WAVES
`
` David H. Staelin
`
`Ann W. Morgenthaler
`
`«
`
`«
`
`Jin Au Kong Momentum Dynamics Corporation
`
`Exhibit 1017
`Page 001
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 001
`
`

`

`
`
`ELECTROMAGNETIC WAVES
`
`
`
`David H. Staelin
`
`Ann W. Morgenthaler
`
`Jin Au Kong
`
`Massachusetts Institute of
`
`Technology
`
`An Alan R. Apt Book
`
`PRENTICE HALL, Upper Saddle River, New Jersey 07458
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 002
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 002
`
`

`

`Library of Congress Cataloging-in-Publication Daw
`Staelin, David H
`Electromagnetic waves / David H. Staelin, Ann W. Morgenithaler, Jin
`Au Kong.
`<m.
`p.
`Includes index.
`ISBN 0-13-225871-4
`1, Electromagnetc waves
`2. Electrodynamics.
`Ann W. O. Kong,Jin Au. OF Title.
`1994
`QOCb6L.ST4
`439,2—de20
`
`I, Morgenthaler,
`
`93-15705
`CrP
`
`Publisher; Alan Apt
`Production Editor: Bayani Mendoza de Leon
`Cover designer: Bruce Kenselaar
`Cover concept: Katharine E. Staelin
`Copy editor: Shirley Michaels
`Prepress buyer: Linda Behrens
`Manufacturing buyer: Dave Dickey
`Supplements editor, Alice Dworkin
`Editorial assistant: Shirley McGuire
`
`©1998 by Prentice-Hall, Inc.
`A Pearson Education Company
`Saddle River, NJ 07458
`Upper
`
`The author and publisher of this book have used their best efforts in preparing this book. These efforts
`include the development, research, and lesting of the theones and formulas to determine their effecveness.
`The author and publisher shall not be liable in any event for incidental or consequential damages in
`connection with, or arising out of, the furnishing, performance, or use of these formulas.
`
`All tights reserved. No part of this book may be
`reproduced, in any form or
`by any means,
`without permission in wnting from the publisher.
`
`=~
`
`ISBN O-13-2258?1-4
`
`Prentice-Hall [international (UK) Limited,London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada Inc., Toronto
`Prentice-Hall Hispanoamericana, S.A., Mexico
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of
`Japan, Lnc., Tokyo
`Pearson Education Asia Pte. Ltd., Singapore
`Editora PrenticeHall do Brasil, Lida., Rio de Janeiro
`
`ISBN 0-13-2258
`
`—$$$—$—_—__
`IAN
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 003
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 003
`
`

`

`
`
`2
`
`RADIATION BY CURRENTS
`AND CHARGESIN
`FREE SPACE
`
`
`2.1 STATIC SOLUTIONS TO MAXWELL’S EQUATIONS
`we consider how electromagnetic disturbances are created and how
`In this chapter,
`we saw that Maxwell's equations pre-
`they propagate in free space.
`In Chapter 1,
`dicted the existence of electromagnetic waves, even in vacuum where no currents
`or
`of interest were of
`were present (though
`sources outside the region
`charges
`course necessary to generate these waves). We shall now consider the case where
`wave
`the sources p and J are not zero,
`resulting in an
`equation.
`inhomogeneous
`Knowledge of the exact current and charge distributions on an
`object(antenna) lo-
`predict the field distributions at every point
`cated in free space is then sufficient to
`in space.
`wefirst discuss the
`wave
`Before developing the inhomogeneous
`equation,
`simpler static field solutions to Maxwell's equations. Static fields do not
`change
`waves. We shall see that the static field solu-
`in time and therefore cannot produce
`are
`tions to a
`quite similar to the dynamic field solutions to the
`particular problem
`same
`problem with a
`time-dependent source, but the dynamic solutions must take
`into account the finite propagation velocity of electomagnetic
`waves. Because in-
`formation does not propagate instantaneously from one
`in space to another,
`point
`ofradiation. Section 2.2
`this retardation effect must be included in the description
`radiating fields, but for now we consider only the simpler
`of
`describes the dynamics
`static fields.
`— 0 in the time-varying Maxwell's equations (1.2.1) 1.2.4),
`[f we let 0/dr
`or let @ — 0 in the ime-harmonic Maxwell's equations (1.4.4)(1.4,7),
`we see that
`the four basic equations of electromagnetism immediately decouple into two
`
`pairs.
`
`46
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 004
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 004
`
`

