Electromgnetics 5- Lesson 5 Vector nlsis Introduction ) hsicl quntities in EM could be sclr (chrge current energ) or ector (EM fields) ) Specifing ector in -D spce requires three numbers depending on the choice of coordinte sstem Howeer EM lws re independent of coordinte sstem of use ) The use of ector nlsis in EM is not necessr (eg Jmes Mwell s originl work) but will led to elegnt formultions 5 Vector lgebr Vector ddition nd subtrction Fig 5- Illustrtion of ector ddition: C B (fter DKC) Dot (inner) product The dot product of two ectors nd B is sclr equl to the product of nd the projection of B on : B B (5) where B [ π ] is the smller included ngle Dot product is commuttie ( B B ) nd distributie [ ( B C) B C ] B Edited b: Shng-D Yng
Electromgnetics 5- Cross (outer) product The cross product of two ectors nd B is ector: B n B B (5) where n is unit ector norml to nd B (in the direction of the right thumb when the four fingers rotte from to B ) B [ π ] is the smller included ngle Cross product is neither commuttie ( B B ) nor ssocitie [ ( B C) ( B) C ] but is distributie [ ( B C) B C ] roduct of three ectors ) Sclr triple product: (5) ( B C) B ( C ) C ( B) Its mgnitude represents the olume of the prllelepiped (Fig 5-) Fig 5- Illustrtion of sclr triple product: ( B C) (fter DKC) ) Vector triple product: (54) ( B C) B( C) C ( B) 5 Orthogonl Coordinte Sstems Definition nd bsic properties Edited b: Shng-D Yng
Electromgnetics 5- point in -D orthogonl coordinte sstem cn be locted s the intersection of three cured mutull perpendiculr surfces represented b { ui constnt i three bse ectors (unit ectors in the directions of coordinte es) { u i } } The stisf with: ui ( i j k) () () (); (55) u j uk ui u j if i j δ ij (56) if i j Eq s (5) (5) cn lso be eluted b liner lgebr formuls if the inoled ectors re represented b the liner combintion of the bse ectors of some orthogonl coordinte sstem: u u u B u B u B u : B B b eq s (55) (56) B ( ) ( B B ) u u u u u u B ( u u ) B ( u u ) B ( u u ) B B B B B B [ ] B B i i (57) i B u u u B (58) B B B Metric coefficients When the coordinte chnges from ( u u u ) to ( u du u u ) the obsertion point moes long the direction of u b differentil length of h du where h denotes the metric coefficient of u The sme rule pplies to u nd u s well Note tht h i onl if u i represents quntit of length Edited b: Shng-D Yng
Electromgnetics 5-4 When the coordinte chnges from ( u u u ) to ( u du u du u du ) the obsertion point moes b differentil displcement of enclosed differentil olume of d i i i u i defining n Crte coordinte sstem ) ( u u u ) ( ) (Fig 5-) ) The bse ectors { i } u i re independent of the obsertion point ) Since { } re ll quntities of length the metric coefficients nd differentil olume re: { h h } h d ddd (59) Fig 5- ositioning of one point in the Crte coordinte sstem (fter DKC) Clindricl coordinte sstem ) ( u u u ) ( r ) (Fig 5-4) ) Two of the bse ectors { } point (Fig 5-4b) chnge with the polr ngle of the obsertion r Edited b: Shng-D Yng
Electromgnetics 5-5 ) When the coordinte chnges from ( r ) to ( r d ) the obsertion point moes long the direction of b differentil length of r d h r The metric coefficients nd differentil olume re (Fig 5-4c): { h r h } h d rdrdd (5) Fig 5-4 () ositioning of one point (b) the two the clindricl coordinte sstem (fter DKC) dependent bse ectors (c) the differentil olume of Clindricl coordinte trnsformtion ) Trnsformtion of position representtion (Fig 5-4b): () Clindricl ( r ) Crte ( ) : r r (5) (b) Crte ( ) clindricl ( r ) : Edited b: Shng-D Yng
