UNIVERSITÀ DI PISA. Math thbackground

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1 UNIVERSITÀ DI ISA Electomagnetc Radatons and Bologcal l Inteactons Lauea Magstale n Bomedcal Engneeng Fst semeste (6 cedts), academc ea 2011/12 of. aolo Nepa p.nepa@et.unp.t Math thbackgound Edted b D. Anda Gualuc 1

2 Lectue Content Coodnate sstems Catesan coodnate sstem Clndcal coodnate sstem Sphecal coodnate sstem Coespondence between coodnate sstems Vectos and Vectos Algeba Bounda condtons 2

3 Catesan coodnate sstem Is a sstem to unquel detemne the poston of a pont n space wth espect to a efeence pont called ogn. Each pont s defned b the ntesecton between thee pependcula p sufaces; the vesos ndcate the decton of adectedas. Each veso s pependcula to one of the above sufaces and passes though the pont. Catesan coodnate sstem (,, ) detemne the vesos decton,, do not change the dectons wth changes of pont n space = + + poston vecto d d dl d d d S = dd ds d = dd ds d = dd Dffeental elements: d S = dd d S = dd d S = dd dv = ddd dl = d+ d+ d 3

4 Clndcal coodnate sstem Clndcal coodnate sstem (t s useful when the studed sstem s smmetc wth espect to as) (,, ),, detemne the veso dectons, change the dectons fom pont to pont = + poston vecto ds = dd ds = dd d ds = dd Dffeental elements: ds = dd ds = dd ds = dd dv = ddd dl = d + d + d 4

5 Sphecal coodnate sstem Sphecal coodnate sstem (t s useful when the studed sstem s smmetc wth espect to ogn) (,, ) denotes the dstance fom the ogn (, ) ndcate the decton of the pont locaton detemne the vesos decton,, change the dectons fom pont to pont = poston vecto ds = d d ds = sn dd ds = dd dv = sn 2ddd dl = d + d + sn d = 2Dffeental elements: 5

6 Eamples S V R 2π π 2 2 S = ds = sndd = 4πR S 0 0 2π π R 4 V = dv = sn dd d = π R V R 2π π 2 M = ( dv ) = (,, ) sndd 1. Sphee suface aea: 2. Sphee volume: 3. Weght: V n S 4. Gauss law: ε eds = Q= dv 0 n S V l L a dl b dl 5. Electostatc feld: b e dl = Φ Φ a b a 6

7 Coespondence between coodnate sstems (I) Catesan Clndcal Sphecal cos sn sn cos snsn cos cos sn sn cos + coscos sn sn + cos sn sn + cossn + cos cos sn 7

8 Coespondence between coodnate sstems (II) Cld Clndcall Catesan Sphecal 2 2 actan( / ) cos sn + cos + sn sn sn cos + cos cos sn 8

9 Coespondence between coodnate sstems (III) Sphecal Catesan Clndcal cos ( / ) 1 tg sn cos + snsn + cos coscos + cos sn sn sn + cos cos sn + + cos cos sn 9

10 Coodnate sstem coespondence eample Electc feld ntenst E ( + + ) Q ( + + ) Q = = / πε ( + + ) 4 πε ( ) Snce chage has sphecal smmet: Catesan coodnates E = Q 4πε 0 2 Sphecal coodnates 10

11 Vectos and Vectos Algeba (I) Vecto: s a geometc element epesented b an oented segment wth an aow at one end. Hasboth thlength th( (magntude) and decton. statng pont magntude decton Vectos addton: a a b a+ b = a + b 2abcos b Multplcaton b a Scala: a a 2a a vecto k scala If: k > 0 k < 0 ka ka 11

12 Scala poduct: a b Vectos and Vectos Algeba (II) a b= abcos opetes: a b= b a a b= 0 f: a b EX: One mpotant phscal applcaton of the scala poduct s the calculaton of wok: F dw = F dl = Fd cos Wok done b constant foce, staght lne moton dl Vecto poduct: a b = ab sn EX: vecto poduct appeas n the calculaton a b opetes: of toque and n the calculaton of the magnetc foce on a movng chage. ( a+ b) c= a c+ b c b a b= a b a b= 0 f: a b a a ( λb) = λ( a b) = ( λu) v a ( b c) = b( a c) c( a b) 2 ( a b ) ( a c ) = a ( b c ) ( a b )( a c ) τ = F = Fsn F = qv B = qvbsn 12

13 Bounda condtons e 1nom n e 1 e 1tang medum2 medum1 Bounda condtons of the tangent electc ( 2) ( 1) e = e n n e = e + e e = e + e 1 1nom 1tang 2 2nom 2tang e 2tang = e 1tang Bounda condtons of the nomal component of electc nducton d = d d d n 1 n 2 = ε e = ε e ε e = ε e n 1 1 n

14 Refeences 1. G. Manaa, A. Monocho,.Nepa, Appunt d CampElettomagnetc 2. J. Slate, N. Fank, Electomagnetsm 3. M. Schwat, ncple of Electodnamcs