`

`Sec. 2.1
`
`Static Solutions to Maxwell's Equations
`
`Faraday’s law and Gauss's law in free space become
`Vx E=0
`
`47
`
`(2.1.1)
`
`(2.1.2)
`V-E=p/e
`and this pair of electrostatic equations, along with appropriate boundary conditions,
`charge density p is assumed to be
`uniquely determines the electric field. The
`specified.
`Likewise, Ampere’s law and Gauss’s magnetic law in free space are
`VxA=Jl
`=0
`
`(2.1.4)
`V-uoH
`where again this pair of magnetostatic equations with boundary conditions specifies
`are
`H completely if J is known. Because E and Af
`decoupled, it is impossible
`to have wave
`propagation, since it is the interaction between the time derivatives of
`E and the space derivatives of H and vice versa which leads to
`electromagnetic
`radiation. We may solve (2.1.1) and (2.1.2) for the electric field by noticing that
`=
`identity V x
`0 holds for any scalar field ®. Since V x E =0,
`the vector
`(V)
`we can write
`
`(2.1.3)
`
`E=-vVo
`(2.1.5)
`where ® is called the scalar electric potential, and the negative sign is chosen
`so that electric field lines point from regions of high potential
`to regions of low
`potential (i.¢., in the direction that a
`posiuve charge would moveif it were
`placed
`in the field). The electric potential is a
`particularly useful quantity because it is
`a scalar which contains complete information aboutthe three components of the
`vector electric field. Substitution of (2.1.5) into (2.1.2) yields the scalar Poisson
`equation in vacuum:
`
`=
`
`V7
`
`—p/ey
`(2.1.6)
`an
`inhomogeneous second-order partial differential equa-
`It is Poisson's equation,
`we shall use
`tion, which we should like to solve, and since it is a linear equation,
`the method of superposition.
`a
`the simplest charge distribution, namely
`point charge g
`Consider, first,
`located at the origin, There are two ways to find the scalar electnc potential;
`the first makes use of Gauss's law directly. The differential form of Gauss’s law
`(2.1.2) may be converted to an
`integral representation by using Gauss's divergence
`theorem (1.6.7) discussed in
`1:
`v=
`ido
`Chapter
`(2.1.7)
`
`$F
`[eG
`¥
`The quantities V, A, A, da, and dv have been defined in Section 1.6.
`
`A
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 005
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 005
`
`

`

`48
`
`Radiation by Currents and Charges in Free Space
`
`Chap. 2
`
`applying
`
`the divergence
`(2.1.8)
`
`A
`
`point
`
`Taking the volume integral of Gauss's law (2.1.2) and
`Efe =
`theorem (2.1.7) yields
`P dv
`v €o
`§
`/,, edu is simply q, the total amount of charge enclosed by the volume V. If
`But
`a
`single point charge in space, then by symmetry the electric field must
`there is only
`be radially directed away from the charge. Therefore, we choose the volume of
`sphere of radius r centered at the origin. The electric field thus
`to be a
`integration
`to A, Hence, we
`is normal to the surface ofthe sphere enclosing q,so E is
`parallel
`just muluply &,, constant at a
`given radius, by the surface area of a
`E-fda = 4nr°E,
`sphere, giving
`(2.1,9)
`A
`f
`Combining (2.1.8) and (2.1.9) gives the static electric field solution for a
`charge:
`
` q
`
`(2.1.10)
`Because E is radially directed and dependent only on r, we expect ©
`to be a
`function of r alone (no angular dependence). Therefore, the equation E =—-V®
`reduces to
`ae
`(21.11)
`g
`E,=->- Hap
`= —
`q
`axer
`where po is an
`constant. We
`usually
`integration
`will be zero
`infinitely far from the
`point charge.
`The second method for finding ®(r) from a
`point charge distribution is more
`mathematical, as it solves the Poisson equation (2.1.6) directly by wnting down
`the Laplacian in spherical coordinates. But first we need a mathematical way to
`express the fact that we have a
`at the origin. We write the charge
`charge singularity
`as
`distribution p(F)
`
`E(r)=F
`
`4reqr?
`
`Integration
`
`over r
`
`yields
`
`d(r)
`
`(r)
`
`+ Go
`
`sh
`(2.1.12)
`
`set Pp to Zero so that the potennai
`
`(2.1.13)
`oM=4qe7)
`is the three-dimensional Dirac delta function. This generalized func-
`where 57(F)
`uion has the following properties:
`(1) &)
`
`(2) &F)
`+00
`
`—to
`of
`Property (3) implies that the delta function has dimensions of m~?,
`
`=0 forr #0,
`+ co for r
`=0,
`PF) dv=1.
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 006
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 006
`
`