Electromgnetics 5-6 tn ( ) r (5) ) Trnsformtion of the ector components: ector V cn be represented in the clindricl coordinte sstem V r r or in the Crte coordinte sstem V where r (5) roof: V ( r r ) ( r ) r ( ) ( ) B obsering Fig 5-4b r Sphericl coordinte sstem ) ( u u ) ( ) u (Fig 5-5) ) ll of the bse ectors { } ngle of the obsertion point (Fig 5-5) chnge with the imuthl ngle nd the polr ) When the coordinte chnges from ( ) to ( d ) the obsertion point moes long the direction of b differentil length of d h When the coordinte chnges from ( ) to ( d) the obsertion point moes long the direction of b differentil length of d h The metric coefficients nd differentil olume re (Fig 5-5b): { h h h } d ddd (54) Sphericl coordinte sstem is useful when the obserer is er fr w from the source Edited b: Shng-D Yng
Electromgnetics 5-7 region Fig 5-5 () ositioning of one point (b) the differentil olume of the sphericl coordinte sstem (fter DKC) Sphericl coordinte trnsformtion ) Trnsformtion of position representtion (Fig 5-5): () Sphericl ( ) Crte ( ) : (55) (b) Crte ( ) clindricl ( ) : tn ( ) tn ( ) (56) Edited b: Shng-D Yng
Electromgnetics 5-8 Edited b: Shng-D Yng ) Trnsformtion of the ector components: ector V cn be represented in the sphericl coordinte sstem V or in the Crte coordinte sstem V where (57) roof: ( ) V ) ( ) ( ) ( B obsering Fig 5-5b <Comment> ) B eq s (5) (57) n rbitrr ector V in the clindricl (sphericl) coordinte sstem is uniquel specified onl if the position of obsertion ie the rible (ribles ) is fied ) Trnsformtion of position representtion cn be regrded s trnsformtion of position ector components () Clindricl: ( ) r mens r r ( ) ( ) r r B eq (5) r r r consistent with eq (5) The dependence on comes from the bse ector r (b) Sphericl: ( ) mens ( ) ( ) B eq (57) consistent with eq (55)
Electromgnetics 5-9 The dependence on nd comes from the bse ector 5 Vector Clculus Grdient: definition nd phsicl mening Fig 5-6 Illustrtion of grdient of sclr field V (fter DKC) For sclr field V ( u u u ) consider two equi-potentil surfces S : V V nd S : V V dv (Fig 5-6) The shortest distnce between n obsertion point S nd the surfce S would be dn where dn ndn is ector pointing to S (thus dn > ) nd norml to S t The spce rte of chnge of the sclr field V long some rbitrr direction ie dv is mimied when // dn The grdient of sclr field V is ector field whose mgnitude nd direction chrcterie the mimum spce rte of increse of V : dv V n (58) dn <Comment> If dv > (the field increses long n ) V // If dv < (the field decreses long n n ) V //( n) V lws points long the direction of field increse Edited b: Shng-D Yng
Electromgnetics 5- Grdient: formuls of elution Consider n rbitrr point S (Fig 5-6) V n l ( ) is: l dn dv dv α α dn The spce rte of increse of the sclr field V long n direction dv In -D coordinte sstem ( u u u ) (59) ( V ) l the chnge of the sclr field dv due to displcement u u u cn be represented b: Eq (59) is equilent to dv ( ) l V V l l V V V dv l l l V d B the representtions of dv nd we he: V l B comprison both sides of the equlit Since the differentil length ( V ) ( ) u u u V must be i due to the chnge of rible V u V u V u l l l u i b smll mount of V du i is i hi dui ( i h is the metric coefficient of u i ) V u V h u h u h u u V u V In Crte coordintes ( u u u ) ( ) { h h } V V h [eq (59)] V V (5) (5) Diergence: definition nd phsicl mening Vector field cn be illustrted b flu lines such tht the field mgnitude is mesured b the number of flu lines psg through unit surfce norml to the ector Edited b: Shng-D Yng
