Uniform Plane Wave Solution to Maxwell’s Equations
Brian K. Hornbuckle
January 27, 2016
1
Maxwell’s Equations
Maxwell’s equations and the constitutive relations describe the behavior of electromagnetic fields.
∇×E=−
∂B
∂t
∇×H=J+
∂D
∂t
(1)
(2)
∇ · D = ρv
(3)
∇·B=0
(4)
D = E
(5)
B = µH
(6)
J = σE
(7)
Here: E is the electric field (V m−1 ); B is the magnetic flux density (T); H is the magnetic
field intensity (A m−1 ); D is the electric flux density (C m−2 ); ρv is the density of “free” electric
charges, or charges that are not initially part of a medium (C m−3 ); is the permittivity of a
medium (F m−1 ); µ is the permeability of a medium (H m−1 ); and σ is the electrical conductivity
of a medium (S m−1 ). Boldface sympbols (like E) are vectors and all of these variables are functions
that, in general, vary in time and space, i.e. f (t, x, y, z).
(1-4) are what we refer to as “Maxwell’s equations” but the constitutive relations (5-7) are
also needed to complete the picture of the general nature of electric and magnetic fields and how
electric and magnetic fields are coupled, that is how electric and magnetic fields are dependent
on each other. (1) is commonly called Faraday’s law which states that a time–varying magnetic
field creates a circulating electric field. (2) is called the Maxwell–Ampere law and tells us that a
circulating magnetic field (6) is created by both a current of electric charges (7) and by a time–
varying electric field (5). Gauss’s law states that electric fields eminate from electric charges in
straight lines (3), while magnetic fields do not originate at points (there are no magnetic charges)
but can be visualized as forming loops that have no beginning or end (4).
From this short discussion you can get a sense of why we say electric and magnetic fields are
coupled and why we can use the term “electromagnetics.” Faraday’s law (1) and the Maxwell–
Ampere law (2) describe how time–varying electric fields produce time–varying magnetic fields,
which produce time–varying electric fields, which produce... In a nutshell, this is why electromagnetic radiation is able to propagate, even in “free space” or a vacuum. Once there is a change in
one quantity (e.g., an electric field) a change in the other quantity (the magnetic field) occurs, and
these changes continue to “propagate” in time.
1
2
Real and Complex Fields
Joseph Fourier determined that all signals can be represented by a linear combination of sinusoids
(sines and cosines). Let’s take advantage of this and think of any electric or magnetic field in terms
of its sinusoidal components. When we use sinusoids to represent electric and magnetic fields it is
mathematically convient to represent “real” fields that we would physically observe and measure
with “complex” or “time–harmonic” fields. For example, an electric field can be written
n
o
(8)
E(t, x, y, z) = < Ẽ(x, y, z) ejωt
√
where < {} takes the real part of the argument, j = −1 is the imaginary number, ω = 2πf is
the radial frequency of the sinusoid, and f is what is commonly referred to as the frequency of
the sinusoid. What we will do now is use Ẽ(x, y, z) to represent an electric field in mathematical
equations (and assume an ejωt time dependence) instead of using E(t, x, y, z),.
Why use the transformation in (8)? When it comes down to it, complex numbers like ejωt are
really just a tool to make the math easier. It is always possible to describe the real world using
only real numbers. However, we can sometimes make it easier for ourselves if we use both real and
imaginary numbers. That is the case here with this complex notation. In (8) the time variable t
has been removed and we are left with a complex field Ẽ that only varies in space. This notation
will also allow us to use Euler’s formula ejθ = cos θ + j sin θ to represent sinsoids and exploit the
unique mathematical properties of the exponential function. When we need the real electric field
again, then we’ll use (8) to find it.
The˜(tilde) symbol is used to distinguish real fields from complex fields. Where should the tilde
be placed? On what variables? When? Textbooks are not consistent on this issue, so there is no
correct answer. Many texts drop the tilde altogether. I use the tilde when I need to remind myself
that the fields are complex (time–harmonic fields), or when teaching others the concept of complex
fields for the first time. I was frustrated when I first learned these concepts when an author or
teacher would drop or forget the tilde. I will try to be consistent with my use of the tilde in the
class notes, but undoubtedly there will be times when I either drop it or put it in a place that you
think is not appropriate. If you get frustrated with my personal use of the tilde, this is probably
a good sign since this means you are thinking carefully about time–harmonic fields and becoming
familiar with the subtleties.
