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Unformatted text preview: Chapter 2 Vector Differential Relations and
Maxwell’s Differential Relations in Free Space . 9mm: To derive the dzﬁerential forms of Maxwell’s equations
in free space from their integral versions postulated in
Chapter 1. —> Requires the divergence theorem and the theorem of Stokes. —a Requires the gradient, divergence, and curl operators of
vector analysis (i.e., derivatives of ﬁelds with respect to
spatial coordinates). ' The diﬁ‘eremialforms of Maxwell’s equations are useful for the
study of: 0 Electric and magnetic ﬁelds in material regions (Chap. 3). * Solution methods for practical electric and magnetic systems
(Chaps. 4 and 5).  Electromagnetic waves and propagation (304352, etc...) ' Begin with some background on the differentiation of vector
ﬁelds... 35 13: Chapter 2 °In many physical problems involving vector ﬁelds, a knowledge
of their rates of change with respect to space, time, and other
parameters is often important. —> Vector derivatives. A t __ ti d : . F __ ' — I
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\\ CQNSERVATIVE PROPERTY n OF grad (F)
 Recall, a
X AN IMPORTHNT wrcékm. PROPERTY oF grad (F) grad( f ) wax—Bi + ayaj + a: J: a a
For? EM new 71:5de x5 : ax y z 55 8mm! (nun/.4 = o
1 oDefme a vector partial differential operator represented by the
symbol V (pronounced del) as follows: 3 a 3
V sala— + ayg— + 825—
Hows FOR ALL “weLL 861mm” SCAM}? Emma/VJ F, x y 2
Then,
:5 And new grad (c) I: A CoNJERWmt/E 5a». a a a
V A grad(f)=Vf=ax—:+ay—f+az——j:
3x 8y dz
0 PROOF 3 .
————— 1.e.,
CONSIDER, P 7 P :9 grad(f) s Vf
facadcryd; =fdr =[r]
P.
’3 . I; * “grad ” and “V ” are often used interchangeably in
__ _ a o o coordinatefree expressions, but V has a speciﬁc meaning in each
" {(ulﬁlzﬁﬁ) — F( M, , “z I L‘ ), speciﬁc coordinate system!
FER THE POI/v73“ 8 (“fl “f, (4;) g! P( my up L13), (N434 V deﬁned only for rectangular coords here!)
‘TH K5, 1: P ~The same symbol V will be used to signify some additional
3? grad (r) . 44 = f df‘ : [43] ° :0, differential operations on vegtors, where different meanings for V
1 If, P must be used in different coordinate systems.
0 3518: Chapter 2 ll — Ema]; Consider a scalar, time—independent ﬁeld in some
region of space given by: T(x,y) = 200x +100y
then, VT 5 grad(T) 23x31 + ayi]: + gala: 3): By 82
= 200 aJr + 100 ay (0) Graph of T = constant. (b) Side view of (a). 3518: Chapter 2 1. 2. 3. Question: Free Space Statics Field: E or B ? (Explain) Sources: pv orJ ? (Why?) How is source “A” related to “B” ? (Discuss) 12 3518: Chapter 2 13
Question: Free Space Statics 1. Field: E or B '? (Explain) 2. Propose a source configuration to generate this ﬁeld. (Hint: Two basic source conﬁgurations are required together.) 3518: Chapter 2 14
m n. E I: E.”  As with scalar fields, the space rate of change of a vector ﬁeld
close to a given point is often of physical interest. The divergence of a vector ﬁeld F(r,t) is a measure of the net
outwardﬂwc of F per unit volume at a point P, i.e., div F 5 lim 3 (flux lines/m3) 8H1 div F E 1 [M + a(FZhIhS) + a(F3hihz)
8112 3113 * What is the physical signiﬁcance
of a divergent ﬁeld? Fig. 2.1: The meaning of div F:
net outward ﬂux per unit volume
as Av —> O. 3518: Chapter 2 15  Recall the use of flux lines to represent variations of vector fields
graphically: Flux lines indicate at each point in a region of space: 1.The direction of a vector ﬁeld.
