TUTORIAL 6. Review of Electrostatic

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Sabina Gilbert
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1 TUTOIAL 6 eview of Electrotatic

2 Outlie Some mathematic Coulomb Law Gau Law Potulatio for electrotatic Electric potetial Poio equatio Boudar coditio Capacitace

3 Some mathematic Del operator A operator work a a vector. Gradiet of a calar How fat a fuctio varie whe it compoet var, ad the directio i the oe of maximum rate of icreae. Divergece of a vector How much a vector pread aroud a poit. Curl of a vector How much a vector circulate aroud a poit. $ x ˆ ˆ z ˆ,, x z x z f f f f f f - f. x ˆ ˆ z ˆ.,, x z + x z, div v $ + v v v v x z x xˆ ŷ ẑ $ v x z v v v x z z

5 Some mathematic Divergece Theorem The urface itegral of the curl of a vector field over a ope urface i equal to the cloed lie itegral of the vector alog the cotour boudig the urface. Stoke Theorem The urface itegral of the curl of a vector field over a ope urface i equal to the cloed lie itegral of the vector alog the cotour boudig the urface. v Adv A d $ $ A d A dl $ $ c Two Null idetitie A curl-free field ca be expreed a a gradiet of a calar field. A divergecele field ca be expreed a a curl of a vector field. $$$$$$$ A

6 Static Electric field Field Static Field Space ditributio of a vector or calar parameter. Field do ot chage with time. Static Electric field Electric charge are at ret, ad electric field do ot chage with time. What ivolved? Charge Electrical field iteit Electrical potetial

7 Force betwee two charge Coulomb law q q F k r ˆ r Where: Electric field iteit due to oe poit charge The Electric field iteit of a poitive charge i i the outward radical directio ad ha a magitude proportioal to the charge ad iverel proportioal to the quare of the ditace from the charge q E a E a q $ p p $

8 Coulomb law A tem of dicrete charge The force o charge Q due to all the other charge i the vector um of the force created b the idividual charge. E = 4πε k= q k k k 3 A tem of cotiuou charge Itegratio E E E v L a a a v dv $$ d $ l dl q q q N q 3 Q q i r q 4 q 5 Q

9 Gau law Surface Itegratio S +Q +Q S Gau law Gau law aert that the total outward flux of the E-field over a cloe urface i free pace i equal to the total charge ecloed i the urface divided b ε E d Q $$ Applicatio Fid electric field ditributio give charge ditributio epeciall for mmetric coditio. Special cae : Normal compoet of the electric field iteit i a cotat over a cloed urface.

10 Where to ue Gau law the E-field of charge ditributio with ome mmetr coditio, uch that the ormal compoet of the electric field iteit i cotat over a ecloed urface e.g. Determie the E-field iteit of a ifiitel log, traight, lie charge of a uiform deit l Q E d ρ dl = ρa L l ide S rer rd E ide S rd l a L ll EE $$$ rrd d $ dz = LE a ˆ E a ˆ d = r r r r ide E arer ar $$$ l = rπrle r E d r ide If the E d = cotat coditio doe t exit, the the Gau law i ot ueful. See the followig example

11 Ez E E Er Er E

12 Example Determie the Electric field caued b a pherical cloud of electro with radium b ad a volume charge ρ = ρ ρ = deit for ad for > b. b

13 b

14 Potulatio of Electrotatic +,-$./ : E $$$$$$$$$$ E +,-$./ $ $ c E d E dl Q $$$$$$$$$$ i the permittivit of free pace Electric field metioed here ol due to tatic charge i free pace Phical meaig of them: Implie that a tatic electric field i ot oleoidal Aert that tatic electric field are irrotatioal. $ E d $ E d l c Q $ i a form of Gau Law mea the calar lie itegral of the tatic electric field iteit aroud a cloe path vaihe. Kichhoff oltage Law

15 Electric potetial Scalar field A curl-free vector field could alwa be expreed a the gradiet of a calar field Due to oe charge E, P E dl P mmo to calculate firt q q a a d q +

16 q q E qa a q $ a E a d E = Ep $ 7 P Ed l E a EE a, P p $ $ q q firt, the appl p It i more commo to calculate the The equatio above to electric field ite E a E a above tell u the p electric $ thefield outward radialofdirectio adpoit ha ac The above tell u the iteit a poitive fid E out calar field: Electric potetial proportioal to the quar the outward radial directio ad adiverel ha a magitude proportioal For cotiuou For dicrete charge: ad iverel proportioal to the quare of the ditace from th The above tell u the electric field iteit of a poitive poit charge i i gradietqof a q alwa aof the ditributed charge: beaexpreed tem dicrete charge the directio ha a magitude proportioal to the charge outward d aradial a ad Similar whe we are v ad iverel proportioal to the quare of the ditace from the dv charge P ial uig Coulomb law to v q k P calculate the electric + 5 ulate firt, the appl the equatio above to k k to a iteit field ddue Gau Law cotiuou charge For cotiuou Aq tem of cotiuou charge Waource to lcalculate E: dl Gau Law ditributed charge: l a d E d l E, v Gau v Law dv Wa to calculate E: E v a v 8 dv $$$ E v Wa E: to calculate d E a dv $$$ EWe ca alo a ue d $$$$$ E law to Gau v lv l Q dl E a dv E a d E a dl $$$ $$$$$ $ E d ue Gau law to We ca alo calculate E: l v L 8 Qaert that the total outw law We ca alo ue Gau law to calculate E:Gau Ed $$ urface i free pace i equal to the tot Q divided b

17 Poio equatio Laplacia Laplacia tad for the divergece of the gradiet of = r a x x + a r + a r = x + + r a x x + a r + a r = Poio Equatio i a ecod order differetial equatio hold at ever poit i pace. Laplace Equatio i the goverig equatio for problem ivolvig a et of coductor = ρ ε =

18 Summar Gradiet: I EM: 6 5,, -. 6 / x z E Divergece: Curl: I EM: Gau law i differetial form Gau theorem: Stoke theorem: I EM: F F Fz x z E da 5 Ed x 5 F 3 3 curl F 54 F 54 E S F d = F da curl C A $ 5 E 478 Gau law it $x,,z E 4 x,,z 4 E d E d q r x,,z

19 Capacitace Defiitio The potetial of a iolated coductor i directl proportioal to the total charge o it. d Q=C The cotat C i the capacitace of the iolated coductor. Procedure Chooe a appropriate coordiate tem Aume charge +Q ad -Q o the coductor B Gau law, fid E form Q Fid potetial differece betwee coductor -Q ad +Q Fid C b Q/

20 Example $+$,---$.$/$$3$4,5+$ $.$$3,78$9$,5$+5$:$,5$;<$=83$ 533+$3+$>3/33,$83$-6+.3-$8-$$+3?3$ 3+?4$$+,-../.3.4/ /5.8-.9:9/859.;<8-.9:9/8;6

21 Solutio to example

22 Solutio to example = Q πεl 5 dr = Q l 7 l5 7 r πεl

23 Some hit for project Idea Uig umerical method to olve itegratio problem. Phical meaig of matrix multiplicatio

24 Some hit for project How to calculate Electrical potetial at a poit Poit to Poit φ = Q 4πε x,,z Surface to Poit φ = 4πε σ Whe charge ditributio i cotat d x,, z $ i S i 4 i dx d $ i x xi i z zi

25 Some hit for project How to calculate Electrical potetial at a poit Dicretio of cotiue urface $ N i S i i d z x i 4,, tal coductig plate with N umber of cell, the i,, z x P i i i i z z x x + +

26 Some hit for project How to calculate Electrical potetial at a poit Matrix repreetatio of et of liear equatio / 4 d z z x x N S $ $ / 4 d z z x x N S N N N $ $ / 4 d z z x x N S N N N $ $ / 4 d z z x x N S N N N $ $ $ $ $ : : : : oductig plate with N umber of cell i be, -. / / / / / l m +

27 Some hit for project What i lm ad how to fid it l m S 4 m xm x $ m $ zm z a, m Midpoit of each mall urface tep tartig pit i Matlab: i=; for m=b-a:-a:a-b for =-b+a:a:b-a _coordi=m; x_coordi=; z_coordi=b; i=i+; ed ed Difficult, But ot impoible. - Godfather II a O -b -b+a -b+3a -b+5a -b+7a b-3a b-a L=b b b-a -b+5a -b+a ed poit x x z d d. d. d. -d. -d. -d -d

28 Some hit for project What i lm ad how to fid it x z d d. d. d i xi i zi. -d. -d j xj j zj. -d -d L, L, L, L, Li,j L,

29 Some hit for project be, -. / / / / / l m + σ i = i l ij

Fig. 1: Streamline coordinates

Fig. 1: Streamline coordinates 1 Equatio of Motio i Streamlie Coordiate Ai A. Soi, MIT 2.25 Advaced Fluid Mechaic Euler equatio expree the relatiohip betwee the velocity ad the preure field i ivicid flow. Writte i term of treamlie coordiate,