3
Maxwell’s Equations in Time–harmonic Form
Time derivatives are found in (1) and (2). Another advantage of the complex or time–harmonic
form is that time derivatives can be reduced to just algebra. For example,
o
n
o
∂E
∂ n
= < Ẽ(x, y, z) ejωt = jω < Ẽ(x, y, z) ejωt
∂t
∂t
(9)
such that ∂/∂t in (1) and (2) can be replaced with jω and they can be rewritten using (5), (6),
and (7) to become
∇ × Ẽ = −jωµ H̃
(10)
and
∇ × H̃ = σ Ẽ + jω Ẽ.
(11)
Look closely at (10) and (11). It would be nice if these two equations were perfectly symmetric,
i.e., if both were in the form of a curl equaling one quantity. To do this, note that the right–
hand side of (11) can be written (σ + jω) Ẽ = jω c Ẽ where we have used a “new” or complex
2
permittivity defined as
σ
(12)
ω
to combine the two terms. Unfortunately virtually all textbooks do not use the c notation to
distinguish the “old” and “new” permittivities, with the new permittivity including the effect of
the electrical conductivity. So to get you into the habit of thinking about this, I will not use the c
notation.
Then (1)-(4) become
∇ × Ẽ = −jωµH̃
(13)
c = − j
∇ × H̃ = jωẼ
(14)
∇ · Ẽ = 0
(15)
∇ · H̃ = 0
(16)
where (5) and (6) have been used to eliminate D̃ and B̃ and it is assumed that ρ˜v = 0. It is
imperative to remember that here represents the “new” complex permittivity that incorporates
the effect of the electrical conductivity. Now we have symmetric equations!
4
Helmholtz Equations
Maxwell’s equations (13)-(16) can be reduced to what are called the Helmholtz equations using
vector identities. It turns out that a propagating sinusoidal wave is a solution to the Helmhotz
equations which is consistent with our previous understanding of the behavior of electromagnetic
radiation and how it propagates as a wave! And this also justifies are desire to represent fields
as sinusoids using (8)! The Helmholtz equations for the time–harmonic forms of the electric and
magnetic fields Ẽ and H̃ assuming an ejωt time dependence are
∇2 Ẽ + k 2 Ẽ = 0
(17)
∇2 H̃ + k 2 H̃ = 0
(18)
and
√
where: k = ω µ is the wave number or propagation constant (rad m−1 ); ω = 2πf is the radial
frequency (rad s−1 ); f is the frequency (s−1 or Hz); µ is the permeability of the material through
which the wave is propagating (H m−1 ); and is the “new” or permittivity of the material which
is complex in general (F m−1 ).
5
Uniform Plane Wave
An electric field of the form Ẽ(z) = x̂Ẽx solves (17) if
+ −jkz
Ẽx = Ẽxo
e
.
(19)
This is a uniform plane wave propagating in the +ẑ direction. Uniform plane waves have uniform
(constant) properties in a plane perpendicular to their direction of propagation. For the uniform
plane wave described by (19) the plane of uniformity is the x–y plane. Exo is complex in general so
it has a magnitude |Exo | and a phase ejφ . Since the electric and magnetic fields are coupled, there
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must be a magnetic field H̃(z) associated with this electric field. The magnetic field can be found
using (13). The magnetic field has only a ŷ component and has the form:
H̃y =
where η =
5.1
p
E˜xo −jkz
e
η
(20)
µ/ is the intrinsic impedance of the material (Ω). Note that (20) also solves (18).
Lossless Medium
The real electric field is found by converting the time harmonic form of the electric field in (19)
using (8). The real electric field has the form
E(t, z) = x̂ |Exo | cos (ωt − kz + φ) .
(21)
This is a wave equation with amplitude |Exo |, radial frequency ω = 2πf (rad s−1 ), frequency f (s−1
or Hz), phase constant or wave number k = 2π/λ, wavelength λ (m), and reference phase φ (rad)
propagating in the +ẑ direction. Note that the magnitude of Exo is the amplitude of the real field
E(t, z) and the phase of Exo is the spatial portion of the phase of E(t, z).
