dc
1
acacac )2()2(jXR
a
lj
a
lZ
acacac )2()2(jXR
a
jl
a
lZ
a
lXR
2
1acac
Since the wire radius a>>, skin depth, the wire ac resistance Rac>>Rdc, the wire dc resistance
38
ac resistance per unit length of a coaxial cable:
We can find the ac resistance of a coaxial cable made of a central solid round conductor and a coaxial annular conductor surrounding it with a dielectric medium filling the region between the conductors by extending the analysis presented for a straight round wire. For a good conductor making the coaxial cable and at high frequencies, as in the case of a straight wire, we can treat the inner and outer conductors behave as a planar surface.
(ac resistance of a coaxial cable of length l)
,
a
lR
2
1ac (recalled expression for the ac resistance of length l of a straight round wire)
a
lR
2
1ac
Concept of finding the ac resistance of a straight wire extended to the inner conductor of radius a and to the outer conductor of radius b
(ac resistance contributed by the inner conductor of length l)
b
lR
2
1ac
(ac resistance contributed by the outer conductor of length l)
lbab
l
a
lR
11
2
1
2
1
2
1ac
ba
l11
2
1cable coaxial a of )1(length unit per Resistance
39
JH
t
BE
t
EE
t
DEJ
t
EEH
t
HE
0
Wave propagation through sea water
Maxwell’s equations (recalled):
The objective of the study is to find out which frequencies of operation (lower ~ 10 kHz or higher ~ 10 GHz) should be preferred from the standpoint of a lower attenuation of the wave propagating through sea water.
40
)(00 Htt
HE
t
EEH
)(0 t
EE
tE
EjjEjEE
))(()( 002
jt
EjjEE
)()( 02 0 E
)(exp ztj RF quantities vary as
EEjjE
20
2 )(
jjj )( 00
t
HE
0 (Maxwell’s equations)
(Maxwell’s equations)
Partial time derivative and curl involving partial derivatives with respect to space coordinates are interchangeable
41
EEjjE
20
2 )(
jjj )(0
yxyxyxyx E
z
E
y
E
x
E,
22
,2
2
,2
2
,2
0
yx
)exp()exp(ˆ)](exp[ˆ)exp(ˆ,,,, zjzEzjEzEE yxyxyxyx
yxyx E
x
E,
22
,2
jjj )(0
Let us examine two situations: one for >> and the other for << with reference to sea-water communication.
water)(sea mho/m4
m/F10854.88181 120
(uniform plane wave supposedly propagating along z)
42
Propagation at lower frequencies
)exp()exp(ˆ)](exp[ˆ)exp(ˆ,,,, zjzEzjEzEE yxyxyxyx
jjj )(0
jj
j )1(0
5101.12
f
water)(sea mho/m4
m/F10854.88181 120
2/10 )1(
j
j
Two alternative ways of putting the propagation constant
)(typically Hz1010kHz10 3f
)(typically Hz1010kHz10 3f
jj
j )1(0
Let us recall the following expression at lower frequencies:
43
20
lflf
jj
j )1(0(recalled at lower frequencies)
(recalled) 101.12 5
f
water)(sea mho/m4
m/F10854.88181 120
)(typically Hz1010kHz10 3f
parts)imaginary and real theg(separatin 0 jj
partsimaginary and real theSeparating
Ignoring the second term under the radical)
(recalled)(obtained by separating the real and imaginary parts)(subscript ‘lf’ referring to lower frequencies)
44
Hz1010GHz10 9f
hfhf0
0 2
jj
Propagation at higher frequencies
Let us recall the following expression at higher frequencies:
)(typically Hz1010GHz10 9f
water)(sea mho/m4
m/F10854.88181 120
110
Expanding binomially and ignoring higher order terms
(subscript ‘hf’ referring to quantities at higher frequencies)
jj
j )2
11(0
2/10 )1(
j
j
45
20
lf
) typicallyfrequency,(lower Hz1010kHz10 3f
Lower attenuation at lower frequencies than at higher frequencies
)2(2
2
2
22skin
00lf
m15
Lower frequencies are preferred to higher frequencies for sea-water communication for lower attenuation as well as for a reasonable antenna size typically of the order of half wavelength in sea water (as compared to the corresponding antenna size in free space).
