8.022 (E&M) Lecture 13. What we learned about magnetism so far

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1 8.0 (E&M) Letue 13 Topis: B s ole in Mawell s equations Veto potential Biot-Savat law and its appliations What we leaned about magnetism so fa Magneti Field B Epeiments: uents in s geneate foes on hages in motion v Foe eeted on hage q with veloity v: F = q B Eplanation: thee must eist a magneti field B Speial Relativity: B is just E seen fom anothe efeene fame Ampee s Law: 4 π B ids = I C enl Appliation: B geneated by uent in a : I B = ˆϕ 1

2 Divegene of B I B = ˆ ϕ Conside the B podued by a of uent: Calulate its divegene in Catesian oodinates: Given y and ˆ ϕ ˆ os y ϕ - ˆ sin ϕ y ˆ - y ˆ = + = = + y + y I y ˆ y ˆ I y y B = - i B = - = 0 + y + y ( + y ) ( + y ) This is a geneal popety of the magneti field: i B = 0 Simila equation fo E: i E = 4πρ The divegene of E is elated to the density of eleti hages The divegene of B must be elated to the density of magneti hages Æ Magneti monopole don t eist (Thee may be magneti monopoles leftove fom the Ealy Univese, but neve obseved epeimentally so fa) G. Siolla MIT 8.0 Letue 13 3 Ampee s law in diffeential fom Apply Stoke s theoem to Ampee s law: C 4π Bds i = I enl 4π Bds = B i ds = J i ds i S S C 4π S B J ids = 0 fo any sufae 4π Æ Ampee s law in diffeential fom: B = J G. Siolla MIT 8.0 Letue 13 4

3 Towad Mawell s equations Let s ollet all the equations in diffeential fom that we found so fa: i E = 4πρ Å Relates E and hage density (ρ) - Gauss i B = 0 Å No magneti monopoles! E = 0 Å E is a onsevative field 4π B = J Å Relates B and its soues (J) - Ampee Not omplete Mawell s equations yet, but we ae getting lose G. Siolla MIT 8.0 Letue 13 5 Veto potential A Definition of potential fo eleti field: (P) = wok needed to move a unit hage fom efeene to P Relationship between φ and E: E = φ Hidden advantage: If E=- φ E 0 beause ( φ ) = 0 φ Can we intodue something sim la fo B? Goal: enfoe div B=0 Sine i f = 0 fo any f, we define φ i B A A is alled veto potential in analogy with φ A is not onneted to wok o enegy (but to angula momentum) G. Siolla MIT 8.0 Letue

4 Non Uniqueness Eletostatis: given a hage distibution and bounday onditions Æ potential φ is uniquely identified Magnetism: does it wok the same fo A? No, thee ae infinite numbe of A oesponding to a single B Eample: B = Bz ˆ. Find A that eates this B f ield. 0 Requiements: Q: what uent eates this B? A A z y B= ˆ = 0 A = yb0 y z A = B ˆ 0 y A Az B y = = 0 Possible solutions: z B 0 A = ( y ˆ + y ˆ) A A y B= z = B 0 y A =...infinit e ot h es! We ae given one oupon to simplify equations when needed G. Siolla MIT 8.0 Letue 13 7 Poisson s equation fo A Eletostatis: E = φ i E = 4 πρ Magnetism: B = A 4 π 4π 4 π A = J ( ia ) A = J B = J / We used the identi ty: A = ( i A ) A (Pset#7) Use you oupon now! φ = 4πρ Poisson's equation 4π Choosing i A=0 A = J G. Siolla MIT 8.0 Letue

5 Solving Poisson s equation fo A 4π How do you so ve A = J l? Think of it in atesian oodinates: 4 π A = J 4π A Y = J Y 4 π A Z = J Z ρ Rem em be Poisson's equation φ = 4πρ and its solution φ = dv V J 1 J Sam e as ou new e quati on if eplae φ A and ρ A = dv V I dl Fo uent flow i ng in a w ie: A = G. Siolla MIT 8.0 Letue 13 9 Biot-Savat Law I dl Find B podued fom uent knowi ng that A =. I dl I dl B= A= = Using the fat that (ab)=a( b)+( a) b: I 1 1 I 1 1 = ( dl ) dl = ( dl ) d + + l 1 ˆ Sine d l =0 and = : I ˆ B = dl G. Siolla MIT 8.0 Letue

6 Biot-Savat Law: illustation Biot-Savat: db I ˆ = dl dl ˆ db db is pependiula to uent and to adial dietion E.g.: if you have dl //, // y Æ B//z G. Siolla MIT 8.0 Letue Appliation of Biot-Savat: B fom loop of uent z y Calulate B eated by a loop of uent Radius: R Distane fom ente of the loop: z Solution on ais Apply Biot-Savat Detemine dietion of db Symmety Æ only omponent // z suvives I B= ( db ) = dl ˆ sin θ z wi e dl ˆ = dl = Rd ϕ ; sin θ = R / ; = R + h I π π IR B = R sin θ d ϕ ˆ z = ˆz B 90-θ π I = ˆ z 0 3 / loop ente G. Siolla MIT ( R z Letue ) 13 R 1 6

7 Appliation of Biot-Savat: B fom solenoid What if we stak a N ings ove a length L? Use esult of single loop + supeposition: π R Single ing: db = (R +z ) 3/ Integate on all ings ( in the midd le of the solenoid) L / π R π ni L / R dz B = nidz = - L / 3/ R ( + z ) - L / 3/ ( R + z ) π ni L = L + 4 R Fo L>>R: = B 4 π ni di G. Siolla MIT 8.0 Letue 13 With n=n/l 13 Solenoid and Ampee s law One an pove that B outside the solenoid is =0 Ampee an be used to simply pove that B does not depend on : 4 π Bdl i = I etang le enl Sine B is //z and pesent only inside the solenoid: 4 π 4 π N 4 π B()L= NI B ( ) = I = ni no dependene on R L G. Siolla MIT 8.0 Letue

8 Solenoid s magneti field: demos Epeted: 4π B = ni I I Can we test this epeimentally? G1: B fom a single using ion filings G13: B fom s G16: B inside solenoid G. Siolla MIT 8.0 Letue Moe demos on magneti fields Moe demos: G14: map B aound a using a ompass G9a: ollapsing solenoid Can you eplain what s happening? G18: Long solenoid Long solenoid with N =760, I=4.5 ma, length = 46 m tun (R=10 Ω, L=18 mh) What is B? 4 4 B π ni π 760 = = 4.5 = i i Veify with Hall pobe Gauss??? G. Siolla MIT 8.0 Letue

9 Thompson s epeiment: vaiation Vaiation on a theme: instead of aneling effets of E and B, one ould tune the fields and measue the adius of uvatue of the eleton beam. Paametes of the poblem: V= 300 V I= 1.4 A R= 5 m Solution: e/m= C/Kg (f: C/Kg) G. Siolla MIT 8.0 Letue Summay and outlook Today: Towad Mawell s equations: Veto Potential: B A 4π ib = 0 and B = J Biot-Savat Law: db I ˆ = dl Net time: What happens when B vaies in time? Faaday s and Lenz s laws and thei appliations G. Siolla MIT 8.0 Letue