Chapter 6: Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws 6.1 Mawell s Displacement Current; Maxwell Equations

Size: px

Start display at page:

Download "Chapter 6: Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws 6.1 Mawell s Displacement Current; Maxwell Equations"

Brian Kennedy
5 years ago
Views:

1 Chape 6: Mawell Equaions, Maosopi Eleomagneism, Conseaion Laws 6. Mawell s Displaemen Cuen; Mawell Equaions The Displaemen Cuen : o fa, we hae he following se of laws : D,, fee H J fee E, an (6. Taking he iegene of H J, we obain fee H J fee (6. J fee if This iolaes he law of onseaion of hage. 6. Mawell s Displaemen Cuen; Mawell Equaions (oninue Mawell obsee ha if we posulae H J fee D, (6.5 whee JD D is alle he isplaemen uen by Mawell, hen, H J fee fee, D J whih is onsisen wih he onseaionofhage. of hage. (6.5 an be wien: H J J, The immeiae signifiane of (6.5 is ha i esablishes a new mehanism o geneae he -fiel, i.e. by a ime-aying E-fiel. Eample of he isplaemen uen: C I eal uen on he wie isplaemen uen in he gap Vsin fee D 6. Mawell s Displaemen Cuen; Mawell Equaions (oninue The Mawell Equaions : In (6., eplaing H J fee wih H J fee D, we hae a new se of equaions alle he Mawell equaions: homogeneous equaions E (6.6 6 D fee inhomogeneous equaions D H J fee These 4 equaions fom he basis of all lassial eleomagnei phenomena. s isusse in Ch. 5, Faaay's law onnes E an. s will be shown in Ch. 7,,(6.6 lea o EM waes. Thus, Mawell's heoy onnes "opis" an "eleomagneism". On he ohe han, he Loenz foe equaion, f E J, onnes "mehanis" an "eleomagneism". 6. Mawell s Displaemen Cuen; Mawell Equaions (oninue eiew of Laws & Equaions Obaine une ai Coniions : ala an eo poenials: E ( E, (, 4 E (b, (e, (, 4 (f J J J Physial laws: ( ( E 7- (a (pp. 4 J ( ( (pp ( Quesion : Whih of he aboe laws/equaions sill hol ue if? Why? 4

2 Fiel enegy: 6. Mawell s Displaemen Cuen; Mawell Equaions (oninue WE ED (4.89 W (5.48 H Foes: f EJ fe E f J ounay oniions: ( D D n fee (4.4 ( E E n ( n (5.86 n ( H H K fee (5.87 Quesion: Whih of he aboe equaions sill hol ue if? Why? 5 6. Veo an ala Poenials Fom he homogeneous Mawell equaions, we may fin a eo poenial an a sala poenial o epesen E an. (6.7 E E E E (6.9 Wih (6.7 an (6.9, we wie he inhomogeneous Mawell equaions (fo auum meum in ems of an as follows E ( (6. J E J (6. in auum Thus, he se of 4 Mawell equaions fo E an hae been eue o ouple equaions fo an Veo an ala Poenials (oninue ( (6. ewie ( (6. J If he poenials an saisfy he Loenz oniion:, (6.4 hen, (6. an (6. ae unouple o gie he equaions: (6.5 J (6.6 Equaions (6.5 an (6.6, une he Loenz oniion, ae equialen in all espes o he Mawell equaions. If an o no saisfy he Loenz oniion, hen hough he gauge g ansfomaion isusse below, we may obain a new se of poenials an, whih saisfy he Loenz oniion Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge Gauge Tansfomaions : (6.7 ewie (6.7 an (6.9: E (6.9 If (, ae ansfome o (, aoing o : an abiay sala funion of an hen an will gie he same E an, i.e. E The ansfomaion efine by (6. an (6. is alle he gauge ansfomaion. The inaiane of E an une suh ansfomaions is alle gauge inaiane. (6. (6. 8