Electromgnetics 5- Fig 5-7 Flu lines of ector field (fter DKC) If represents the directed flow densit ds represents the totl flow oer n open S surfce S For olume V enclosed b closed surfce S the outwrd flu ds will be positie(negtie) onl if the olume contins flow source(k) The diergence of ector field is sclr field chrcteriing the net outwrd flu per unit olume: lim which is used to chrcterie the flow source(k) quntittiel ds (5) S S Diergence: formuls of elution In the Crte coordinte sstem consider ector field ( ) nd n infinitesiml cuboid respectiel (Fig 5-8) centered t ) with side lengths ( Fig 5-8 differentil olume in the Crte coordinte sstem used to derie eq (5) (fter DKC) Edited b: Shng-D Yng
Electromgnetics 5- Edited b: Shng-D Yng ) On the front fce : S the outwrd flu is S s d F where s d lthough the ector field ) ( is function of position we cn pproimte it b constnt ector ) ( on the infinitesiml surfce S of the cuboid ( ) ( ) F B the first-order Tlor series pproimtion ( ) ( ) ( ) ( ) ( ) ( ) F Since is the olume of the cuboid (denoted b ) we he: ( ) ( ) ( ) F ) On the bck fce : S the outwrd flu is S s d F where s d We pproimte ) ( b constnt ector ) ( on the infinitesiml surfce S of the cuboid ( )( ) F B the first-order Tlor series pproimtion ( ) ( ) ( ) ( ) ( ) ( ) F The totl outwrd flu for the front nd bck surfces S nd S becomes: ( ) F F
Electromgnetics 5- ) The sme strteg cn be used for the remining four surfces of the cuboid The totl 6 outwrd flu for the cuboid becomes: ds Fn S n ( ) B eq (5) the diergence of ( ) t ) is ( lim ds (5) For other orthogonl coordinte sstems eq (5) is generlied to eq (-) of the tetbook S Diergence theorem The definition of diergence [eq (5)] implies tht the totl outwrd flu of ector field oer closed surfce S is equl to the olume integrl of the diergence of the ector field oer the olume V enclosed b S : ds S ( ) V d (54) This fct cn be shown b subdiiding the olume V into mn smll res where the contributions of flu from the internl surfces of djcent smll elements will cncel with one nother (Fig 5-9) Fig 5-9 Subdiided olumes for proof of the diergence theorem (fter DKC) Edited b: Shng-D Yng
Electromgnetics 5-4 Curl: definition nd phsicl mening If the ector field represents field of force the work done b the force in moing some object round closed pth (contour) C (ie the energ obtined b the object when treling long C ) is: Circultion C (55) consertie force produces no circultion becuse for n contour C In other words consertie force does not drie objects circulrl If non-consertie force hs nonero circultion for n infinitesiml contour C round point it forms orte source t tht dries circulting flows To quntittiel mesure the strength nd direction of orte source we define the curl of ector field s ector field whose () mgnitude represents the net circultion per unit re nd () direction is the norml C direction n of the differentil contour C m (with re s ) which is oriented to mimie the circultion n lim s Cm s (56) Curl: formuls of elution The circultion per unit re of ector field long n rbitrril oriented contour C u (with re su nd unit norml ector u ) is: C u lim ( ) u (57) s su u In the Crte coordinte sstem we cn derie the -component of b considering Edited b: Shng-D Yng
Electromgnetics 5-5 ector field ( ) nd n infinitesiml rectngulr contour C centered t ) with unit norml ector nd side lengths ( respectiel (Fig 5-) Fig 5- differentil re in Crte coordintes used to derie -component of eq (5) (fter DKC) ) On the pth ie { ) [ ] } the work done b the ( force is W where lthough the ector field ( ) function of position we cn pproimte it b constnt ector ) on the infinitesiml pth W ( ) ( ) series pproimtion ( ) ( ) is ( B the first-order Tlor ( ) W ( ) ( ) ( ) Since is the re of the rectngle (denoted b s ) we he: W ( ) ( ) s ( ) ) On the pth ie { ) [ ] } ( the work done b the force is W where We pproimte ( ) b constnt Edited b: Shng-D Yng