2. The magnitude of a vector ﬁeld. ~If a vector field F is representable by a continuous system of
unbroken ﬂux lines in a volume region: => The region is source free, or equivalently,
2: F is divergenceless, i.e., div F = 0 (Fig. 2.2 (a )).  If the ﬂux plot of F contains broken or discontinuous ﬂux lines: => The region contains sources of the ﬁeld flux, i.e., => F has a nonzero divergence in that region (Fig. 2.2 (b )). Arbitrary
closed (b) Fig. 2.2: (a) A vector ﬁeld F in a sourcefree region (as many
ﬂux lines enter S as leave it). (b) A vector ﬁeld in a region
containing sources (S possessing net outward flux). 3518: Chapter 2 16 17
Divergence — Speciﬁc Coordinate Systems t Note: div F in cartesian coordinates: . aF BFy 6F
1. Cartesian coordinates: dwF = 8; + 33:— + 5:: BE: BF) an
+._+_ =>divFEV~F diVF = 8x By 82 ‘  v . * ' 6‘ ’ ,9 Hi . 97 ‘ v
2’ Circular cylindrical coordinates: The notatlons le F ‘and V F are used mterchangeqb'ly in
coordinatefree expressmns, 1314; V has speCIﬁc vector deﬁnition in cartesian coordinate system. div F = Limpp) + iﬂ + 838
P P p 890 z  mama Find the divergence of each: 3. Spherical coordinates: (b) L = ap([{ /p ) 1 ~a—(ngin9) + 1 aF‘” mm as ”may divF = i r2 (rzFr) + 24w * div F = scalarvalued ﬁeld. 1 ““:.;*_.i7" \_ \:T\Test closed
‘ surface S * div F has diﬁ‘erentforms in different coordinate systems ! 3*WhatisdivLatp :0? 3518: Chapter 2 ' 3518: Chapter 2 Flux Plots and Divergence (lid 18 3513: Chapter 2 19
MW  If F(r, t ) is “wellbehaved” in some region of space, then: j (div F) dv 5;} F~ds
V S is true for the closed surface S bounding any volume V. Fig. 2.3: Geometry of a typical volume V bounded by a closed surface 5 used
in relation to divergence theorem. =The volume integral of ((div F) dv ) taken throughout any V
equals the net ﬂux of F out of the closed surface S bounding V. N.B: “wellbehaved” => F and VF are continuous in V and on S. The divergence theorem (also known as Gauss’s theorem) is
mainly used to transform surface and volume integrals. => Required for establishing several useful equations and theorems of electromagnetic theory.
3518: Chapter 2 20
Divergence Theorem — Maxwell’s Equations The differential, or point, forms of two of Maxwell’s equations
can be derived from their corresponding integral forms by using
the divergence theorem. 1. Gauss's Law for Magnetic Fields  Apply the divergence theorem to the MaxwellGauss integral
law for magnetic fields: 953615 = 0 => [(divawv = 0
S V must hold true for any V. Therefore, the differential form of
Gauss’s law for magnetic ﬁelds is given by: VB=O => A differential equation that must hold everywhere! Note: VB=O z: B is divergence free. => B is solenoidal. => The flux plot of any B ﬁeld must consist of closed
lines. 35 1 8: Chapter 2 21
2. Gauss’s Law for Electrig Fields in Free Space 0 Apply the divergence theorem to the Maxwell—Gauss integral
law for electric fields: rﬁeoEds = [pvdv
S V => V()deoE v=vfpvdv => V[V~(£OE)—pv]dv = 0 must hold true for any V. Therefore, the differential form of
Gauss’s law for electric ﬁelds is given by: V‘(%E) = pv z: A dﬁerential equation that must hold everywhere! *The divergence of electric flux (80E) at any point in a region
is precisely equal to the volume electric charge density there. :> The ﬂux sources of E ﬁelds are electric charges! z: Electric ﬁeld lines can only originate or terminate on
electric charges. 3518: Chapter 2 22
V, gm of a Vector; Eigld ~The curl of a vector ﬁeld F(r, t) is a pointwise measure of the circulation of F around a closed path, per unit area bounded by the
path contour. * Pointwise circulation 27 The circulation of a vector ﬁeld F around a closed path 5 is
deﬁned as: Circulation of F around a E i det
t z: F has nonzero curl at a point if the circulation of F about a
differential path a that encircles the point is nonzero. 0 For a vector field F(r, t):
curl F => an oriented circulation 2; avector result! (Explain!) z: curl F is defined componentwise, as follows : 43nd:
[curlFL s Alim 1—— .l'i—)O A51. Hence, curlFE al[curlF]1 + az[curlF]2 + a3[curlF]3 3518: Chapter 2 23 Note: The component surfaces Asi are ﬂat (i.e., planar) with positive normal directions given by ai. The positive integration
sense about t is given by the right~hand rule (R.H.R.). As] —> d5 Integration
sense
Ilfcurl F11 (ui)// Fig. 2.4: A closed line 6 bounding the vanishing area
AS], used in deﬁning the a; component of curl F at P. Curl — Differential Deﬁnition oThe diﬁerential expression for the curl of a vector field F in generalized coordinates is given by the following determinant
form: 31 32 33
mm 3111 hlhz
a a a 3518: Chapter 2 24
Aside: curl F also written rot F (“rotation of F”). * What is the physical signiﬁcance of a rotational ﬁeld? Rotational field => Nonconservative ﬁeld! Irrotational ﬁeld = Conservative ﬁeld! oRecall, curl F at a point is vector whose magnitude is an
indication of the net circulation of F per unit area about the point. 0 Examples? Paddle wheel I Fig. 2.5: A velocity ﬁeld in a ﬂuid, with an interpretation of
its curl from the rotation of a small paddle wheel. 35 1 B: Chapter 2 25 Curl — Speciﬁc Coordinate Systems 1. Cartesian coordinates: BF
curlF =ax[%—l:f—%:_y] + ayPFx _y_2_] + 34 y 3F; 2. Circular cylindrical coordinates: 3172 GP BF 3F
“““F = apli‘sa‘ 35’] * “tire232i
l_i1. F _.l_§fé 3. Spherical coordinates: r sine 8 “““F = 3’ [i(F¢sin9) .332] + ae[ 1 8F, 6 )]
3¢ 3 BF}
7km — ] 2 curl F has diﬁ‘erent forms in dijferem coordinate systems. Aside: In cartesian coordinates: curl F :— V x F (check 3) * The notations “curl” and “V x ” are used interchangeably in
coordinate—free expressions (. ..be careful!) 3518: Chapter 2 26
— M Long thick wire problem (radius a; current I). chncg: H I? : (3(0.