hfhf0
0 2
jj
0hf
0hf 2
2
0lf
lf
0
0
hf
lf 22
2
2 ( referring to lower frequencies)
3
hf
lf 10
water)(sea mho/m4
m/F10854.88181 120
)2( f
46
Plasma oscillationRefractive index of ionosphere
Space-charge waves on an electron beam
Cyclotron waves on an electron beam
Wave propagation in a medium of charged particles
47
Consider an ensemble of electrons and positive ions maintaining overall charge neutrality
Consider the physical displacement (perturbation) of electrons (which are much more mobile than positive ions) from their equilibrium position to a small extent
Space-charge electric field in the direction of the displacement of negatively charged electrons providing a restoring force
Overshoot of electrons
Restoring force again coming into play
Oscillation of electrons about their mean position at the natural angular frequency― electron plasma frequency.
Displacement of electron layers by an amount and the resulting space-charge restoring force
Plasma oscillation
48
zs
zs
zs aaaE
000
)(2
)(2
E z
0
0
md
d te E z
2
2
d
d tE z
2
20
0
0
0
d
d tp
2
22
0s
0s
zs aE
0
0
0
p
Displacement of electron layers by an amount and the resulting space-charge restoring force
s = surface charge density
0 = volume charge density
= cross-sectional area perpendicular to displacement
(before perturbation) (after perturbation)
(electric field due to positive-end layer)
(force equation)
me /
(electric field due to negative-end layer)
49
d
d tp
2
22
)(0
0
p
)exp()exp( tjBtjA pp
tBAjtBAtjtBtjtA pppppp sin)(cos)()sin(cos)sin(cos
tDtC pp sincos
tD p sin
0at 0 t 0C
Set
Solving
(A and B are constants)
(C = A+B and D = j(A-B) are constants)
Solution indicating that the electrons oscillate about their mean position with an angular frequency of oscillation p called the plasma frequency (electron):
0
0
p
50
The ionosphere is located at the heights of 50-300 km from the earth.The ionosphere consists of electrons and ionsThe ionization in the ionosphere is caused by solar radiation (ultraviolet, soft X-ray, -particle (helium atoms from which the electrons are knocked off), etc.)The sky-wave propagation utilizes the phenomenon of total internal reflection from the ionosphere, which, in turn, is dependent on the value of its refractive index. We can find value of the refractive index of the ionosphere by studying the interaction of electromagnetic waves with the ionosphere.
Refractive index of ionosphere
The current density in a medium constituted by charged particles in a free- space medium consists of the convection current density (unlike the conduction current density in a conducting medium) and the displacement current density.
t
DJ
densitycurrent Convection
t
DvJ
The relation between the convection current density J, volume charge density and velocity v of charged particles (electrons) has been obtained earlier (in Chapter 3 while studying Child-Langmuir’s law) as:
.vJ
vJ
We can write the relation in vector form as
51
t
Ev
t
DvJ
000
Eedt
vdm
jt
ED
0
Perturbed electron current density in the ionosphere under the influence of an interactive electromagnetic wave
Perturbed convection current density obtainable from the relation J = v (electrons not having a dc beam velocity unlike in an electron beam)
Displacement current density
Time dependence as tjexp
EjvJ
00
)ionosphere theof electronsfor 0(
)importance oforder second of is which (ignoring
)(
)(
0
11
1
10
111
10
1110
1
110
111111
vz
vv
t
vz
vv
t
v
z
vv
z
vv
t
v
z
vvv
t
vz
vvv
t
v
z
vv
t
v
z
v
dt
dz
t
v
dt
dv
Eet
vm
Force equation:(current density equation)
Current density equation:
52
EjvJ
00 (current density equation) (recalled)
Eet
vm
(force density equation) (recalled)
j
Ev
jt
Eevmj
electron)an of ratio mass-to-(charge m
e
)/
1(]11
[2
00000
Ejjj
EjJ
(recalled)EjJ p
)1(2
2
0
0
0
0
0
p
EjJ
effective )1(2
2
0effective
p
2
2
effective, 1)(
prn
Refractive index n of the ionosphere
2
2
0effectiveeffective, 1/
pr
(refractive index of the ionosphere)
531101101 vvvJ
Current density equation
Space-charge waves on an electron beam
It is of interest to study the interaction of electromagnetic waves with a charge flow, for instance, a beam of electrons, which is a constituent of many practical electron devices such as vacuum electron devices like microwave tubes and charge accelerators. We are going to show that an electron beam supports two space-charge waves. Coupling of space-charge waves with electromagnetic waves is of relevance to understanding the behaviour of practical electron devices. Our study will be restricted here to finding the phase velocities of space-charge waves supported by an electron beam.