By taking the time derivative of the argument of the cosine function in (21) it can be shown
that the phase velocity up is
dz
ω
1
up =
= =√ .
(22)
dt
k
µ
The permeability µ and permittivity can also be written µ = µr µo and = r o , respectively,
where µr and r are the relative permeability and permittivity of the material, and µo and o
are the permeability and permittivity of free space. Virtually all materials relevant to microwave
remote sensing have relative permeabilities of unity. We will always assume that µr = 1 and that
µ = µo . The relative permittivities of materials relevant to microwave remote sensing can vary in
magnitude from 1 to 80 and is complex in general. “Free space” is a term used for a vacuum. Air
has a relative permittivity of approximately unity, so it can be approximated as free space. The
phase velocity in free space, c, is found by substituting into (22).
c=
ω
1
=√
ko
µo o
(23)
Here ko is the free space wave number. Evaluation of (23) gives c = 2.9979 × 108 m s−1 .
Using (22), (23), and other definitions (e.g. up = λf ), the following relationships can be written:
ko =
ω
2π
=
c
λo
λo
λ= √
r
c
up = √
r
√
ω
ω
k=
= √ = ko r
up
c/ r
where λo = c/f is the free space wavelength.
4
(24)
(25)
(26)
(27)
The general relationship between the electric and magnetic fields is given as follows:
1
k̂ × Ẽ
η
(28)
Ẽ = −η k̂ × H̃
(29)
H̃ =
where k̂ denotes the direction of propagation.
5.2
Lossy Medium
When the medium through which an electromagnetic wave is propagating is lossy, the magnitude
of the wave is attenuated. Lossy mediums have wave numbers k that are complex: k = k 0 − jk 00 ,
where k 0 = < {k} is the real part of the wavenumber, and k 00 = = {k} is the imaginary part. To see
the effect of a complex wavenumber, consider the real time–dependent field of the uniform plane
wave in (19).
n
o
0
00
E(t, z) = < x̂ Exo e−j(k −jk )z ejωt
(30)
The result is:
E(t, z) = x̂ |Exo | e−k
00
z
cos ωt − k 0 z + φ
(31)
The imaginary part of the wavenumber k 00 attenuates the wave. The real part of the wavenumber k 0
acts just as before as the phase constant and determines the wavelength of the wave as it propagates
in the medium, as well as the propagation velocity up .
Since we assume that µ = µo (µr = 1) in microwave remote sensing, then the relative permittivity r of a lossy medium must also be complex: r = 0r − j00r . Note that the relative permittivity
is also sometimes called the dielectric constant. When µr = 1,
√
k 0 = ko |< { r } |
√
k 00 = ko |= { r } |.
(32)
(33)
√
When evaluating r care must be taken since r is a complex number. It is more convenient, and
√
more physical, to define the index of refraction n = r = n0 − jn00 . Using this definition,
k 0 = ko |< {n} | = ko n0
(34)
00
(35)
k = ko |= {n} | = ko n
00
The refractive index and relative permittivity are related as follows:
0r = (n0 )2 − (n00 )2
(36)
00r = 2n0 n00
(37)
n0 =
n00 =
sp
(0r )2 + (00r )2 + 0r
2
sp
(0r )2 + (00r )2 − 0r
2
(38)
(39)
The advantage of using the index of refraction to characterize a medium instead of the relative
permittivity (dielectric constant) is shown by (36) through (39). When using the relative permittivity, the phase constant k 0 is a function of both the real and imaginary parts of r . Furthermore,
5
the attenuation k 00 is also a function of both the real and imaginary parts of r . If we use the
refractive index, then the phase constant k 0 is only a function of n0 , and the attenuation k 00 is only
a function of n00 . Hence the real part of the refractive index determines the wavelength of the wave
in the medium (λ = λo /n0 ) and the phase velocity (up = c/n0 ), and the imaginary part of the
refractive index determines the attenuation (k 00 = ko n00 ).