`

`Sec. 2.1
`
`Static Solutions to Maxwell's Equations
`
`49
`
`We can now rewrite (2.1.6) using the spherical form of the
`Laplacian operator
`and the charge distribution given by (2.1.13). Since p is spherically symmetric, the
`are zero, and we find that only the radial
`angular derivatives of ® in the Laplacian
`derivative contributes:
`
`1d (6) =-29H)
`r dr?
`# 0, the delta function is zero
`
`€o
`
`For r
`
`by property (1):
`=
`
`a g
`
`9 r¥#0
`rit?)
`=
`By inspection, (2.1.15) has the solution ®
`K/r + 09, where again Do may be
`—
`set to zero so that @(r
`00)=0, The constant A is determined by integrating
`over a
`of radius r > 0
`both sides of (2.1.14)
`=
`sphere
`rdo
`Fr
`r
`rsinddp
`2x
`1
`ad?
`sre)
`ues
`ely
`|
`[af
`ar
`rao
`rsinodd +8) =—4
`ve
`Bi
`[
`-|
`|
`0
`0
`0
`&q
`€o
`where the last equality follows from properties (1) and (3) of the delta function.
`on the left side of (2.1.16) is trivial since the
`the 6 and ¢ integrations
`Performing
`integrand is independent of both coordinates. Thus, (2.1.16) becomes
`r
`‘2
`Le ey dear?
`o rdr?
`€0
`dividing by 47 gives
`q
`
`(2.1.14)
`
`(2.1.15)
`
`(2.1.16)
`
`(2.1.17)
`
`(2.1.18)
`
`Integratingthe left side of (2.1.17) by parts and
`oo
`ia
`d
`d
`
`arear Mae
`
`rs (r®) ft
`-f[
`
`where the surface term [the first term on the left side of (2,1.18)] is zero because
`=
`d(K)/dr =0. The second term isan
`so
`integral of a total derivative,
`d(r&’)/dr
`q
`Pere 4n€
`point charge located at the origin, the scalar potential
`
`We conclude, then, that for a
`is
`
`
`om=—
`4m eqr
`where r =
`to an observer at Fr.
`is the distance from the charge (at the
`|7|
`origin)
`Both approaches therefore give the same result for the potential of a
`point charge
`located at the origin.
`
`(2.1.19)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 007
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 007
`
`

`

`i]
`
`Radiation by Currents and Charges in Free Space
`
`Chap, 2
`
`If the point charge
`(2.1.19) becomes
`
`is not located at the origin, but is instead found at
`
`7", then
`
`=
`
`oF)
`
`=,
`4req|F —F!
`measures the distance between the observerat 7 and the point charge
`where |F —F'|
`we have an incremental! volume év’ filled
`at 7’. Andif, instead of a
`point charge,
`=
`with charge 6g‘
`p(F)dv’, then this incremental amount of charge will give rise
`to the incremental] potential
`
`=
`
`5O(7F)
`
`2.1,20
`
`)
`
`(2.1.21)
`
`pr)dv'——
`4req|F -F'|
`We now
`integrate the potential created by each infinitesimal amount of charge
`over each ofthe infinitesimal volume elements Sv’ where
`to find
`p(F') #
`(2.1.20)
`the total electric potential: oF) =
`eae
`[E45
`4ne\F —F|
`as the scalar Poisson integral
`or the superposition in-
`This integral is designated
`are
`tegral. The latter name comes from the fact that since Maxwell's
`equations
`linear, we can
`sum the potentials contributed by each incremental amount
`simply
`charge acting alone to get the total potential. Notice that all of the fF depen-
`of
`a function
`so that ® is
`over dv’ in (2.1.21),
`dence disappears after integration
`only
`of the observer coordinate r. (In performingthis integral, r is kept constant.) The
`scalar potential may be computed by using the superposition integral if p is known,
`and then E can be determined by taking the negative gradient of ©. Figure 2.1 is
`a
`pictorial representation of the
`superposition integral.
`The static magnetic field equations (2.1.3) and (2.1.4) give rise to a similar
`.
`=
`x
`vector Poisson
`Because V
`V' (V
`A) =0 is
`and the identity
`8
`=
`equation.
`we can
`as the curl of an unknown
`any vector
`express B
`A,
`oH
`
`obeyedby
`vector A
`
`(2.1.22)
`B=wH=VxA
`to
`where A is called the vector
`potential. As usual, it is necessary ultimately
`check that (2.1.22) does not conflict with any of the other Maxwell’s equations.
`Substitution of (2.1.22) into (2.1.3) gives
`Vx (Vx A) spol
`which may be simplified by the vector
`identity (1.3.6)
`Vx(VxA=V(V-AD—- VA
`(2.1.24)
`to be
`we are free to choose its divergence
`Because only the curl of A is specified,
`we wish; this choice is known as
`setting the gauge. To seethat (2.1.22)
`anything
`
`(2.1.23)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 008
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 008
`
`

