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chapter2 - Chapter 2 Vector Differential Relations and...

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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 (304-352, 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 -------------_--_------- ---t----i d -: . F __ ' — I ——=—- - £3. 4;) - $(________fiw;~j 5M) =.= ORDINHRY DERIVHTIVE or f to. 1m: 0L. PROVIDED 77-m7' 7715 “Lin/7' Ex/svs'. x» "‘ Lmrr Ems-rs " Q waved—Meme: & fay/7f. 9‘ DO f E 3-5 Have Gnu: Qttecnou? ' Eu. Af:f(u.+Au.)- f(u.) Ly / (cum) f/u'lAu) f ALL V“) 3518: Chapter 2 . F: = amen: o LET 7c: FUAJ} f: £04.); 6' : ECU.) LET - fcr) f(X’ 7’ 2‘) "‘ I = VEcroR urn; (00835. x, 3,-3“ Ml : f;d_. + F3; THEAI, <10. du. LL 3f(x,1,z) = M (f(x+Ax,:1,£)-f(x,1,z) ) Ax-bo AX JggL = 3:. g + pi 9" du. du. du. 3(- Is BE A game ? a! JCfXQ) = df X9 + fx d9 ’K- 00 1’ mo 3: Ham: W6 DIRfCWO/V? d“, du. d“ 3X 9" SIMILAR EXPRESSIONS 5R i AND i? 8'; a: _A_/._§. cg : SCALAR. . 445. IF f H43 gun/113995.; PARTIAL Demwmrar or A7' 4mg— fle W, Wan/.- a‘f : air 3X33 639x 3513: Chapter 2 3513: Chapter 2 H 11$ 1' ES lrE'l! ' The space rate of change of a scalar field close to a given point is often of physical interest. The vector differential operator known as the gradient can be used to characterize the direction and magnitude of the maximum space rate of change of a scalar field. . LE-r f: (“may ‘43) 8.5 :9 "Carmen/rd $14007” " Scam»? H‘Lb. 7716 m or qu‘hlux) .De-Pcms or! The Coon-t an“l:‘~0 “3 , 8m- Hni 35M we WIN Lmfl 0R SURFACES: NE. df‘ 5 (mm. (HA/v66 m 1". 35 1 B: Chapter 2 dn : Swot-r57 (Le. ,L) b/mcc PAM [‘3 7a ,5"; - Tue Ftp/m 3‘ mo 33 ARE Lama A VEC'rbR Elm/mg d1 RPM on! .5} ma 5'2. Rear LL I all z dl,a. _. —l 4— dizgz + digs” Tye Moo/v7" Er die: if du, +_af duz + 3'? eiu.3 gala: flames IN ROM 5"" au‘ 3&3 B 71: If“ But all, z: hldu.’ / c112: hzduz/ (£13 7-: I13 0/113 (in-916cm +97%! + at 9 5:6; I :0" 2. $3 3 7. * EXPRESS df' A: A "DOT anauc‘r” oF Two Vac-mks: “(e-Sf. + %f + e) w WWW W CALL Tins @unth/TY THE 62.4mm?" 0F THC vacr/M/ F, 0/? SIMPLY and (F)- 8 =‘.> dn = dfl cos 9 E 73': Shim D/s-zmas F4014 2M5: 73:53 1.2., [3134(9) = E Them Space- 2472:“ | '3'; or cums-E aF Par/‘3. 6-241);ch —- Spec/n: (can. 5737-01: _L. CART€SIM COORDINW'E'S -‘ and“): 3x32; + 93%; +9131; . PROPERTIES OF 324 (F) g 2. C/LmDP/mt. CDoRDmm-res : __I_ /. grad (F) .J— 7?) Ecpnu - VALuE SURFACES 3? ‘4’? I - 5 pp 5: P P aw SAME E -VAl.uE 5: arms 7-25”: I F 3 mu u ’ 3. SPHCRICFL. Coolant/rams: dF=o => agle)-d_£=o $63467)_Ld£ - 3__rad(F)=2~§_E +3.1. +0. 1 3f (fl- 0"! 1-16.: MM EauI—Vnmi SuRmcf.’) r 3f 0 Fgg " rsma A; R d ‘ Z. Tm: Vecm 3m 0‘) Guys Bow 7m: HASH/724D: X- b/Frekavr ER”: Fok EHCH C0021) 3/ ., RN13 771E DIRECT/ON of we HflXmfiL SPACE PM of CHflNGE of F E M Pom-r MI I! PEG/cal. fl/VLLfi 3mm cg /fl~o (P). dF : [Marfldlcose m => G/f‘ mnxtnun («ME/V 5=O° 5&6?er Jae. 10 q 6‘ ,, \\ 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 coordinate-free 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 source-free 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 coordinate-free 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 = scalar-valued 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 “well-behaved” 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: “well-behaved” => F and V-F 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 Maxwell-Gauss integral law for magnetic fields: 953-615 = 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: V-B=O => A differential equation that must hold everywhere! Note: V-B=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: rfieoE-ds = [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 component-wise, 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 => Non-conservative 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’] * “tire-232i 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 “well-behaved” 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— B-dt = {Lds +4—f EOE-d5 (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 Maxwell-Faraday 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: VX-B—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 tune-static (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- fi-u 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(V-F)—V2F Provides a useful relationship between the curl- curl and Laplacian operators for divergenceless fields: Vx(VxF)=—V2F ifV-F=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+F-H (l3)Vx(F+G)==VxF+VxG (4)17 X(G+H)= FXG+FXH l4)V(fg) ==ng+gi (5)Fx(GxH)= G(H-F)—H(F-G) (15)v (jF): F~Vf+f(V-F) (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>gfiF-dt=fs(me-d- (2|)Vx(VxF)zV(V~F)—V1F (22) v x (ng) == fo Vg (9) {firm ~11: ,= I, H W: + (W) - MM 110) gfisL/Vg—gm-ds=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.: =fy-ceyg)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 fur-man- fizw I‘T INTEGRAL flcozen: :7 5’6ch Marc-yam. 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, 2-11. 3513: Chapter 2 ...
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