Electromgnetics 5-6 Edited b: Shng-D Yng ector ) ( on the infinitesiml pth ( ) ( ) W B the first-order Tlor series pproimtion ( ) ( ) ( ) ( ) ( ) ( ) s W The totl work done long the pth nd pth becomes: ( ) s W W ) The sme strteg cn be pplied to the remining pth nd pth 4 The totl circultion due to contour C becomes: ( ) s 4 B eq (57) the -component of is: ( ) ( ) lim C s s We cn further derie the - nd -component of b emining the circultion due to contour C nd C respectiel s result in the Crte coordinte sstem cn be formulted s: (58) For other orthogonl coordinte sstems eq (58) is generlied to eq (-7) of the tetbook
Electromgnetics 5-7 Stokes theorem The definition of curl [eq (56)] implies tht the totl circultion of ector field oer contour C is equl to the surfce integrl of the curl of the ector field oer the open surfce S bounded b C : C ( ) S ds (59) This fct cn be shown b subdiiding the open surfce S into mn smll res where the contributions from the internl boundries of djcent smll elements will cncel with one nother (Fig 5-) Fig 5- Subdiided res for proof of Stokes theorem (fter DKC) <Comment> Grdient diergence nd curl re ll point functions describing locl field behiors Lplcin: definition nd phsicl mening Grdient diergence nd curl re ll first-order differentil opertors In EM theor howeer we need to del with second-order derities of sclr nd ector fields Lplcin of sclr field V is nother sclr field defined s: ( ) V V (5) To show the mening of Lplcin tke sclr function of gle rible f () s n emple Edited b: Shng-D Yng
Electromgnetics 5-8 d f d lim df f ( / ) f ( / ) lim ; d f ( / ) f ( / ) f ( ) f ( ) f ( ) f ( ) lim f ( ) f ( ) lim f ( ) ( f f ) d f This mens tht the second-order deritie of f () ie describes the difference d between the field lue f nd the erge field lue f of its surrounding points s result the sclr Lplcin of sclr field of multiple ribles V ie similr mening V hs the Lplcin of ector field is nother ector field defined s: (5) ( ) Lplcin: formuls of elution In the Crte coordinte sstem we cn substitute eq s (5) (5) into eq (5) to obtin: V V V V (5) Similrl we cn substitute eq s (5) (5) (58) into eq (5) to obtin: ( ) ( ) ( ) (5) Lplcin formuls for clindricl nd sphericl coordintes cn be found in the inside of bck coer of the tetbook Null identities Two identities inoling with repeted del ( ) opertions re importnt for the concept of Edited b: Shng-D Yng
Electromgnetics 5-9 potentil functions (DKC Ch Ch6): ( ) V (54) consertie (curl-free) ector field cn be epressed s the grdient of sclr field (electrosttic potentil) ( ) (55) solenoi (diergence-free) ector field cn be epressed s the curl of nother ector field (mgnetosttic potentil) Eq s (54) (55) cn be esil proen in the Crte coordinte sstem Helmholt s theorem (decomposition) ector field F is uniquel determined if both its diergence nd curl re specified eerwhere s result we will introduce electric nd mgnetic ector fields b specifing their diergence nd curl (fundmentl postultes) first ector field F cn be decomposed into: ) The curl-free (irrottionl) component F i with Fi g Fi where g represents the flow source generting F B eq (54) F i V where V represents the sclr potentil of F ) The diergence-free (solenoi) component F s with Fs Fs G Edited b: Shng-D Yng
Electromgnetics 5- where G represents the orte source generting F B eq (55) where represents the ector potentil of F F s s result F F F V (56) i s ie ector field cn lso be determined b specifing its sclr nd ector potentils Edited b: Shng-D Yng