27m."
AHPERe’s Law :> 8¢ =
L >0.
#7:? 6’ P
y.
0
gﬂ.‘ : Q<¢L
7711‘ ‘X’ COHPHRE To CuRJZEMr Davyrry g"
OVER PRGELEH QEG/ON I 3518:Chapter2 Rotatignal Fields — Elux Plgts * Discuss the “curl” of this ﬁeld. ‘33 27 F "r 3513: Chapter 2 Rotational Fields — Flux Plots Stokes’s Theorem . If F(r, t) is “wellbehaved” in some region of space, then: M
g
_
*“
""'""'——___...._._"__."" “”57”” f (curl F)ds = fwd:
_* S l
M
““—
—_
_— holds for every closed line t in the region, if S is a surface
_ “NE”? bounded by a.
———
u
m
m
m
m.“
—:———.—_'._.__.'_"""" “”‘"‘NT
u
m“
“‘ l S K (pasmve side) lntémﬁ‘ sense of * Discuss the “curl” of this ﬁeld. fem“ Fig. 2.6: Geometry of a typical open surface S bounded by the closed path (used in relation to
Stokes’s theorem. =9The net flux of (curl F) through any open surface S equals the
net circulation of F around the closed path a bounding S.  Stokes’s theorem is primarily a mathematical tool — used to
transform surface and contour integrals. . Useful in BM. theory?
3518: Chapter 2 3518: Chapter 2 ' 30
Stgkes’s Theorem — Maxwell’s Equations oThe differential forms of two of Maxwell’s equations can be derived from their corresponding integral forms by using Stokes’s
theorem. 1. Fargday’s Law £12m = 7%] Bcds (foranyS)
t S . A dzﬁ‘erential equation that must hold everywhere! :9 The curl of E at any point is exactly equal to the time rate of
decrease of B at that point. 3518: Chapter 2 31
2. Ampg‘re’s Law gEree§pace2 —l— Bdt = {Lds +4—f EOEd5 (for anyS)
”0% S dt 5 3 f[(V></JJIB) — J — §;(&)E)]ds = O (for any S)
S  A diﬁerential equation that must hold everywhere! => The curl of (B/po) at any point is equal to the sum of the
electric and displacement current dengue; there. 35 1 B: Chapter 2 32 re ' a ic u ions 01f the electric and magnetic fields in free space are static, the operator 8/&t appearing in the MaxwellFaraday law and in the
MaxwelLAmpere law should be set to zero: Vx E = 0 Must hold at any point!