3 6. Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge (oninue Loenz Gauge : ny se of an une he gauge anfomaion gies he same ( (6. E an. Hene, ( (6. J If he oiginal (, o no saisfy he Loenz oniion, we may hoose a gauge funion an eman ha h he new (, saisfy: ( This hen unouples an o gie he same equaions as in (6.5 (6.5 an (6.6: J (6.6 Using an, we obain fom ( he equaion fo : ( Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge (oninue Coulomb Gauge : (also alle aiaion gauge, ansese gauge, o solenoi gauge In he Coulomb gauge, we hae (6. (6. (6. hen, (6. J (6.4 To unouple an, we wie J J J an eman Jl [ Jl is alle longiuinal o ioaional uen] J [ J is alle ansese o solenoial uen] We may onsu J an J fom J as follows: J J l 4 l J ( ( 4 J l ee poof a he en (6.7 of his seion. ( Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge (oninue ewie (6.8: (6.8 If (, aleay saisfy he Loenz oniion, a esie gauge ansfomaion wih gien by he equaion: (6. an pesee he Loenz oniion. ll (, in his esie lass ae sai o belong o he Loenz gauge. ge The Loenz gauge ge is ommonly use beause i gies he se of equaions [(6.5 an (6.6] whih ea an on equal fooings. Fuhemoe, as will illbe shown in Eqs. (9 an (4 of Ch., (6.5 an (6.6 as well as he Loenz oniion hae he same fom in all ineial fames. Opional 6. Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge (oninue ewie (6.:. The soluion is (, (, alle he insananeous (6 4 Coulomb poenial (6. J ( In Jl, eplaing wi anuse 4 J h (6. an, we obain l (6.8 J ub. Jl fom (6.8 ino ( Jl J (6.4 The las em on he H of (6.4 is hen anelle by J l o esul in an equaion fo unouple fom : J (6.

4 Opional 6. Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge (oninue Disussion: (i J J oes no lea o ime aiaion of hage ensiy [see (].. onibues only o he nea fiels.. aiaion fiels ae gien by alone. (ii. Coulomb gauge allows sepaaion of "nea" an "aiaion" fiels. (iii The Coulomb gauge is ofen use when hee is no soue. Then, an saisfies he homogeneous equaion. wih he fiels gien by E, (6. Opional 6. Gauge Tansfomaions, Loenz Gauge, Coulomb Gauge (oninue Poblem: Poe Jl [in (6.7] J[in (6.8] J ( Poof : J J (6.8 4 J ( J ( [ 4 [ ] ( J ( ( ( ( J J J ( J ( (by he iegene hm. ( J ( 4 ( ( 4 ( J J J ( J [ 4 4 J ( ] J QED l J 4 Jl by (6.7 ( 4 Chap.. Fomal oluion of Eleosai ounay-value Poblem (oninue Fomal oluion of Eleosai ounay-value Poblem : ( The epession ( is appliable only o 4 unboune spae. y Geen's heoem, we may genealize i o an epession fo boune spae wih pesibe bounay oniions. Consie a geneal eleosai bounay-alue poblem: ( ( / wih ( s ( fo on ( Geen's n ieniy: s ( ( ( ( ( ( ( ( s a (.5 (5 n n In (.5, le ( be he soluion of ( wih aiable (i.e. Φ(. Le ( G D(,, whee G D(, is he Geen funion saisfying G D (, 4 ( wih G D (, fo on ( ubsiuion of ( an ( ino (.5 gies 5 Chap.. Fomal oluion of Eleosai ounay-value Poblem (oninue 4 ( ( [ ( GD(, GD(, ( ] [ ( GD(, GD(, s ( ] a n n Thus, we obain on GD (, G 4 D 4 s n s ( ( (, ( a (.44 (.44 epesses he soluion of he geneal eleosai poblem in ( in ems of he soluion GD(, of he poin soue poblem in ( an he bounay alue ( s of on. To ealuae (.44, we fis sole ( fo GD(,, hen subsiue GD(,, (, s ino (.44. I is ofen simple o sole G D (, fom ( han soling iely fom (, beause ( has he simple b.. of G D (, on. ppliaions of (.44 an be foun in Chs. an. The poblem below gies an appliaion wihou he nee o sole ( fo G(,. 6