`

`Sec. 2.1
`
`Static Solutions to Maxwell's Equations
`
`51
`
`Source p (F")
`
` He
`
`Figure 2.1 Charge distribution and observer coordinates.
`
`is arbitrary, we
`is satisfied even if the divergence of
`A =A+Vw where
`let
`V x A =¥V xA since the curl
`is any scalar. Taking the curl of both sides
`gives
`of the gradient of any scalar is zero. Thus A is not
`unique. For static fields, we
`simplify (2.1.24) by letting
`
`(2.1.25)
`which is called the Coulomb gauge. Combining (2.1.23), (2.1.24) and (2.1.25)
`gives the vector Poisson equation for vacuum
`
`V-A=0
`
`which may
`
`V7A=—poJ
`also be written as the three scalar
`=
`
`equations
`
`(2.1.26)
`
`—pod;
`V?*Aj
`where i =x, y,z.' Because we can transform the
`previously derived scalar Pois-
`son
`equation (2.1.6) into each of the Cartesian components of the vector Poisson
`—
`J;, and €9 +
`equation (2.1.26) by making the substitutions @ —
`A;, p
`1/19,
`where i can be x, y, or z, We can
`immediately find the vector solution to (2.1.26)
`a
`by similarly transforming the superposition integral (2.1.21), yielding
`superposi-
`tion integral for each Cartesian componentof the static vector
`potential:
`dv’
`=
`Modi’)
`
`Ai?)
`
`v
`
`4afF—F|
`
`1. Note that this separation of a vector differential equarion into three scalar equations is straight-
`forward only in Cartesian coordinates; for example, V7A, # —jzo/, in spherical coordinates,
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 009
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 009
`
`

`

`52
`
`Radiation by Currents and Charges in Free Space
`
`Chap. 2
`
`i stands for a Cartesian componentof the field, and we may
`Again, the subscript
`make the notation more compact using
`vector notation:
`2.1.27)
`
`A®) =
`
`—
`
`[eee
`dor |r
`we can
`in a
`This means that if we know the current
`problem,
`distribution
`to get oH 2 Therefore,
`in theory
`calculate A directly and then take its curl
`magnetic fields in a statics problem if we know the
`we can find the electric or
`an
`or current distribution simply by performing
`integration. Although this
`charge
`integra] may be difficult to evaluate, a more serious
`also exists, as the fields
`problem
`themselves may alter the source distributions, Fortunately, for most of the simple
`problems solved in this text, the charge and/or current distributions are known
`or may be approximated quite accurately.
`exactly
`
`"|
`
`2.2 RADIATION BY DYNAMIC CURRENTS AND CHARGES
`If we now consider the fully dynamic
`form of Maxwell's equationsin free space,
`gradient of a scalar potential because
`we see that we can no
`longer Jet E be the¢
`we may again express oH
`as
`E=-dB/at 40. But since V-
`Vx
`uo =0,
`potential; (2.1.22) is still a valid equation.
`the curl of a vector
`Substitution of
`(2.1.22) into Faraday’s law (1.2.1) gives
`vx E+
`
`)=0
`which means that E + @A/dr (instead of E alone) may be expressedas the gradi-
`ent of a scalar
`
`(2.2.1)
`
`—
`E+
`
`fe)
`
`«2.
`(2.2.2)
`
`0A
`—=-+-V
`a
`since VY x V@=0 for any
`®, This equation has the correctstatic limit:
`scalar
`—
`=-—V¢®. Ampere’s law (1.2.2) can now be
`as d/dr
`O, (2.2.2) reduces
`E
`to
`—
`~ woe
`rewritten with substitutions for H and £
`using (2.1.22) and (2.2.2):
`Vx (Vx A)= Hed
`(2.2.3)
`=
`ay
`Motors
`a®
`A
`
`($)
`we arnve ar the useful Biof-Savart aw, which relates the magnetic field
`__ 2. Using this
`procedure,
`as an intermediary:
`H directly to the current J without the vector potential
`-
`I) xF-F) 4,
`Fone
`Snir —FPP
`
`“40
`
`[
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 010
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 010
`
`

`

`Sec. 2.2
`
`Radiation by Dynamic Currents and Charges
`
`53
`
`After use of the identity (2.1.24) and rearrangement of the terms in (2.2.3),
`we find:
`A+ poco
`=—pol +V¥
`(2.2.4)
`—
`a
`—
`—
`2VA
`
`am
`
`)
`we let
`
`5
`
`aA
`1L0€0 a2
`(v
`Since we are free to choose the gauge (divergence of A),
`=
`Pid Supkjer
`eNOS
`which is called the Lorentz gauge. Combining (2.2.4) and (2.2.5)
`vector Helmholtz equation:
`inhomogeneous
`—
`Loeo——at
`V7A
`vA
`we have a wave
`If J =0,
`equation (the homogeneous Helmholtz
`equation
`cussed in Section 1.3) which is consistent with the fact that both E and A
`obey
`wave
`equations in source-free vacuum. If E and H obey
`wave
`equations, then A
`In a similar fashion, we may
`Gauss's law (2.1.2) with (2.2.2) to yield
`and ® will also.
`VO + —
`(2.2.7)
`A)=
`=
`-£
`av.
`0
`A from (2.2.7), we obtain
`Again using the Lorentz gauge(2.2.5) to eliminate V
`the inhomogeneous scalar Helmholtz equation for the scalar potential ©:
`—
`=
`am
`poeo—>
`at
`are time-harmonic sources, then the scalar and vector Helmholtz equa-
`If p and J
`as
`tions can also be expressed
`
`i
`(2.2.5)
`
`now
`
`gives the
`
`(2.2.6)
`
`dis-
`
`=
`
`z
`—poJ
`
`-
`
`Vb
`
`—P/€o
`
`(2.2.8)
`
`=
`=
`
`—Hod
`
`(2.2.9)
`(2.2.10)
`
`+ wr
`LeA
`VA
`VW+ w*Uperd
`—0/€o
`nature of
`and p is made explicit. Notice that both
`where the phasor
`A, ©, J,
`or
`(2.2.9)}-(2.2.10) reduce
`inhomogeneous Helmholtz equations (2.2.6)and (2.2.8)
`+
`to the static Poisson equations (2.1.26) and (2.1.6) respectively in the limit 0/dr
`or
`0 (w— 0), The dynamic solution to (2.2.9)
`the static
`(2.2.10) is thus simply
`a factor which accounts for the finite propagation
`time of the
`solution modified by
`—
`waves
`or retardation is |F
`through space;this time delay
`F'|/c.
`electromagnetic
`We would now like to find solutions to the inhomogeneous
`Helmholtz equa-
`to the
`superposition integrals (2.1.21) and (2.1.27). For
`tions analogous
`simplic-
`we shall consider time-harmonic fields since
`form the basis for more
`ity,
`they
`arbitrary solutions. We first demonstrate that if the static point charge g at the
`=
`starts to oscillate with time dependence g(f)
`then the static
`q coswy,
`origin
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 011
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 011
`
`

