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Plane Wave Propagation in Lossless Media
(Intensive Reading)
1 Maxwell's equations
James Clark Maxwell (1831-1879) postulated a set of laws which define the relationship
between the electric field E, the magnetic field H and the sources of these fields: ρ (volume
density of free charges) and currents, J (surface density of free currents). These laws can be
written in terms of the four equations known as Maxwell's equations. Maxwell's equations in
differential form are shown in the 1st column of Table 1. These equations can be used to
explain and predict all macroscopic electromagnetic phenomena and form the foundation of
electromagnetic theory. Vectors E(r , t ) = E( x, y, z , t ) and H (r , t ) = H ( x, y , z , t ) are the
instantaneous electric and magnetic field intensities, which are functions of position
r = ( x, y, z ) and also an arbitrary function of time t. ∇ × E is the curl of the electric field, and
∂
∂
∂
∇ ≡ xˆ +
yˆ + zˆ . The term ε = ε r ε 0 is known as the permittivity of the medium, while
∂x
∂y
∂z
μ = μ r μ 0 is the permeability. ε r and μ r are the relative permittivity and relative
permeability respectively, and are properties of the specific medium in which the
electromagnetic fields exist. The terms ε 0 = 8.854 × 10 −12 and μ 0 = 4π × 10 −7 are the freespace permittivity and free-space permeability.
General time-varying form
∂ B(r , t )
∂t
∂ H (r, t )
= −μ
∂t
∂ D(r , t )
∇ × H (r , t ) = J (r , t ) +
∂t
∂ E(r , t )
= J (r , t ) + ε
∂t
Electrostatic case
Magnetostatic case
∇ × E(r, t ) = −
∇ ⋅ D(r , t ) = ∇ ⋅ εE(r, t ) = ρ(r, t )
∇ × E(r ) = 0
∇ × H (r ) = J (r )
∇ ⋅ D(r ) = ∇ ⋅ εE(r ) = ρ(r )
∇ ⋅ B(r , t ) = ∇ ⋅ μH(r, t ) = 0
∇ ⋅ B(r ) = ∇ ⋅ μH (r ) = 0
Table 1 Maxwell's equations in differential form and fundamental postulates of electrostatics
and magnetostatics.
A stationary (steady, static) charge or current does not vary with time, i.e. ρ or J is not a
function of time. Note the following:
• A stationary charge gives rise to a stationary electric field, but no magnetic field. The
study of these phenomena is called electrostatics. It is therefore possible for a
stationary E to exist without the presence of H in the same space (caused by a steady
ρ, with J = 0).
1
•
Stationary currents cause stationary magnetic fields, but no electric fields
(magnetostatics). It is therefore also possible for a steady H to exist alone without an
E (caused by a steady J, with ρ = 0).
The fundamental postulates for electrostatics and magnetostatics are shown in the 2nd and 3rd
columns of Table 1. It is obvious that Maxwell's equations simplify to the fundamental
relations for electrostatic and magnetostatic models in non-time varying cases by setting all
∂( )
terms equal to zero.
∂t
Also note the following for dynamic case (i.e. time varying fields):
• If the fields are changing with time, it is impossible for E or H to exist separately.
• A changing E produces an H; a changing H produces an E. We refer to these as
electromagnetic (EM) fields.
• A changing H induces a changing E, not only in the region of the change, but also in
the surrounding region. Changing fields in surrounding regions induce fields in an
even more distant region, so that energy can be propagated outwards. More about this
in Chapters 1 and 2.
• EM waves require no material to support them, i.e. they can exist in the atmosphere,
empty space, etc.
• EM waves can also be guided by guiding systems, e.g. co-axial cables and other types
of transmission lines. More about this in Chapter 4.
In a homogeneous, source-free region, J = 0 and ρ = 0 , while ε and μ are constant
everywhere in the medium. Maxwell's equations then reduce to the relations shown in Table
2.
∇ × E(r, t ) = −μ
∂ H (r , t )
∂t
∇ × H (r , t ) = ε
∂ E(r, t )
∂t
∇ ⋅ E (r , t ) = 0
∇ ⋅ H (r , t ) = 0
Table 2 Maxwell's equations in a homogeneous, source-free region.
2 Time-harmonic fields
If we assume that vector (or scalar) fields have a sinusoidal time dependence (also called time
harmonic fields), we may use phasor notation. Using a cosine reference, the phasor quantities
are defined by
A (r , t ) = Re[ A(r ) e jωt ]
(1)
We then have the relations between the instantaneous (time-dependent) and phasor quantities
as shown in Table 3.