5 6.4 Geen s Funion fo he Wae Equaion (6.5 an (6.6 hae he basi fom: 4 f (, (6. in fee spae. We assume he spae is unboune ( infinie an sole (6. by he Geen funion meho. We fis obain he Geen funion fom ee ne page ( G(,,, 4 ( ( (6.4 Fo a poin soue in unboune an G ( G an isoopi mei um, i is onenien o ansfom he oigins of spae an G G ime o he soue poin a an, so G G (,,, (, ha G epens only upon an. whee,, an. Thus, (6.4 gies (, (, 4 ( ( ( G G 7 Chap.. Laplae Equaion in pheial Cooinaes ( sin sin sin U( Le ( P( Q ( UQ Q PQ U sin P UP sin sin sin Muliply pyby UPQ Diiing all ems by sin q, we see ha he - epenene is isolae wihin his em. o ( his em mus be a onsan. Le i be (. P sin [ U U sin (sin Q ] (. P Q The j-epenene is isolae wihin his em, m so his em mus be a onsan. Le i be -m Geen s Funion fo he Wae Equaion (oninue ewie (: (, (, 4 ( ( G G Pefoming a Fouie ansfom in, we obain (, (, 4 (, (6.7 G G i G (, G (, e ( whee (, i G G (, e (4 In he limi i, (6.7 akes he fom of fhe Poisson equaion wih a poin soue a. Hene, G(, 4 ( lim G (, (6.8 Noe: Jakson efines k / (p. 4 an enoes G(, by G k ( (p. 44. Hee, we eain he noaion as an eplii emine ha G (, is an -spae quaniy Geen s Funion fo he Wae Equaion (oninue Fo, (6.7 eues o: G (, G (,. i i (, e e G(, G(, G (5 If,,(5 is also a ali soluion fo sine i eues o as [as equie by (6.8]. Hene, fo, we hae G (, G (, G (,, (6.9 i subje o he oniion. In (6.9, G (, e (6.4 i ( i e G (, G (, e [fom (4] ( ub. fo an fo ino (6.4, we obai bin [ ( G (,;, ] G :eae Geen funion G : aane Geen funion (6.4 (6.44

6 6.4 Geen s Funion fo he Wae Equaion (oninue We hae obaine soluions: [ ] ( G (, ;, (6.44 fo he equaion: ( G(,,, 4 ( ( (6.4 The soluion G iniaes ha an effe obsee a (, is ause by he aion of a poin soue a isane away a an ealie ime /. This is a physial soluion beause he ime of he ause ( peees he ime of he effe (. Fo he G soluion, howee, he ime of he ause ( woul be afe he ime of he effe (. This is no physially possible. Thus, "ausaliy" equies ha we eje he G soluion an se, in (5 o (6.9. Then, he physial soluion of [ ( ] (6.4 is G G. 6.4 Geen s Funion fo he Wae Equaion (oninue Going bak o he basi fom of (6.5 an (6.6: 4 f (, (6. This equaion has a isibue soue f(,. ine we aleay hae he soluion G fo a poin soue a (,, he soluion fo in (6. is, by he piniple of linea supeposiion, f (, (, G (,,, f(, e (6.47 = [ ( ]/ p q whee he noaion implies ha quaniies in he bakes e (inluing he posiion eo ae o be ealuae a he eae i me:. We an eify ha (6.47 is he soluion by sub. (, G (,,, f(, ino (6. an use (6.4. [see M&W, pp fo an alenaie eiaion of (6.47] Disussion: (i ewie (6.47: 6.4 Geen s Funion fo he Wae Equaion (oninue f(, e (, (6.47 (6.47 is ali fo unboune spae (see p. 44, boom. If hee ae bounay sufaes, bounay oniions mus be onsiee in oe o aoun fo soues on he bounay. simila siuaion an be foun in eleosais, whee he soluion ( ( 4 (. is ali fo unboune spae, while he soluion 4 D 4 ( ( G (, ( G (, a (.44 applies o a finie olume wih bounay effes aoune fo by he seon em on he H. s n D 6.4 Geen s Funion fo he Wae Equaion (oninue (ii ewie he Geen funion: G [ ( ]/ This is he signal obsee a (, ue o he aion of a ela funion soue a (,. uh a soue has equal omponens in all fequenies. If he meium is ispesie (i.e. wae spee ays wih he fequeny, omponens of he signal will popagae a iffeen spees an eah a iffeen imes. Thus, he signal obsee a will be a pulse of finie uaion, ahe han a ela funion of ime + as in G. This eplains why he soluion fo G is ali only fo he fee spae o a non-ispesie meium [see p. 4 (op an p. 45] in whih all he wae omponens popagae p owa a he same spee an onsequenly eah a he same insan of ime. (iii The elaion beween obsee'simeanhes he eae ime, ( /, iniaes ha a signal fom he hage aels a spee owa he obsee, inepenen of he moion of he hage (Einsein's posulae. 4