`

`54
`
`Radiation by Currents and Charges in Free Space
`
`Chap. 2
`
`=
`
`can be replaced with the time-harmonic potential phasor
`~kr
`potential &,
`g/4mepr
`the chain rule and the spherical Laplacian,we
`find
`= t
`= %,¢7!*"". Using
`~kKo
`—2jk
`Vo=
`2
`=,
`am,
`(>
`(ve.
`ie
`*)-
`The second term in this expression vanishes since ®, is inversely proportiona! to
`r.and we conclude that
`=
`
`(2.2.11)
`
`equation
`
`26
`e/*"
`(V2, —kd,)
`If we nowsubstitute (2.2.11) into the inhomogeneous time-harmonic wave
`we find that
`(2.2.10),
`—
`=- 2 8H
`
`£0
`
`(V2@,
`which reduces to the
`
`simpler equation
`
`k2,) e+ wen be"
`=
`(¥20,) e-*" =
`
`0
`0
`ae
`=2
`if the dispersionrelation in free space k?
`holds. The second equality in
`@jz9éy
`=
`at 7
`0, the only value for which the delta
`(2.2.12) follows because e~/*" is unity
`function is nonzero. Therefore, (2.2.12) simply reduces to the static scalar Poisson
`for a
`point charge and we have shown that the potential phasor
`equation
`
`= gir
`4ireor
`an oscillat-
`is a valid solution to
`(2.2.10) in free space when the source p is just
`=
`ing point charge. Superposition of the incremental charge elements 5g’
`p(F’)dv'
`a
`over the volume V‘, which encloses any arbitrary charge distribution, gives
`dy-
`=
`similar to the static version (2.1.21):
`namic superposition integral
`3
`dv
`'
`ges
`by)
`(2.2.14)
`_
`=jHF-F'l
`|
`ef)
`[
`Therefore, (2.2.14) is a
`general solution to the inhomogeneous time-harmonic
`vector Hetholtz equa-
`scalar Helmholtz equation (2.2.10). The inhomogeneous
`u.
`tion may be written down by inspection from (2.2.14) by noting
`(2.2.9) is
`identical to (2.2.10) when the substitutions p+ J;, ®— A;, and €&) +
`1/o
`are made. Recombiningthe three scalar equations into the more com-
`(i =x, y,z)
`vector Poisson integral?
`pact vector notation gives the dynamic
`=
`dv
`;
`(2.2.15)
`Af)
`HolF’)
`e7jir-F|
`[SSin?"
`Notice that in the static limit (w= =0), the dynamic superposition integrals
`(2.2.14) and (2.2.15) reduce to the static Poisson integrals (2.1.21) and (2.1.27).
`3, Although we have chosen to derive time-harmonic superposition integrals, it should be realized
`thar time-dependent forms of these integrals also exist. We omit their lengthy derivations, but include
`
`e#r 537)
`
`PF)
`=
`
`(2.2.12)
`
`(2.2.13)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 012
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 012
`
`