The magnitude of the electric field in a lossy medium can be found from (31).
|Ẽ| = |Exo | e−k
00
z
(40)
When |Ẽ| is e−1 = 37% of its original magnitude |Exo |, then z = δs = 1/k 00 , where δs is called
the skin depth of the material. The skin depth is a measure of how far a wave can propagate into
a medium. Waves are attenuated quickly in a medium that has a small skin depth. When the
medium is lossless (k 00 = 0) then the skin depth is infinite and the wave propagates indefinitely.
Metals have high electrical conductivities, and hence 00r is large, which generally leads to a large
k 00 . Metals therefor have very small skin depths.
6
Propagation of Power
The Poynting vector S is defined as
S=E×H
(41)
and has units of W m−2 . It represents the instantaneous power per unit area carried by an
electromagnetic wave. The direction of the Poynting vector is k̂, the direction of propagation.
The time–average power density is
o
1 n
< Ẽ × H̃∗
(42)
2
where ∗ denotes the complex conjugate. The time–average power density is the quantity that
is measured by a microwave radiometer. In passive remote sensing, this time–average power is
used to infer specific properties of the source of the electromagnetic radiation. In reality, only a
specific component of the time–average power is measured as determined by the polarization of the
radiometer’s antenna.
For a wave propagating in ẑ in a lossless medium (= {n} = 0),
Sav =
Sav (z) = ẑ
|Ẽ|2
|E˜xo |2 + |E˜yo |2
= ẑ
2η
2η
(43)
For a wave propagating in ẑ in a lossy medium (= {n} =
6 0),
Sav (z) = ẑ
|Ẽ(z = 0)|2
00
cos (θη ) e−2 k z
2|η|
(44)
where η = |η|ejθη .
7
Power Attenuation
As electromagnetic radiation travels through a lossy medium, it is attenuated according to the
electrical properties of the medium, specifically the imaginary part of the refractive index, n. For
a uniform plane wave propagating in +ẑ,
Sav (z) =
o
1 n
|Ẽ(z = 0)|2
00
< Ẽ × H̃∗ =
cos (θη ) e−2 k z
2
2|η|
6
(45)
where again k 00 = ko n00 . At any point z = z 0 , the ratio of the time–average power at z = z 0 , Sav (z 0 ),
to the time–average power at z = 0, Sav (0), is:
Sav (z 0 )
=
Sav (0)
|Ẽ(z=0)|2
2|η|
cos (θη ) e−2 k
|Ẽ(z=0)|2
2|η|
00
z0
= e−αz
0
(46)
cos (θη ) e0
where α = 2 k 00 is the power attenuation constant of the medium.
In general, any power ratio like the one in (46) can be expressed in units of either nepers (Np)
or decibels (dB). Consider a power ratio P2 /P1 where P2 < P1 (the power has been attenuated).
To find the attenuation in nepers, simply take the absolute value of the natural logarithm of the
power ratio.
P2 N = ln (47)
P1
N has units of Np and the two powers are related through
P2 = P1 e−N
(48)
(Note that since P2 < P1 , ln (P2 /P1 ) < 0.) To find the attenuation in decibels, take the base–ten
logarithm of the power ratio, multiply by ten, and take the absolute value.
P2 N = 10 log P
(49)
1
Here N has units of dB. To convert between nepers and decibels, think of what an attenuation of
P2 /P1 = e−1 ≈ 37% (when α = 1 Np m−1 and z = 1 m) would be in decibels.
P2 P2 N = ln = ln(e−1 ) = 1 Np = 10 log = 10 log(e−1 ) ≈ 4.343 dB
P
P
1
(50)
1
The power attenuation constant, α, for a specific medium can be found in references with
units of either Np m−1 or dB m−1 . To compute the actual attenuation of power using (46), the
units of dB m−1 must be converted to Np m−1 using (50). Carefully observe how α has been
defined. If the field attenuation constant, k 00 , is given with units of dB m−1 , then a conversion of
1 Np m−1 = 8.686 dB m−1 instead of (50) must be used to find k 00 in nepers per meter since
10 log P2 = 10 log(e−2×1 ) = 20 log(e−1 ) ≈ 8.686 dB.
P (51)
1
References
Ulaby, F. T., Fundamentals of Applied Electromagnetics, Prentice Hall, Upper Saddle River, NJ,
1997.
Ulaby, F. T., R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive, vol. 1,
Artech House, Norwood, MA, 1981.
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