7 6.4 Geen s Funion fo he Wae Equaion (oninue f (, e (i The soluion in (47: (, (47 is ue o he soue f. Moe geneally, we may a o his soluion a omplemenay funion in(,, whih is any soluion of he homogeneous wae equaion: Thus, in geneal, he soluion of 4 f(, f(, an be wien (, (, e in (6.45 Fo eample, in(, an be a plane wae inien on a ielei f(, obje while e is he wae geneae by he inue uens an hages in he ielei obje (eae in Ch eae oluion fo he Fiels (6.5 ewie J (6.6 Eah Caesian omponen of (6.5 an (6.6 is in he fom of + (6.. ssuming fee spae an supeposing he Geen funion G fom all poins in he isibue ib soues an J, we obain (, (, (, 4 J (, (,, ( J (, Noe: an eue o (.7 an (5., espeiely, in he sai limi, i.e. when an J ae inepenen of ime. e eae oluion fo he Fiels (oninue The fiels E an an be epesse in ems of an. We may also epess E an iely in ems of J, by oneing he Mawell equaions ino equaions fo E an in he fom of (6.. E Mawell equaions E in fee spae J E J E E ( E E J E E E J J J E ( J J (6.49 ( eae oluion fo he Fiels (oninue (6.49 an (6.5 ae in he same fom as (6.. ssuming infinie spae an apply he Geen funion G, we obain J E (, 4 J (6.5 4 e J (, 4 J (6.5 e 4 (6.5 an (6.5 an be onee ino he Jefimenko fomulae [see (6.55 an (6.56], whih epliily show he euion o he sai equaions (.5 an (5.4. 8

8 Giffih.. Jefimenko s Equaions eae poenials: (, J(, V(, an (, 4 4 E V ˆ ˆ V [ ] 4 J (, ( 4 4 ˆ ˆ E [ ] 4 4 J ˆ ˆ J [ ] 4 The ime-epenen genealizaion of Coulomb s law. J 9 Giffih Jefimenko s Equaions (ii eae poenials: (, J(, V(, an (, 4 4 J (, [ ] J J 4 4 ˆ J J ˆ an ( J The ime-epenen genealizaion of [ ] ˆ 4 J he io-aa law. These wo equaions ae of limie uiliy, bu hey poie a saisfying sense of losue o he heoy. 6.7 Poyning s Theoem an Conseaion of Enegy an Momenum fo a ysem of Pailes an Eleomagnei Fiels w W w w f, f, f The ae of wok one by he E-fiel on hage pailes insie a olume V is gien by f ( E J H D f E J E ( E H E D H E E H H E H J E E H E D H JE EH E DH (6.5 ae of onesion of EM enegy ino mehanial an hemal enegies. 6.7 Poyning s Theoem (oninue JE EH E DH ewie (6.5: The ems E an D H in he inegan an be inepee physially if we make he following assumpions: ssumpion :The meium is linea wih negligible ispesion an negligible losses. We an hen wie (easons gien in Ch. 7 of leue noes D (, E (,, (, H (, E D, ED H H. (6 ssumpion : The fiel enegy ensiy fo sai fiels u ED H (6.6 E D epesens he fiel enegy ensiy een fo ime- epenen fiels. Fom (6 an (6.6, we hae u ae of hange of E D H fiel enegy ensiy (7

6.7 Poyning s Theoem (oninue JE EH E D H E DH JE u EH (6.7 ewie (6.5: ub. u fo, we obain u JE (6.

7 as u JE EH a n E meh fiel V E s na Emeh Efiel s a [ Poyning's heoem] (6.

fiel enegy insie V. Then, by onseaion of enegy, is he powe/uni aea.

7 Poyning s Theoem (oninue V Magniue of Poyning eo (alulae by C. Y.

7 Poyning s Theoem (oninue seay sae J E u sn a J E s n a I iui V iew: I fiel iew: E J E, E JE s

7 Poyning s Theoem (oninue Conseaion of Linea Momenum of Combine ysem of Pailes an Fiels : Wie

9 6.7 Poyning s Theoem (oninue JE EH E D H E DH JE u EH (6.7 ewie (6.5: ub. u fo, we obain u JE (6.8 whee, E H, is alle he Poyning eo. The meaning of beomes lea if we wie (6.7 as u JE EH a n E meh fiel V E s na Emeh Efiel s a [ Poyning's heoem] (6. n whee Emeh is he oal mehanial/hemal enegies insie V (no pailes moe in o ou of V an Efiel he oal fiel enegy insie V. Then, by onseaion of enegy, is he powe/uni aea. fiel Eample : powe lines sie iew I +V I 6.7 Poyning s Theoem (oninue V Magniue of Poyning eo (alulae by C. Y. Kao oss seional iew E Noe: E H= ( E ( H [ ( E H ] V E j j j j j j j V 4 Eample : a DC iui 6.7 Poyning s Theoem (oninue seay sae J E u sn a J E s n a I iui V iew: I fiel iew: E J E, E JE s n a Powe flows ou of baey. J Powe JE ansmission s n a by Poyning Powe flows eo ino esiso Poyning s Theoem (oninue Conseaion of Linea Momenum of Combine ysem of Pailes an Fiels : Wie own he Mawell equaions in he auum meium: E E EE J E EE E [ f EJ EE E ] [ E E E E ] E This em, whih equals, is ae fo lae manipulaion. ub. he epession fo he foe ensiy f ino Newon's n law: we obain meh meh meh P f P P E : oal momenum of all pailes in V. E E E E (6.6 6