`

`Sec. 2.3
`
`Radiation from a Hertzian Dipole
`
`55
`
`are found from the superposition integrals (2.2.14) and
`Once ®(F) and
`A(F)
`(2.2.15), the electnc and magnetic fields may be calculated by applying (2.1.22)
`and (2.2.2),
`
`2.3 RADIATION FROM A HERTZIAN DIPOLE
`source is called a Hertzian dipole and is illustrated in Fig-
`The simplest radiating
`a distance ¢ which contain
`ure
`It consists of two reservoirs separated by
`2,2(a).
`amounts of charge; the charge oscillates back and forth between
`equal and opposite
`w. The
`charge reservoirs are located on the 2-axis at
`the reservoirs at
`frequency
`an electric dipole with dipole
`moment
`z= +d/2, constituting
`
`(2.3.1)
`p=igd
`— o© but the
`gd product remains finite, we have a
`
`In the limit that d + 0 and g
`Hertzian dipole.
`Conservation of charge dictates that a current J must flow between the two
`=
`or J =
`charge reservoirs as the
`jwg in time-
`charge oscillates, where J
`0q/0rt
`harmonic form.
`(We shall find the electric and magnetic fields for the Hertzian
`at a
`w, where we understand that for an
`arbitrary frequency
`single frequency
`dipole
`we can superpose single-frequency solutions.) Therefore, the current
`spectrum
`density J is
`
`(2.3.2)
`J=21d3@)
`where we check the dimensions to make sure that (2,3,2) makes sense. Because
`the delta function has the dimensions of m~*, Jd
`5°(7) has dimensions of A/m?*
`which is also the proper dimension for J. We find (2.3.2) by considering the
`to have finite (though small) width w, length /, and depth d as shown
`dipole
`in Figure 2.2(b). Because the current flows in the 2 direction, J=i1/lw=
`~
`zIid/lwd. But lwd is the volumeofthe dipole, and as /, w,d— 0, 1//wd
`5° (F), giving the result (2.3.2).
`If we substitute the current distribution for a Hertzian dipole (2.3.2) into the
`
`the time-dependent superposition integrals for @(F,1) and A(F,t) for completeness:
`er) =
`du
`_
`4neqiF —7|
`e@t-F-Fi/e)
`[.
`Modst = F=File) ay
`4e,1)=
`we
`[
`4n|F-—F'|
`p(r’, 1) =
`as
`Re{p(F') e/“], then the retarded charge den-
`If the charge density oscillates sinusoidally
`—F
`=
`pF) e/=-/**-7) is observed in the integrand of (2.2.14) (with the e/“
`sity o@'.1 —|F
`l/c)
`factor implicit) since k =
`a/c. Therefore, the time-dependent equations above yield the correct time-
`harmonic equations when a
`sinusoidally varying source is chosen.
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 013
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 013
`
`

`

`56
`
`Radiation by Currents and Chargesin Free Space
`
`Chap. 2
`
`potential (2.2.15),
`
`we find that the vector
`
`potential
`
`for the
`
`expression for the vector
`dipole is
`
` Figure 2.2 Hertzian dipole.
`
`=
`
`AP =2
`
`elke
`
`(2.3.3)
`
`
`Id
`a
`because all of the current is concentrated at the origin.* Since A varies with r, it
`is natural to use
`as defined in Figure 2.3. The
`spherical coordinates r, 9, and >
`z-unit vector may be written as
`
`which may
`
`also
`
`% =Fcosé —Asind
`be seen from Figure 2.3, and therefore
`—
`=
`(F. cos @
`6 sin@) Hold et
`A(r, 0)
`4orr
`
`(2.3.4)
`
`(2.3.5)
`
`4. Mathematically, the integration
`
`is carried out to obtain (2.3.3).
`
`£O =F)\8F)du'= fF)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 014
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 014
`
`

`

`Sec. 2.3
`
`Radiation from a Henzian Dipole
`
`57
`
`Ne
`
` (b)
`
`Figure 2.3 Spherical coordinates.
`
`
`
`(2.3.6)
`
`(2.3.7)
`
`#0:
`
`\"
`
`es
`|se088
`
`(2.3.8)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 015
`
`Wecan find
`directly by taking the curl of A in spherical coordinates
`sr
`_ 7
`Fr
`ré
`sinod
`H=—VxaA= 7
`d/dg
`Hor’ sin@
`93/3é
`a/dr
`Ho
`
`rAg
`4
`orsindA,
`7
`are zero for the Hertzian dipole.
`H=¢o—
`a
`wi
`a
`a
`(4)
`¢ anr *
`Fea
`'
`re az
`+
`ats
`
`J
`1+— 0
`|
`We can find E from H using Ampere's law (1.2.2) where J =O for r
`-
`=
`1
`E=—VxdH
`J @€o
`Ho jkId i,
`oes ee
`maar
`+6/1+—
`jkr
`
`where 0/0 and
`
`Ay
`
`a
`
`oss
`
`-jkr
`
`}.|
`A
`
`L
`1
`(=)
`——
`Jee Naked
`ng
`in@
`
`+(3
`a)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 015
`
`

