VEKTORANAY Kursvecka 3 GAU THEOREM and TOKE THEOREM Kaptel 6 7 dor 51 82
TARGET PROBEM Do magnetc monopoles est? EECTRIC FIED MAGNETIC FIED N +? 1
TARGET PROBEM et s consder some EECTRIC CHARGE 2 - + + 1 and two closed surfaces, 1 and 2 If B d 1 does not contan an charge. It has no sources and no snks: no feld lnes destroed and no feld lnes created nsde 1 1 E d = 2 do contan a charge. It has a snk. Feld lnes are destroed nsde 2 2 E d then magnetc monopoles est!! To calculate ths ntegral we need: to ntroduce the dvergence of a vector feld A, dv A the Gauss theorem one of the Mawell equatons: A d = V dvadv dv B = 2
THE DIVERGENCE In cartesan coordnates, the dvergence of a vector feld s: DEFINITION dva + + A (1) It s a measure of how much the feld dverges (or converges) from (to) a pont. et s suppose that A s the veloct feld of the water n a pool et s make a hole n the pool Far from the hole, the veloct s almost constant. The dvergence s almost ero. Close to the hole the veloct s changng a lot. The dvergence s hgh. If there s no hole (and no source ) dv A = The dvergence s also a measure of sources or snks (ths concept wll be more clear at the end of the lesson) 3
A d = V dvadv THE GAU THEOREM where s a closed surface that forms the boundar of the volume V and A s a contnuousl dfferentable vector feld defned on V. (2) ˆn 2 d 2 V =f 2 (,) 2 ˆn 1 d 1 1 =f 1 (,) d d p dd = d nˆ eˆ = d eˆ 2 2 2 dd = d nˆ eˆ = d eˆ 1 1 1 4
THE GAU THEOREM PROOF V dvadv = + + = V ddd + + ddd ddd ddd V V V et s calculate the last term: f2 (, ) V f (, ) p 1 p [ (,, (, )) (,, (, ))] ddd = dd d = A f A f dd = 2 1 2 1 dd s the projecton on p of the small element surfaces on d 1 and d 2. dd = eˆ nˆ d = eˆ nˆ d Therefore: 1 1 2 2 = A (, f, (, )) eˆ nd ˆ + A(, f, (, )) eˆ nd ˆ = Aeˆ nd ˆ 2 2 2 1 1 1 Whch means: V dv = A ˆ ˆ e nd (3) 5
THE GAU THEOREM PROOF In the same wa we get: V V dv = A ˆ ˆ e nd dv = A ˆ ˆ e nd (4) (5) Addng together equatons (3), (4) and (5) we fnall obtan: = + + = dvadv ddd ddd ddd V V V V A eˆ nd ˆ + A eˆ nd ˆ + A eˆ nd ˆ = A d 6
PROOF THE GAU THEOREM What f we consder a more complcated volume? V V 2 We dvde the volume V n smaller and smpler volumes V = V1+ V2 +... = V V dvadv = dvadv = V V 1 A d = A d 7
PHYICA INTERPRETATION uppose that vr ( ) s the veloct feld of a flud (homogeneous and ncompressble) et s appl the Gauss theorem to a volume V of the flud vd = V dvvdv () It s the flud volume per second [m 3 /s] that moves out (n) from the closed surface If there are no snks and no sources, then the amount of flud that enters n s equal to the amount of flud that ets from. Ths mples that the flow v d s ero. Therefore, dv( v ) = dv( v ) = dv( v ) < dv( v ) > No snk and no source flu s destroed and there s a snk flu s created and there s a source 8
TARGET PROBEM Do magnetc monopoles est? If B d then magnetc monopoles est!! We can use the Gauss theorem and the Mawell equaton to calculate B d Gauss B d = Mawell dvb = V dvbdv B d = Magnetc monopoles do NOT est 9
WHICH TATEMENT I WRONG? 1- The dvergence of a vector feld s a scalar (ellow) 2- The dvergence s related to a measurement of the flu (red) 3- The Gauss theorem translates a surface ntegral nto a volume ntegral (green) 4- The Gauss theorem can be appled also to a non closed surface (blue) 1
VEKTORANAY TOKE THEOREM
TARGET PROBEM A current I flows n a conductor How can we calculate the magnetc feld? I B We need: Defnton of the curl (or rotor) of a vector feld rot A The tokes theorem A dr = rota d A law that relates the current wth the magnetc feld: the fourth Mawell s equaton (wth statc electrc feld): rotb = μ j 11
THE CUR rot A DEFINITION (n cartesan coordnate) rot A eˆ ˆ ˆ e e A = =,, A A A rot stands for rotaton In fact, the curl s a measure of how much the drecton of a vector feld changes n space,.e. how much the feld rotates. In ever pont of the space, rota s a vector whose length and drecton charactere the rotaton of the feld A. The drecton s the as of rotaton of A The magntude s the magntude of rotaton of A 12
THE CUR rota EXAMPE A(,, ) = (,,) Evdent rotaton rota = (,, 2) Drecton: the drecton s the as of rotaton,.e. perpendcular to the plane of the fgure The sgn (negatve, n ths case) s determned b the rght-hand rule Magntude: the amount of rotaton In ths eample, t s constant and ndependent of the poston,.e. the amount of rotaton s the same at an pont. 13