2
Instantaneous
Phasor
A(r , t ) = aˆ A cos[ωt + kf ( x, y, z ) + φ]
A(r, t ) = aˆ A sin[ωt + kf ( x, y, z ) + φ]
A(r ) = aˆ A e jφ e j k f ( x , y , z )
A(r ) = aˆ (− j ) A e jφ e j k f ( x , y , z )
= aˆ A e j (φ−π / 2) e j k f ( x , y , z )
Table 3 General relations between instantaneous field expressions and their phasors forms.
Note that phasors are not physical quantities – they are mathematical expressions that
simplify the mathematics. Instead of using the expression for the time-varying vectors, we
may use the phasor forms to do the calculations, and whenever we need the physical
instantaneous quantities, we simply use (1) to obtain them.
Let
E(r, t ) = Re[E(r ) e jωt ]
(2)
H (r, t ) = Re[H (r ) e jωt ] .
E(r ) and H(r ) now represent the phasors of the time-harmonic fields E(r , t ) and H (r , t ) . If
we substitute (2) into the expressions provided in column 1 of Table 1 or Table 2 and
simplify, we obtain Maxwell’s equations for phasors in medium with permittivity ε = ε r ε 0
and permeability μ = μ r μ 0 as shown in Table 3.
Time-harmonic form
(with sources)
Time-harmonic form
(homogeneous, source-free)
∇ × E(r ) = − j ωμ H (r )
∇ × E(r ) = − j ωμ H (r )
(i)
∇ × H (r ) = J (r ) + j ωε E(r )
∇ × H (r ) = j ωε E(r )
(ii)
∇ ⋅ D(r ) = ρ(r )
∇ ⋅ E(r ) = 0
(iii)
∇ ⋅ B(r ) = 0
∇ ⋅ H (r ) = 0
(iv)
Table 3 Maxwell's equations in terms of phasors.
3 The wave equation
Take the curl of (i), use (iii) and substitute (ii):
LHS : ∇ × ∇ × E = ∇(∇iE) − ∇ 2 E = 0 − ∇ 2 E
RHS : − j ωμ(∇ × Η ) = − j ωμ( j ωεE) = ω μεE
2
(3)
(4)
Equating the (3) and (4) gives the homogeneous wave equation:
∇ 2 E(r ) + k 2 E(r ) = 0 ,
3
(5)
where k = ω με is the wavenumber which is a function of frequency and the material
properties ε = ε r ε0 and μ = μ r μ0 . Equation (5) is known as the homogeneous vector
Helmholtz equation. Consider the simple special case of spatial variation in one dimension
only (z direction) and with the electric field polarised in the x direction, i.e. E(r ) = xˆ Ex ( z ) .
The wave equation reduces to
d 2 Ex ( z )
+ k 2 Ex ( z ) = 0
(6)
2
dz
The solution of the differential equation in (6) is
Ex ( z ) = E0+ e − j k z + E0− e+ j k z = E x+ ( z ) + Ex− ( z )
(7)
The time-dependent solutions are then obtained from
Ex ( z , t ) = Re[ E x ( z ) e jωt ]
= Re[( E0+ e − j k z + E0− e + j k z ) e jωt ]
= E0+
cos(ωt − kz ) +
E0−
(8)
cos(ωt + kz )
Consider the first term on the right hand side of (8). It has been plotted for several values of t
in Figure 1. At successive times, the curve travels in the positive z direction. If we fix our
attention to a particular point on the curve, we set cos(ωt − kz ) = C1 , or (ωt − kz ) = C 2 . The
position of this point is therefore given by
z = (ωt − C2 ) / k
(9)
and its velocity by
dz ω
up =
= .
(10)
dt k
Figure 1 Wave travelling in positive z direction E x+ ( z ) = E x0 cos(ωt − kz ) for several values of
t.
4
The term in (10) is called the phase velocity. The wave is therefore travelling at a velocity of
u p in the +z direction, and we call it a travelling wave.