`

`55
`
`Radiation by Currents and Chargesin Free Space
`
`Chap, 2
`
`vector for the Hertzian dipole from these
`
`We can also find the complex Poynting
`electric and magnetic fields:
`S=ExH"*
`
`anr
`
`kid\*
`
`|,
`
`=
`
`M0
`
`DAE og
`
`(2.3.9)
`
`I
`
`itis
`
`{eli (a) Jove
`-
`
`Ge) [=
`6(ae)
`From the complex Poynting vector, we may find the time-averaged powerin the
`usual way:
`sin’ @
`
`(S)
`
`
`No
`
`(2.3.10)
`
`e(S}=
`z
` uel
`=5Re
`m2
`—
`0), these fields are valid for all r. For
`In the Hertzian dipole limit (d
`approximate the current as constant over thedipole
`a short dipole where we can
`are valid for r>>d. The electric field E(r,@, 1)
`is dis-
`length, (2.3.7)(2.3.9)
`Figure 2.4 at four instants of time, Were we to watch the field pattems
`in
`played
`were
`it would appear that new
`evolve in time,
`dipole pattems
`constantly being
`formed at r =
`to propagate outward once it
`0 and that each pattern would begin
`was about a
`wavelength from the origin, until far from the source the fields would
`It is evident that for r
`<A/2m (kr < 1), the field resem-
`waves.
`resemble plane
`bles the static field of an electric dipole, and that for r > A/2z it resembles a
`wave with gradually diminishing amplitude,
`In the region where
`uniform plane
`r *
`, the field lines appear to detach from the dipole field structure and begin
`4/20,
`propagating. We examine the exact electric and magnetic field solutions (2.3.8)
`and (2.3.7) in the limit where kr < | (the nearfield zone) and also where kr > 1
`(the farfield zone).
`we consider the nearfield case where kr <1. Using the dispersion
`First,
`=
`we write
`relation k
`and (2.3.1),
`w.//ip€p
`Id =
`=
`(2.3.11)
`jwp
`jwqd
`jkp//uoeo
`Substituting (2.3.11) into the expressions for E (2.3.8) and H (2.3.7) and keeping
`terms (highest powers of 1/kr) gives the following expressions
`only the
`leading
`for the fields:
`(2.3.12)
`(kr <1)
`(?2cosO+Osind)
`sin@ (kr <1)
`(2.3.13)
`
`=
`
`
`E=p
`H=o
`i?
`— |
`in the nearfield zone.) Equation (2.3.12) is the
`(Note that e~/*
`nearfield
`expression for the electric field, and (2.3.13) is the induction field. We note that H,
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 016
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 016
`
`

`

`Sec. 7.3
`
`Radiation from a Hertzian Dipole
`
`59
`
` Figure 2.4 Electric fields for a Hertzian dipole
`
`as a function of time.
`
`to
`to
`when
`1/r?, is much smaller than E (proportional
`which is proportional
`1/r>)
`+
`so that the near field is dominated by electric field (capacitive) effects.
`€A/2,
`vector (2.3.9) in the near field are
`The terms that dominate the complex Poynting
`1/r° and may be seen to be purely imaginary. This means that
`to
`proportional
`although electric energy is stored in the near field, there is no net power flow out of
`the nearfield region, consistent with the fact that there is no wave
`near
`propagation
`the dipole. All of these observations are
`completely consistent with the quasistatic
`solutions to the electric dipole, and we make the additional observation that in the
`=
`=
`0), the magnetic field vanishes identically, and the
`completely static limit (w
`&
`dipole is completely capacitive.
`By contrast, in the farfield zone (kr > 1), we
`terms of order 1/(kr)*
`neglect
`(or smaller) in (2.3.7) and (2.3.8), resulting in electric and magnetic fields which
`both have 1/r dependence:
`E=0 inka e/* sind
`4nr
`
`(kr > 1)
`
`(2.3.14)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 017
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 017
`
`