=> Any static E ﬁeld is irrotational (conservative). (Don’t forget: V ' (80E) = pv must also hold at any point.) 2. Ampﬁm’s Law: VXB—z
#0 J => The curl of a static B ﬁeld at every point in space is
proportional to the current density J there. :> B circulates “around” all current regions! (Don’tforget: V'B = 0 must also hold at any point.) 351 B: Chapter 2 33 V 1 Ci rl 1'! iv r r'
 For f = ﬁr, t) sufﬁciently smooth:
div( grad(f)) = V (Vf) = sz  In generalized coordinates: _ l a [22/23 6/ a ml 6)“ 6 121122 (if)
v (Vf) — [11/12/13 [6111 (T a“! + (91‘; [12 6111 + 6313 h} alla =9 Laplacian operator: l 6 Izz/t3 0 8 hshl 0 a [11/12 6
V _=. + + 6
[21/12/23 0111 kl 5a; 0112 111 an; an, [13 u} 4
N
Ill * Laplacian operator familiar? (Recall Laplace’s Equation.)  The Laplacian operator is useful for obtain both tunestatic
(Chaps. 4 and 5) and time—varying EM—ﬁeld solutions. 3513: Chapter 2 1. Cartesian coordinates: V'(Vf)EV2f= 51f 52f 52f
a? + a +7 6e M 2. Cylindrical coordinates: 34 1 5 6f 1 61f 02f
V2 EV'V =——— —— ————— —
f f 100p pap)+p’6¢2+5z2
3. Spherical coordinates:
1 5 (3f 1 5 0f 1 ff
2 _‘__ Z_
Vf‘ﬁ ar('a 7+) #5111030 (”9519+ r sinleadﬁ
* Does V7— apply to vectors? Yes:
a 13/2, 6 a 1 6 it): a
2 = 2 3
VF_/zlhzh3[6ul</zl au,)+ x (alFl + 32F; + 33173) 6112 l [12 6112 )+ 6113 ( Izlhz a
[[3 6113 3518: Chapter 2 35
 e.g., in cartesian coordinates: VZF = axsz, + I’VIF, + I‘VZF, *Y‘f=(5% 3;? 5.95%)
+339: +§§s+§§i) 915...)? 9‘)
*9*(5:r* 335'“???  Note; There are two deﬁnitions for the Laplacian operator, one for scalar ﬁelds and one for vector ﬁelds, i.e., the deﬁnition of V2 is context dependent! 3518: Chapter 2 36
gm Qurl Qperatgr 0 For F = F(r, t) sufﬁciently smooth:
curl (curl F): V x (V x F)  In generalized coordinates: I
gels
r—‘l
S‘s
ﬁu
A
gm
”‘53
1
E?
Q}
531
V
l___l
L—V‘J Vx(VxF)=u{a (3’52_E 3 ﬂ_ﬂ
a} 6x 6} dz dz 5:: 3518: Chapter 2 37
 Note: It can be shown that the following vector identity holds: Vx (VxF)=V(VF)—V2F Provides a useful relationship between the curl curl and
Laplacian operators for divergenceless ﬁelds: Vx(VxF)=—V2F ifVF=O * The two identities above hold in all three coordinate Systems! Summary of vector Identiﬁes
M Algebrair Diﬂnmtml
(1) >r G: G F (II)VU+g)=Vf+V§
(2)FxG=—GXF (l2 (F+G==V' F+VG
(3)17 (G+H)=F G+FH (l3)Vx(F+G)==VxF+VxG
(4)17 X(G+H)= FXG+FXH l4)V(fg) ==ng+gi
(5)Fx(GxH)= G(HF)—H(FG) (15)v (jF): F~Vf+f(VF)
(6)F (GxH): G (HxF)=H(FxG) (16V) m(FxG) m=~G (VxF)— F (VxG) (l7)Vx(fF)= (VﬂxF+f(VxF)
Integral (18)v Vf= V’f
7 par: v.“ (19)v(er).—.0
)3st s I” D (20)Vx(Vf)=O
8>gﬁFdt=fs(med (2)Vx(VxF)zV(V~F)—V1F (22) v x (ng) == fo Vg
(9) {firm ~11: ,= I, H W: + (W)  MM 110) gﬁsL/Vg—gmds=f,<fvzg—gvzf>dv 3518: Chapter 2 38 Green’s integral theorems are really specialized versions of the
divergence theorem. ° Green ’sﬁrst integral theorem: 5135mm 'd = f, [fV‘g + (W)  were .5‘
an, til(r173) = (QM(ya) +F(g"a) (55; mi)
=$ ﬁksr INML mam j Ln 1?: FY3; f (#23). 4.: =fyceyg)dv;  Green ’3 second (symmetric) integral theorem: 5f; w; gi) ds = f, thg —gv2mv —— LET G = 317:). f (Syria; =flfgyzr+(gg).(yr)]ar
V ,—.——————__ J' /
/ ,
‘7 furman ﬁzw I‘T INTEGRAL ﬂcozen: :7 5’6ch Marcyam. 7759904 / ‘X‘ ASSQMING ALL Faucﬂoﬂf “WEI—L'Ee‘h’ﬁI/ED H /N V
RAID ON A" 3518: Chapter 2 39
 Green’s integral theorems are important in proving the uniqueness
of vector ﬁelds that satisfy Maxwell’s equations. 0 Why is uniqueness important? 0 Consider the Helmholtz theorem: The Helmholtz theorem shows that the speciﬁcation of both the
divergence and the curl of a vector ﬁeld in a region of space
along with particular boundary condition on the boundary of the
region, are sufﬁcient to make the vector ﬁeld unique. Egteg Maxwell’s equations specify the divergence and the curl
of both E and B in a region, so that these relationships, together with appropriate boundary conditions, can be expected
to yield unique ﬁeld solutions for E and B! h r —  Study complete chapter except for Sections: 2—6, 2—9, 2—10, 211. 3513: Chapter 2 ...
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