000
10
10
10
vJ
vvv
JJJ
vJ (convection current density)
Subscript 0 refers to unperturbed quantities
Subscript 1 refers to perturbed quantities
11011000101010 ))(( vvvvvvJJ
100 vJ
Let us consider a large cross-sectional area of the electron beam perpendicular to the beam flow along z over which the beam velocity v, volume charge density and current density J remain constant:
0/
0//
z
yxOne-dimensional beam flow
54
10101 vvJ
zv
z
v
z
J
1
01
01 011
tz
J
zv
z
v
t
1
01
01
z
v
zv
t
1
01
01
z
v
101D
zv
t
0D
)0(
t
J
(continuity equation)
1101101 vvvJ
Taking perturbed quantities much smaller than unperturbed quantities so that we can ignore the product of perturbed quantities 1v1
Taking partial derivative with respect to z
(to be recalled later in the analysis)
0/
0//
z
yx
55
ss EEm
e
dt
dv 1
z
vv
t
v
z
vv
z
vv
t
vz
vvv
t
v
z
vv
t
v
z
v
dt
dz
t
v
dt
dv
10
111
10
1
110
111111 )(
sEz
vv
t
v
1
01
sEv 1D
zv
t
0D
seEdt
dvm 1
Force equation Space-charge electric field was introduced in course of the deduction of electron plasma frequency of oscillation in a space-charge neutralised medium consisting of electrons and relatively immobile positive ions. In an ideal vacuum, it is not possible for electrons to move to form an electron beam because of mutual repulsive force between the advancing and the following electrons. However, in a practical vacuum electron beam device like a microwave tube where the vacuum is not ideal, the presence of charge neutralising positive ions makes it possible for the electrons to move and form an electron beam. The space-charge field Es comes into play for any perturbation in the position of electrons from their equilibrium position.
(ignoring higher order term, being the product of perturbed quantities: zvv /11
(to be recalled later in the analysis)
56
z
v
101D
sEv 1D
z
EE
zv
zz
v ss
00101
012 DDD
z
Es
012D
0
1
z
Es
12
10
01
0
0
0
101
2D
p
pjD
]D[ 0
zv
t
2
0
0p
0
0
p
22D p
(recalled)
Performing D operation zv
t
0D
(recalled)
(/z, partial derivative with respect to z, and D involving /z. partial derivative with respect to z, and /t, partial derivative with respect to t, being interchangeable)
(following from Maxwell’s equation: )
0
E
(for a uniform plane wave propagating along z: )
0// yx
57
)(exp ztj Perturbed quantities vary as
)(D 000 vjvjjz
vt
pv 0
(dispersion relation of space-charge waves)
pjD
jz
jt
)( 0vjj p
pjD
(recalled)
58
0vvp
p
The upper sign Fast space-charge wave
The lower sign Slow space-charge wave
pep
v
0
(dispersion relation of space-charge waves) (recalled)
constant)n propagatio (beam 0
ev
constant)n propagatio (plasma 0
pp
v
pv 0
00)( vvvp
p
00)( vvvp
p
(phase velocity of fast space-charge wave)
(phase velocity of slow space-charge wave)
59
Cyclotron waves on an electron beam
An electron beam supports cyclotron waves in the presence of a dc magnetic field in the direction of beam flow, provided there also exist the components of beam velocity transverse to the magnetic field. The interaction of electromagnetic waves with an electron beam supporting cyclotron waves forms the basis of understanding the behaviour of electron beam devices such as the gyrotron for the generation of electromagnetic waves in the millimetre-wave frequency range, which finds application in industrial heating, material processing, plasma heating for thermonuclear power generation as well as in domestic microwave ovens.