`

`60
`
`Radiation by Currents and Charges in Free Space
`ew! sin@ (kr > 1)
`© nt
`
`2?
`Chap.
`(2.3.15)
`
`SEPG
`H=¢2
`as
`Wenotefirst that since tl.
`tatic electric field
`1/r? away from the
`decays
`as
`1/r, the near field rapidly becomes unimportant
`and the far fields decay
`origin
`r. The electric and magnetic far fields
`in comparison with the far field for large
`are
`orthogonal, and have an
`have identical
`amplitude ratio
`spatial dependencies,
`=
`no, the impedance of free space. These were
`just the properties of uni-
`Ex / Ag
`waves discussed in Section 1.3, with the exception that the plane
`waves
`form
`plane
`1 had constant amplitudes in space. Forthe farfield solutions (2.3.14)
`in Chapter
`and (2.3.15), the amplitude gradually decreases as ]/r away from the origin, bulif
`we are
`observing the waveal very large distances (r >> 4/270 ), we see whatlocally
`resembles a uniform plane
`wave
`we are far
`in the +r direction.
`(If
`propagating
`as the wave
`enough away, the 1/r factor doesn’t change significantly
`propagates
`wave segmentovera
`outward. We also look at the plane
`sufficiently small range of
`=
`@ so that the sin @ factoris relatively constant, especially
`near @
`2/2.)
`We can calculate the time-averaged power (W/m?) found in this wave from
`the complex Poynting vector:
`a
`ae
`ees
`S=Ex H*=F 19
`OP
`|
`
`IP
`sin? @
`
`
`
`1m
`
`—-
`4r
`|kld/?
`
`in
`
`*g
`
`f
`
`2.3.16
`
`)
`
`It follows that
`
`(5)
`
`p=
`
`=
`
`=F —
`S|}
`e{s1
`JEP
`= =#7
`=5R
`
`
`which is identical with the exact
`expression (2.3.10) calculated from the complete
`Hertzian dipole fields. Thus (2.3.16) is valid everywhere in space, which is not
`over a
`must be conserved.
`surprising since power
`Integration of (2.3.16)
`spherical
`shell of radius r must be a constant independentof r, since this integration yields
`the total power emitted by the dipole. The sin?@ dependence of the Poynting
`yector meansthat most of the poweris radiated to the sides
`the dipole; there is
`of
`no radiation along the dipole axis. Figure 2.5 illustrates the E, Hi, and § vectors
`as well as the electric field and Poynting
`vector
`for the far field of a Hertzian dipole,
`as functions of @.
`magnitudes
`The total power P radiated by the dipole may be calculated by integrating
`the average powerradiated per unit area, over the surface of a
`sphere enclosing
`(S),
`as described above:
`the
`dipole,
`
`cj
`
`rsindd¢(S,)
`(3)-Ada=[
`d@ sin?b= ——
`jkId?
`ae
`To
`tiet [
`
`an
`
`(2.3.17)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 018
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 018
`
`

`

`Sec. 2.3
`
`Radiation from a Hartzian Dipole
`
`61
`
`The total power radiated thus increases as the square of the frequency and as the
`moment.
`square of the dipole
`part of an
`Sometimesit is useful to think of the dipole element as
`equivalent
`in Chapter 9. We define the
`in much greater depth
`circuit; this concept is pursued
`radiation resistance of any radiating element as the total power radiated by that
`one half the magnitude of the current squared; for the Hertzian
`element divided by
`wefind
`dipole,
`
` Figure 2.5 Radiation from a Hertzian dipole.
`
`y?
`i
`=
`=
`if we
`120m 2. Physically, this resistor R.4 would
`377 2
`approximate np
`dissipate the same total power P that the
`with current / radiates.
`dipole
`
`(2.3.18)
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 019
`
`= © Kay? = 20(kd?
`
`67
`
`P W
`
`=
`
`Raa
`
`Momentum Dynamics Corporation
`Exhibit 1017
`Page 019
`
`

`

`G2
`
`Radiation by Currents and Charges in Free Space
`
`Chap. 2
`
`We also define the gain G(@,@) of a
`radiating element as the ratio of the
`,
`@) to the power density radiated
`power density (W/m?) radiated in the direction (f
`=
`source
`an
`emitting uniform power flux
`incall
`by
`isotropic
`P/4ir?
`(Sisouopic)
`directions. Clearly, if the radiating elementis
`isotropic, the gain is | for all angles.
`For the Hertzian dipole,
`
`=
`
`Gié, @)
`
`=
`
`a
`
`(2.3.19)
`
`3 23
`(S(6,¢,7))
`3
`Paar
`as can be seen
`by combining (2.3.16) and (2.3.17). The radiation pattern for
`a source is determined by the gain
`and is a
`plot of G(8,) over all angles but
`normalized so its maximum value is unity. Since G(@,@) is
`of ¢
`independent
`the radiation pattern is symmetric around the Z-axis. The
`for the Hertzian dipole,
`at @ =
`7/2 (in the x—-y plane), and has a null at
`dipole radiates most
`strongly
`@ =
`0,2 (along the z-axis). Thus, the radiation pattern looks like a
`doughnut with
`a
`plots the gain for a
`vanishingly small hole, oriented along the Z-axis. Figure 2.5
`Hertzian dipole.
`If we
`integrate the gain
`always find that
`
`over the solid angle dQ
`sin6de@G(é, ¢) = 4x
`
`‘in
`
`ao
`
`n
`
`=
`
`sin@d@dé, we should
`
`(2.3.20)
`
`|
`[
`0
`0
`no matter what type of radiating element we
`of (S,)
`choose, since the integral
`to the total power P of the source.
`over the surface of any sphere is equal
`Equa-
`tion (2.3.20) is readily derived from (2.3.17) and the definition of gain (2.3.19); it
`may be easily verified for the Hertzian dipole.
`We also define Go to be the maximum gain of the antenna. For the Hertzian
`=
`=
`1.5 whereas for an
`1. Thus the dipole is 50
`isotropic radiator, Gg
`dipole, Gy
`percent more directive than an
`antenna (which isn’t directive at all).
`In
`isotropic
`Chapter 9 we shall discuss how to make antennas that are much more directive than
`the Hertzian dipole—useful if one wants to maximize the power t