10 ime aeage aiaion pessue I / ewie (6.6: 6.7 Poyning s Theoem (oninue H Pmeh ( E [ E E E E ] Define g E H [eleomagnei momenum ensiy ] (6.8 P fiel g [oal eleomagnei momenum in V ] (6.6 an hen be wien (see p.6 by iegene hm. meh Pfiel s ( P T T n a (6. n ( P meh P fie l s Tn a V a (8 whee T T is he Mawell sess enso efine as T E E ( EE (6. Noe: y Newon's law, only P (no meh Pfiel is he foe on V Poyning s Theoem (oninue Poblem : plane wae is inien nomally fom fee spae ono a fla sufae an is oally absobe. Fin he foe on he sufae. oluion: Consie he olume enlose by. On he lef sie, we hae nez (,, E insananeous E ( E, Ey, fiels on he (, y, n z lef sufae Pfiel E ( E E Ey y Tn E ye y Ey y ( E ( E ( E ez ( E ez [ ] aea of lef sufae ( P meh Pfiel ( s Tna F P meh E ez insananeous F ( E : eo aiaion pessue z z Poyning e e 8 insananeous enegy ensiy 6.7 Poyning s Theoem (oninue lenaie soluion o poblem : ssume he plane wae has a finie oss seion an a finie lengh. We may hen enlose he full een of he wae wihin sufae (see figue. Thee is no fiel on he sufae. Hene, fo olume V, eleomagnei ( Pmeh Pfiel s T na momenum ensiy V F P P meh fiel g, z whee g EH z [by (6.8 an (6.9] P e eause he wae aels a spee an i is oally absobe, he eleomag nei momenum P fiel in V eeases a he ae g. F P ez F ( P ez E e z Noe: This meho oes no equie he absobing maeial o be fla. Quesion: The aiaion pessue is ue o he J foe. How? Poyning s Theoem (oninue Poblem : spheial paile in he oue spae wih aius, mass M, an ensiy M =.5 kg/m absobs all he sunligh i ineeps. Fo wha alue of oes he sun s aiaion foe (F on he paile balane he sun s gaiaional foe (F G. ime-aeage aiaion I: sunligh inensiy oluion: pessue (see pob. I / (aeage powe/uni aea a he paile I P 4 F E G: gaiaional ons. ( Nm /kg M : sun s mass (.99 kg GM M GM 4 F M G P F 7 F G 7.7 6mGM m > > 7 FG = F if =.7 m < < P : oal powe aiae by sun (.9 6 W : isane o sun 4 fom Haliay, esnik, an Walke

11 6.9 Poyning s Theoem fo Hamoni Fiels; Fiel Definiions of Impeane an miane Phasos : In linea equaions, hamoni quaniies an be epesene by omple aiables as follows: E (, E ( (, D D ( (, ( i e e H (, H ( J (, J ( (, ( eal omple (alle he phaso I is assume h a he LH is gien by he eal pa ofhe he H Poyning s Theoem fo Hamoni Fiels (oninue epesenaion i of Time -eage Quaniies by Phasos : To epess nonlinea quaniies by phasos, suh as he pou of fh hamoni quaniies, ii we wie he quaniies ii as 4 e i i i i i i E (, e [ E ( ] [ E ( e E( e ] J (, e [ J ( e ] [ J ( e J( e ] Then, J (, E (, i i [ J( E ( J ( E( J ( E ( e J( E( e ] i e [ J ( E( J( E( e ] an he ime aeage an be wien in ems of phasos as J (, E (, e [ J( E ( ], assuming is eal (9 imilay, E (, H (, e [ E ( H ( ] ( * Poyning s Theoem fo Hamoni Fiels (oninue Mawell Equaions in Tems of Phasos : In ems of phasos, he Mawell equaions an be wien: (, ( E (, (, E ( i ( D (, (, D ( ( (, (, H J D (, H ( J ( id ( Comple Poyning's Theoem : Using he phaso epesenaion of Mawell equaions, we obain i ( EH H E J E [ EH ied ] ( i( EH ED H (6. 4 ewie (6.: 6.9 Poyning s Theoem fo Hamoni Fiels (oninue [ J E EH ED H ] This equaion gies he omple Poyning heoem: J E i w w na e m s whee ( i ( (6. (6.4 EH [alle he omple Poyning eo] (6. an he eal pa of is he ime-aeage powe [see (]. In (6.4, w an w ae efine as e m w e 4 ED 4 E The eal pa of we ( wm is he ime w m 4 H 4 H g ( gy y aeage E ( fiel enegy ensiy. (6. If an ae boh eal, he eal pa of (6.4 gies e[ J E] se[ n] a, whih is he ounepa of (6.7 appliable o onsan-ampliueampliue hamoni fiels (fo whih he fiel enegy emains onsan. 44