Let us consider the electron beam along z and a uniform dc magnetic field present in the region of beam flow, also along z. The components of Lorentz forces exist transverse to the longitudinal magnetic field. Let us treat the problem in rectangular system of coordinates for a large cross-sectional area of the beam perpendicular to the axis of the beam and magnetic field (that is perpendicular to z direction)
ss EEm
e
dt
dv 1
(force equation for an electron subject to space-charge electric field)
(recalled)subject to Lorentz force instead of force due to space-charge field Es
yyy
xxx
BvBvm
e
dt
dv
BvBvm
e
dt
dv
)()(
)()(
111
111
charge electronic theof
sign negative thecarrying /me
Bve
force Lorentz
60
BBc
yyy
xxx
BvBvm
e
dt
dv
BvBvm
e
dt
dv
)()(
)()(
111
111
(recalled)
yy
xx
Bvv
Bvv
)(D
)(D
11
11
zv
t
0D
Recalling the same approach as used to present the expressions in terms of the operator D
X
Z
Y
zzyyxx avavavv
1111
xv1
yv1
zv1 zzz aBaBB
zyx
zyx
zyx
BBB
vvv
aaa
Bv 1111
BBBB zyx ,0,0
xcxy
ycyx
vBvv
vBvv
111
111
D
D
(electron cyclotron frequency introduced in Chapter 4)
(magnetic field B is along z)
61
(Dispersion relation for cyclotron waves)pv 0
xcy
ycx
vv
vv
11
11
D
D
(recalled)
ycx vv 11D xcy vv 11D
Performing D operation
zv
t
0D
xcy vv 11D 22D c
(recalled)
xcxccycx vvvv 12
1112 )(DD
22D p
00 cv
Compare it with (recalled)
Led earlier to the derivation of the dispersion relation for space-charge waves
(Dispersion relation for space-charge waves)
Following the same approach we can write the dispersion relation of cyclotron waves simply by replacing p with c
62
(Dispersion relation for cyclotron waves)pv 0
00 cv (Dispersion relation for space-charge waves)
Led earlier to the derivation of the phase velocities of fast and slow space-charge waves
00)( vvvp
p
00)( vvvp
p
(phase velocity of fast space-charge wave)
(phase velocity of slow space-charge wave)
and
Following the same approach we can write the phase velocities of cyclotron waves simply by replacing p with c
00)( vvvc
p
00)( vvvc
p
(phase velocity of fast cyclotron wave)
(phase velocity of slow cyclotron wave)
and
63
Space-charge and cyclotron waves
000 vvvvpp
p
0vvp
p
Space-charge waves Cyclotron waves
Upper sign for the fast wave and lower sign for the slow wave
00 pv 00 cv
000 vvvvcc
p
0vvc
p
Bc 0
0
p
6464
Summarising NotesWave equationsone in electric field and the other in magnetic fieldhave been derived with the help of those two Maxwell’s equations in which these field quantities are coupled.
Solutions to the wave equations in electric and magnetic fields have been obtained for uniform, plane electromagnetic waves propagating through an unbounded free-space medium.
With the help of Maxwell’s equations and the solutions to wave equations in electric and magnetic fields, it has been established, with reference to uniform, plane electromagnetic waves propagating through an unbounded free-space medium, that
there exists no components of electric and magnetic fields in the direction of propagation;
directions of the electric field, magnetic field and wave propagation are mutually perpendicular to one another;
transverse electromagnetic (TEM) mode of propagation is supported by the unbounded medium;
intrinsic impedance and phase velocity of the wave are each related to the permeability and the permittivity of free space, the wave phase velocity being the speed of light c.
6565
Concepts of the skin depth, surface resistance and ac or RF resistance of a medium have been developed by studying the propagation of uniform, plane electromagnetic waves propagating through a semi-infinite conducting medium, considering such waves to be incident from a free-space region to a planar conducting surface extending to infinity.
Ratio of the ac or RF resistance to dc resistance of a conducting wire of circular cross section becomes equal to the ratio of wire radius to twice skin depth, considering the wire to be of high conductivity and/or wave frequencies to be very high such that the skin depth of the conductor becomes small enough compared to the wire radius to make the planar conductor approximation for the conducting wire of circular cross section.
Lower frequencies, say, ~10 kHz is preferred to higher frequencies, say, ~10 GHz, in view of comparatively lower attenuation of waves at such lower frequencies for sea-water communication as revealed by studying propagation through unbounded sea-water of finite conductivity and permittivity.
Study of wave propagation through a medium of charged particles give the concepts of sky-wave propagations through ionosphere as well as those of space-charge waves and cyclotron waves on an electron beam.
Readers are encouraged to go through Chapter 6 of the book for more topics and more worked-out examples and review questions.
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