The second term in (8) represents a wave travelling in the negative z direction with the same
velocity. The magnetic field is related to the electric field by
E + ( z)
H y+ ( z ) = x
η
(11)
Ex− ( z )
−
H y ( z) = −
η
where η = μ / ε is the intrinsic impedance of the medium measured in Ω. Since the magnetic
field is completely specified by the electric field and the media permittivity and permeability,
it is redundant; and therefore, it is customary to perform our analysis in terms of only the
electric field. Note that the magnetic field is perpendicular to the electric field, and that both
the electric and magnetic field vectors are perpendicular to the direction of propagation (see
Figure 2). The distance between two successive E-field or H-field maxima is called the
wavelength λ, and is given by λ = 2π / k = u p / f . At any instant of time, the surface on which
the phase of the electromagnetic field is a constant is defined by (ωt − kz ) = constant , which
in space corresponds to the equation of a plane defined by z = constant. i.e., a plane parallel to
the xy-plane. Waves of this type are known as "Uniform Plane Waves" or just "Plane Waves".
Figure 2 Graphical representation of a plane wave.
If the medium in which the wave is propagating is free space, we have
μ = μ 0 = 4π × 10−7
ε = ε 0 = 8.854 × 10−12
2π c
λ = λ0 =
=
k0
f
k = k0 = ω μ 0 ε 0
up =
1
μ0 ε0
= c ≈ 3 × 108
5
η = η0 =
μ0
= 120π
ε0
(12)
4 General description for plane waves in lossless media
Consider a plane wave propagating in the direction â n , as shown in Figure 3.
Figure 3: Plane wave propagating in direction â n .
Assume that the electric and magnetic fields polarised in the ê and ĥ directions respectively.
Note that ê , ĥ and â n must be mutually perpendicular, and they follow the right-hand rule.
Let the position vector r = ( x, y , z ) be the point where the field is evaluated. For a lossless
medium (like free space), the wavenumber is equal to the phase constant, i.e. k = β . If the
electric field has an amplitude of E 0 , we may define the following relations:
Phasor expression for the electric field
ˆ
E(r ) = eˆ E0 e − jk an ⋅r
(13)
= eˆ E0 e jφ e − jβ an ⋅r
ˆ
where E 0 = E 0 e j φ . The term ( aˆ n ⋅ r ) = OP in Figure 3. The magnetic field associated with
the electric field given in equation (13) is again given by
1
H (r ) = aˆ n × E
η
E
ˆ
(14)
= (aˆ n × eˆ ) 0 e − jk an ⋅r
η
E0
e jφ e − jβ an ⋅r
η
The instantaneous electric and magnetic fields are then given by
E(r, t ) = Re ⎡⎣E(r ) e jωt ⎤⎦ = eˆ E0 cos[ωt − β(aˆ n ⋅ r ) + φ]
and
E0
H (r , t ) = Re ⎡⎣ H (r ) e jωt ⎤⎦ = (aˆ n × eˆ )
cos[ωt − β(aˆ n ⋅ r ) + φ] .
η
= (aˆ n × eˆ )
6
ˆ
(15)
(16)
Alternatively, if the magnetic field phasor
ˆ
ˆ
H (r ) = hˆ H 0 e − j k (an ⋅r ) = hˆ H 0 e jψ e − j β (an ⋅r )
is known, the electric field may be calculated from
E(r ) = −η aˆ n × H
ˆ
= (hˆ × aˆ ) η H e jψ e − j β (an ⋅r ) .
(17)
(18)
0
n
We may also summarise the calculation of wave parameters:
The phase constant is obtained from
β = k = ω με
where ε = ε r ε 0 and μ = μ r μ 0 . The wavelength is calculated from
2π u p
λ=
=
f
β
while the phase velocity is given by
1
ω
up =
= .
με β
The intrinsic impedance1 of the medium is
η= μ/ε .
(19)
(20)
(21)
(22)
Example 1: A uniform plane wave with E = xˆ E x propagates in the +z-direction in a lossless
medium with ε r = 4 and μ r = 1 . Assume that E x is sinusoidal with a frequency of 100 MHz
and that it has a maximum value of 10-4 V/m at t = 0 and z = 18 m.
(a) Calculate the wavelength and the phase velocity, and find expressions for the
instantaneous electric and magnetic field intensities.
(b) Determine the positions where E x is a positive maximum at the time instant t = 10 −8 s.
Solution:
(a) Given:
aˆ n = zˆ
(aˆ n ⋅ r ) = z
eˆ = xˆ
E0 = 10−4
V/m
Therefore:
β = k = ω με = ω μ0 ε0 μ r ε r =
1
2π × 108
ω
μr εr ≈
c
3 × 108
4 = 4π / 3 rad/m
Intrinsic impedance:
For non-magnetic dielectric materials μ r = 1 , so that η = μ / ε = μ 0 / ε0 ε r = η0 / ε r . Free
space or vacuum thus has the highest intrinsic impedance. Note that the intrinsic impedance of a
medium is not in indication of opposition to flow of power; it is merely the ratio between the amplitude
of the electric and magnetic fields. Any medium which is lossless will not inhibit the flow of power
through it, i.e. there is no loss of power as a wave passes through a lossless medium. However,
depending on the intrinsic impedance, two waves carrying the same amount of power will have
different electric-magnetic field ratios.