12 6.9 Poyning s Theoem fo Hamoni Fiels (oninue Fiel Definiion of Impeane : Ii We now apply he omple Poyning's heoem V i i Z o a -eminal iui. Daw a lose sufae n suouning he iui. Le Ii be he inpu uen, geneal iui V i be he in pu olage, an le he inpu enegy Ii flow be onfine o a small aea i. Then, V C i i L IV i i na (6.5 i n an he omple Poyning's heoem [(6.4] a speifi eample J E i we wm sn a an be wien: i i J E e m n i i IV i w w, a I Z whee Z is he impeane of he iui efine as aiaion loss Z Vi 4 e m I i w w a i I J E n i i ix : esisane, X : eaane (6.7 ( Homewok of Chap. 6 Poblems: 8,,, 5, 9 Quiz: De., 6.9 Poyning s Theoem fo Hamoni Fiels (oninue ewie: V Z i 4 e m I i w w a i i I J E n i ssume J E an,, ae all eal. peial ase: Negle he aiaion loss em: a n i geneal efiniion of he impeane P 4 i( W Z m W e of a iui in ems of he powe loss I i an he fiel enegy in he iui an he fiel enegy in he iui P E [ohmi loss] Wm wm We we [ E f iel enegy] whee [ -fiel enegy] [ -f an Wm We posiie eaane; Wm We negaie eaane. This epession fo Z is useful fo miowae iui suies. Opional 6.6 Deiaion of he Equaions of Maosopi Eleomagneism We limi he sope of ou onsieaion of e. 6.6 o a geneal isussion of he aeaging meho an he eiaion of (6.65. Miosopially, he mae is ompose of eleons an nulei, in whih he spaial aiaions of hage/uen isibuion funions an eleomagnei fiel funions ou oehe oe aomi isanes (of he oe of - m. These funions an be egae as sums of l ela funions. Howee, maosopi insumens only measue he aeage quaniy. Hene hee is a nee o eelop an aeaging meho o eue miosopially fluuaing funions o maosopially smooh funions, an heeby obain a se of maosopi Mawell equaions

13 Opional 6.6 Deiaion of he Equaions of Maosopi Eleomagneism (oninue If we eplae eah ela funion, e.g. (, in he miosopi L isibuion funion (of hages, e. ( wih a smooh funion f ( (see f ( figue subje o he oniion f( an if he wih L of f( is muh geae han he aomi 8 isanes (e.g. L m, hen he sum of many suh funions (eah epesening a ela funion in he miosopi isibuion funion will beome a smooh funion epesening he spaially aeage miosopi isibuion funion. This is he meho use in e fo he eiaion of maosopi equaions. Opional 6.6 Deiaion of he Equaions of Maosopi Eleomagneism (oninue We may look a he aboe aeaging poeue as follows. ela funion ( geneaes a smooh funion f (. Thus, fo a isibuion ib i funion F( ompose of a lage numbe of poin soues (ela funions, he esponse [enoe by F( ] will be he supeposiion of he esponses fom all poins: F( f( F(... spaial aeage of F( In he inegan, eplaing wih, we obain (6.65: F ( f ( F (, ( whee f ( is now a smooh funion enee a. s an eample, we le F( ( an sub. i ino (6.65 f ( f( ( f ( Thus, we hae eoee ou assumpion ha he ela funion ( geneaes a smooh funion f ( enee a. 49 5

Control Volume Derivation

Control Volume Derivation School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass