Chapter 3. Electric Flux Density, Gauss s Law and Divergence

Transcription

1 Chapter 3. Electric Flu Denity, Gau aw and Diergence Hayt; 9/7/009; Electric Flu Denity Faraday Eperiment Cncentric phere filled with dielectric material. + i gien t the inner phere. - i induced at the inner urface f the uter phere. Faraday cncluin: There wa mething diplaced frm the inner t the uter phere independent f the dielectric material. It i called diplacement, diplacement flu, r electric flu. The electric flu Ψ i prduced by the charge Ψ = The electric flu denity D i defined a flu per unit area. Due t ymmetry the electric flu i unifrmly ditributed in the gap. At the inner phere : D( r = a) = a ˆ r 4πa At the uter phere : D( r = b) = a ˆ r 4πb Fr a r b : Dr = a ˆ r 4πr Relatin between D and E Reduce the inner radiu t er while retaining the charge. Pint charge. The electric flu denity frm a pint charge D = a ˆ r 4πr (1) Cmpare thi with the electric field frm a pint charge E = a ˆ r (Free pace) 4πε r Therefre, in free pace D = ε E (Free pace) ()

2 Hayt; 9/7/009; 3- E and D in the preence f lume charge ( r ') d' E ( r ) = a ' ˆ V R (Free pace) (3) 4πεR ( r ') d' D( r ) = a ' ˆ V R (4) 4π R E and D inide dielectric material Faraday reult hw that (1) i applicable inide dielectric., (4) i al applicable inide dielectric. But (3) cannt be ued inide dielectric Cmplicated relatin between E and D in thi cae. 3. Gau aw Faraday eperimental reult can be generalied t Gau aw a The electric flu paing thrugh any cled urface i equal t the ttal charge encled by that urface Charge encled by a urface

3 Hayt; 9/7/009; 3-3 The incremental urface element i repreented by a ectr, It i parallel t an utward nrmal at the urface. At a pint P n the urface Incremental urface element = Δ Electric flu denity = D Δ. The electric flu cring the urface i ΔΨ = D, nrmalδ = D Δ : Nte D de nt cr the urface, parallel The ttal electric flu cring the encled urface Ψ= ΔΨ= D Δ encled urface The mathematical frm f Gau aw D Ψ= Δ = The ttal charge can hae different frm = n fr eeral pint charge = dl fr line charge = d fr urface charge l = d fr lume charge Gau aw D d = d = l A pint charge at the rigin f the pherical crdinate elect a phere f radiu a a a Gauian urface. The electric field intenity i E = a ˆ r 4πε r Uing the cntitutie relatin, D = a ˆ r 4πr D = ε E, At the urface f the phere D ˆ = a r 4πa The differential area n the phere d = a inθ dθ dφaˆ r The left ide f the Gau law D π π d ˆ in ˆ in a a d d a d d 4π a θ θ φ r r φ 0 θ 04π θ θ φ = = A epected frm Faraday eperiment

4 Hayt; 9/7/009; Applicatin f Gau aw: me ymmetrical charge ditributin Gau law D d = Find D when i gien. Gau law cannt be ued withut the ymmetry Nt eay if the Gauian urface i nt chen martly. 1. D d D d r 0 n the Gauian urface. D i nrmal t the urface D i tangential t the urface. D d 0, but D = cntant. Eample A pint charge at the rigin The electric flu in radial directin. Gauian urface huld be a phere f radiu a centered at the rigin t make the electric flu nrmal t the phere. A phere meet the tw requirement. D d D d D φ= π θ= π a inθ d θ d φ 4π a D phere φ= 0 θ= 0 Frm Gau law D = 4πa ince a i arbitrary, thi can be etended t 3-D pace ˆ D = a r and E = a 4πr 4 ˆ r πε r Eample An infinite line charge alng -ai Check the ymmetry f the field 1. Of what ariable D i a functin?. Which cmpnent f D are preent? We epect D frm an infinite line charge t be 1. N change f D alng φ and.. D i in radial directin nly D = D( ) aˆ Gauian urface huld be a cylindrical urface. It huld be cled by plane urface at the tp and the bttm. D = D( ) aˆ

5 The cled urface integral i D d D d 0 d 0 d D π + + = d φ d = D π ide tp bttm = 0 φ = 0 Hayt; 9/7/009; 3-5 Therefre frm the Gau law D = D π π π E = πε Eample An infinite caial cable Etremely difficult t le by Culmb law. Aume a urface charge denity at the inner cnductr. Frm ymmetry we knw D = D( ) aˆ The Gauian urface huld be a circular cylinder f length and radiu. Then, the cled urface integral becme, D d = D π The encled charge i π = adφd = πa = 0 φ = 0 Therefre a D = aˆ aˆ π = πa : It lk the ame a fr the line charge. urface charge denitie The electric flu tart frm the inner cnductr and end at the uter cnductr. The ttal charge at the uter cnductr i a uter = πa, inner, uter =, inner b πb uter, Other chice f the Gauian urface A Gauian urface f radiu > b D = 0 ince the ttal charge i er A Gauian urface f radiu < a D = 0 ince n net charge inide cnductr

6 Hayt; 9/7/009; Applicatin f Gau aw: Differential Vlume Element If n ymmetry, che ery mall Gauian urface. D i aumed cntant n the Gauian urface. A pint P i urrunded by a mall rectangular b with ide Δ, Δy and Δ. D at P i gien by D = D = D aˆ + D aˆ + D aˆ. y y Apply Gau law n the urface D d = = + frnt + back left right tp bttm Δ Dfrnt Δfrnt Dfrnt ΔyΔaˆ frnt D, frnt ΔyΔ D + ΔyΔ Δ Aumed cntant. Frm Taylr erie, D, frnt D+ Δ Dback Δback Dback ΔyΔ( aˆ ) D back, back ΔyΔ D ΔyΔ + Δ Δ Δ frnt back y

7 Hayt; 9/7/009; 3-7 imilarly y + Δ Δ y Δ right, and left y + Δ Δ Δ tp bttm y Therefre y D d + + ΔΔyΔ = y (7) y + + Charge inide y Δ i the lume encled by the b. Thi equatin i mre accurate fr 0 (8) 3.5 Diergence Fr 0 Eq. (7) becme D d y lim = + + = lim = 0 y 0 Vlume charge denity The abe equatin can be eparated int tw. D d y (1) lim = y A relatin between partial deriatie and cled urface integral. It can be applied t any ectr functin. The left ide i called diergence f D r di D. The diergence f the ectr flu denity D i the utflw f flu frm a mall cled urface per unit lume a the lume hrink t er.

8 Diergence in different crdinate ytem y di D = + + y 1 di ( ) 1 φ D = D + + φ φ di D = ( r Dr ) + ( inθ Dθ ) + r r r inθ θ rinθ φ Hayt; 9/7/009; 3-8 In rectangular crdinate In cylindrical crdinate In pherical crdinate () y Gau law y + + = 3.6 Mawell Firt Equatin(Electrtatic) ummarie preiu reult D d di D = lim : Mathematical definitin f diergence 0 y di D = + + y di D = : Diergence in Carteian crdinate : Gau law Different eprein f Gau law D d = D d = Gau law in integral frm lim D d = lim 0 0 di D = Gau law in pint frm

9 Eample A pint charge at the rigin D = a ˆ r 4πr Hayt; 9/7/009; 3-9 In pherical crdinate φ di D = ( r Dr ) + ( inθ Dθ ) + r r r inθ θ r inθ φ ince D = D = 0, θ di D = φ 1 r r = 0 r 4π r, fr r The Vectr Operatr and the Diergence Therem The del peratr i defined a aˆ ˆ ˆ + ay + a y Uing the cncept f dt prduct y D = aˆ ˆ ˆ ( ˆ ˆ ˆ + ay + a Da + Dyay + Da) + + y y We nte that In general, D D i jut equal t di D in Carteian crdinate. repreent di D in any crdinate ytem. Diergence Therem Frm Gau law D d = d Dd V V Therefre the diergence therem tate D d = Dd V The integral f the nrmal cmpnent f a ectr field er a cled urface i equal t the integral f the diergence f thi ectr field thrughut the lume encled by the urface. y Prf: Cnider a differential lume ΔV j bunded by j Frm the definitin f dia A V = A d e j Δ Add all ΔV j M N j j j O P N N lim A ΔVj lim A d A d ΔV j j M ΔV j j j = P = M j = P e j M N O P Integral at the eternal urface Ad V e j An internal urface i hared by tw adjacent lume. (Oppite urface nrmal)

10 Hayt; 9/7/009; 3-10