THE CUR rota PHYICA INTERPRETATION Consder the rotaton of a rgd bod around the -as P The coordnates of a pont P on the bod located at the dstance a from the -as and at = changes n tme: () t = acosωt ( t) = asnωt = a The veloct of the pont P s: v () t = aωsn ωt = ω() t v () t = aωcos ωt = ω() t v = ω, ω, v = Therefore rot v = (,, 2ω ) ( ) ω = 1 2 rot v 14
THE TOKE THEOREM A dr = rota d where A s a vector feld, s a closed curve and s a surface whose boundar s defned b. A must be contnuousl dfferentable on ˆn 15
PROOF THE TOKE THEOREM Fve steps: 1. We dvde n man smaller (nfntesmal) surfaces: = 2. We project on: the -plane the -plane the -plane 1 2 3 3. We prove the tokes theorem on (the onl dffcult part) 4. We add the results for the projectons together and we obtan the tokes theorem on 5. We add the results for together and we obtan the tokes theorem on 16
THE TOKE THEOREM PROOF et s consder the plane surface located n the -plane (.e. =constant= ) wth boundar defned b the curve eˆ et s calculate A dr = A dr A (,, ) d + A(,, ) d + A(,, ) d Term 1 Term 2 Term 3 Term 3 = (=constant! d=) Term 1 2 =g() A (,, ) d b a = A d= + 1 2 1 2 (,, ) A (,, ) d+ A(,, ) d= A (, f ( ), ) d+ A (, g( ), ) d= a b a 1 b =f() 17
THE TOKE THEOREM PROOF [ ] b b b = A (, f ( ), ) d A (, g( ), ) d= A (, f ( ), ) A (, g( ), ) d= a a a (,, ) dd dd dd b f ( ) b g( ) = = a g( ) a f ( ) Therefore we get: Term 1 A (,, ) d= dd In a smlar wa: Term 2 Addng Term 1, 1 Term 2 A (,, ) d= dd Term 2 and Term 3: 3 A dr = dd It s the -component of rota!! 18
THE TOKE THEOREM o can rewrte t as: A dr = ( rota) dd = ( rota) eˆ d In a smlar wa we have: A dr = ( rota) eˆ d A dr = ( rota) eˆ d dd = eˆ nd ˆ = eˆ d Now let s add everthng together: A dr + A dr + A dr = A dr ( rota) eˆ d + ( rota) eˆ d + ( rota) eˆ d = rota d 19
PROOF A dr = rota d But we are nterested n the whole. o we add these small contrbutons altogether: THE TOKE THEOREM 1 2 3 rota d = rota d = A dr = A dr A dr = rot A d 2
TARGET PROBEM Now we can easl calculate the magnetc feld B at a dstance a from the conductor! Ampere s law rotb Where j s the current denst: = μ j I B B dr = rotb d = μ j d = μ j d = μ I tokes Ampere I = j d 21
THE GREEN FORMUA IN THE PANE THEOREM (7.1 n the tetbook) D Q P dd = Pd + Qd ( ) PROOF We can start from tokes theorem A dr = rot A d A dr = ( A ) ( ) d + Ad + Ad = Ad + Ad But we are n a plane, so we can assume A=(A,A,) rot A d = eˆ ˆ edd eˆ eˆ eˆ A A =1 D A dd = A d + A d ( ) whch s the Green formula for P=A and Q=A 22
CUR FREE FIED AND CAAR POTENTIA DEFINITION: A vector feld A s curl free f rota=. ometmes called rrotatonal THEOREM (7.5 n the tetbook) rota= has a scalar potental φ, A=gradφ. PROOF (1) rot A = Adr = rotad = If the crculaton s ero, then the feld s conservatve and has a scalar potental. ee theorem 4.5 n the tetbook. (2) A = gradφ rot A rot grad rot φ φ, φ, φ = = = eˆ ˆ ˆ e e φ φ φ φ φ =, K, K = (,,) (or see a proof on net week) 23
OENOIDA FIED AND VECTOR POTENTIA DEFINITION: A vector feld B s called solenodal f dvb = DEFINITION: The vector feld B has a vector potental A f, B = rota THEOREM (7.7 n the tetbook) B has a vector potental A, B = rota dvb = PROOF (1) B has a vector potental B rota = dvb = dv( rota) = (2) dvb = et s tr to fnd a soluton A to the equaton B = rota We start lookng for a partcular soluton A* of ths knd: ( (,, ), (,, ),) * * * A= A A 24
CUR FREE FIED AND CAAR POTENTIA PROOF Assumng B=rotΑ we obtan: * * = B A(,, ) = B( d,, ) + F (, ) * * = B A(,, ) = B( d,, ) + G (, ) * * B B F G = B d+ d = B But dvb= B B B + = B F G d + = B = B (,, ) B (,, ) F G = B (,, ) A soluton to ths equaton s: F(, ) = G (, ) = B (,, ) d The general soluton can be found usng : ( ) (,, ) (,, ), (,, ), * A= B d B d B d ( ) B=rotΑ * * * rot A A = B B = A A = gradψ A = A + grad ψ 25
WHICH TATEMENT I WRONG? 1- The curl of a vector feld s a scalar (ellow) 2- The curl s related to the lne ntegral of a feld along a closed surface (red) 3- tokes theorem translates a lne ntegral nto a surface ntegral (green) 4- The tokes theorem can be appled onl to a closed curve (blue) 26