7
λ = 2π / β = 1.5 m
η = μ / ε = 60π Ω
E(r ) = eˆ E0 e jφ e − jk a n ⋅r = xˆ 10 −4 e jφ e − j (4 π / 3) z
ˆ
4π
⎛
⎞
E(r, t ) = Re ⎣⎡E(r ) e jωt ⎤⎦ = xˆ 10−4 cos ⎜ 2π × 108 t −
z + φ ⎟ V/m
3
⎝
⎠
The cosine function has a maximum when its argument equals zero. Thus with t = 0 and
z = 18 m, we find that
2π × 10 8 (0) −
4π 1
+φ=0
3 8
Thus φ = π / 6 rad
4π
π⎞
⎛
E( z , t ) = xˆ 10−4 cos ⎜ 2π × 108 t −
z + ⎟ V/m
3
6⎠
⎝
H( z) =
1
aˆ n × E = yˆ 5.305 × 10−7 e jπ / 6 e − j (4 π / 3) z
η
A/m
4π
π⎞
⎛
H ( z , t ) = Re ⎡⎣ H ( z ) e jωt ⎤⎦ = yˆ 5.305 × 10−7 cos ⎜ 2π × 108 t −
z + ⎟ A/m
3
6⎠
⎝
See Figure 4 for a graphical representation of the instantaneous fields at t = 0.
Figure 4: E and H fields of the plane wave at the time instant t = 0. (Example 1)
8
(b) The cosine function has positive maxima when its argument is equal to
± 2nπ , n = 0,1,2, .
Thus
4π
π
2π × 10 8 (10 −8 ) −
z M + = ±2nπ
3
6
⇒ z M = 1.625 ∓ 1.5n = 1.625 ∓ 1.5λ ( m )
Example 2
A uniform sinusoidal plane wave with the following expression for the instantaneous
magnetic field propagates in air:
1 ⎞
⎛ 1
H ( x, z , t ) = ⎜ −
xˆ +
zˆ ⎟ cos(ωt − 6 x − 8 z ) A/m .
20π ⎠
⎝ 15π
Calculate β, λ and ω, and find an expression for the instantaneous electric field intensity.
Solution:
1 ⎞ − j (6 x +8 z )
⎛ 1
H ( x, z ) = ⎜ −
xˆ +
zˆ e
20π ⎟⎠
⎝ 15π
A/m
ˆ
H (r ) = hˆ H 0 e − j β (a n ⋅r )
2
2
1 1
1
1
⎛ 1 ⎞ ⎛ 1 ⎞
H0 = ⎜ −
+ 2 =
⎟ +⎜
⎟ =
2
π 15
12π
20
⎝ 15π ⎠ ⎝ 20π ⎠
1 ⎞ 1
⎛ 1
hˆ = ⎜ −
xˆ +
zˆ /
= (−0.8, 0, 0.6)
20π ⎟⎠ 12π
⎝ 15π
β(aˆ n ⋅ r ) = 6 x + 8 z
Let aˆ n = (a x , a y , a z ) . Then β(aˆ n ⋅ r ) = β a x x + β a y y + β a z z = 6 x + 8 z
⇒ β ax = 6
β ay = 0
β az = 8
⇒ ax = 6 / β
ay = 0
az = 8 / β
But
ax + a y + az = 1
2
2
2
⇒ (6 / β) 2 + (8 / β) 2 = 1
⇒ β = 10 rad/m
aˆ n = (0.6, 0, 0.8)
λ = 2π / β = 0.2π m
η = η0 = 120π Ω
ω = β u p = β c = 3 × 109
rad/s
9
E(r ) = −η aˆ n × H
E( x , z ) = −
120π
[(0.6, 0, 0.8) × (−0.8, 0, 0.6)] e − j ( 6 x +8 z ) = yˆ 10 e − j ( 6 x +8 z )
12π
E( x, z , t ) = Re ⎡⎣E( x, z ) e jωt ⎤⎦ = yˆ 10 cos(3 × 109 t − 6 x − 8 z )
10