Gauss s Law: Circuits

Transcription

1 Gauss s Law: Cicuits Can we have excess chage inside in steady state? E suface nˆ A q inside E nˆ A E nˆ A left _ suface ight _ suface q inside 1

2 Gauss s Law: Junction Between two Wies n 2 <n 1 u 2 <u 1 i 1 =i 2 n 1 Au 1 E 1 = n 2 Au 2 E 2 nu E E E nu 2 2 E suface nˆ A q inside q ( E A E A) inside 1 2 Thee is negative chage along the inteface! 2

3 Gauss s Law fo Magnetism Dipoles: Electic field: + and chages can be sepaated Magnetic field: no monopoles Suppose magnetic dipole consists of two magnetic monopoles, each poducing a magnetic field simila to the electic field. One cannot sepaate them total magnetic chage is zeo. E suface nˆ A q inside o suface Bnˆ A Bnˆ A Gauss s law fo magnetism 4

4 Pattens of Magnetic Field in Space Is thee cuent passing though these egions? Thee must be a elationship between the measuements of the magnetic field along a closed path and cuent flowing though the enclosed aea. Ampee s law 6

5 Quantifying the Magnetic Field Patten 2I Bwie 4 Culy chaacte intoduce: B dl Bdl 4 2 I dl 2I 2 4 Bdl I Simila to Gauss s law (Q/ ) Will it wok fo any cicula path of adius? 7

6 A Noncicula Path Bdl I Need to compae B dl Bdl dl 1 1 dl 2 2 B1 dl 1 and B2 dl2 2I B1 4 B I B 1 2 B2 dl2 B2dl2 B1 dl1 2 1 B2 dl2 B1 dl1 8

7 Cuents Outside the Path Bdl I Need to compae B1 dl 1 and B2 dl2 dl 1 1 dl 2 1 B2 B1 2 2 B dl B dl Bdl fo cuents outside the path 9

8 Thee Cuent-Caying Wies B dl I 1 1 B2 dl I2 B 3 dl B 1 B2 B3 dl I1 I2 Ampee s law B dl I inside _ path 1

9 Ampèe s Law B dl I inside _ path All the cuents in the univese contibute to B but only the ones inside the path esult in nonzeo path integal Ampee s law is almost equivalent to the Biot-Savat law: but Ampee s law is elativistically coect 11

10 Ampee s Law: A Solenoid B dl I inside _ path Numbe of wies: (N/L)d What is B dl on sides? B outside is vey small B dl Bd Bd IN / Ld B IN L Unifom: same B no matte whee is the path (solenoid) 15

11 Ampee s Law: A Tooid B dl I inside _ path Symmety: B path B 2 IN B 2NI 4 Is magnetic field constant acoss the tooid? 16

12 Maxwell s Equations Thee equations: Gauss s law fo electicity Gauss s law fo magnetism Ampee s law fo magnetism E nda ˆ BndA ˆ B dl q inside I inside _ path Is anything missing? Ampee s law fo electicity E d l (incomplete) 19

13 Chapte 23 Faaday s Law

14 Maxwell s Equations (incomplete) E nda ˆ q inside Gauss s law fo electicity BndA ˆ E dl B dl I inside _ path Gauss s law fo magnetism Incomplete vesion of Faaday s law Ampee s law (Incomplete Ampee-Maxwell law) Fist two: integals ove a suface Second two: integals along a path Incomplete: no time dependence 33

15 Electic and Magnetic Fields Electic and magnetic fields ae inteconnected Moving chages make magnetic fields Magnetic fields affect moving chages Electic and magnetic fields depend on efeence fame Change efeence fame: magnetic field is conveted into electic and vise vesa. Today we e looking at time vaying magnetic field: It can poduce electic field! (magnetic induction) 35

16 Cuents Due to Magnetic Foces Metal ba F qe qv B m F e v B F m polaization Static equilibium: F ( e) E ( e) v B E vb Metal is in static equilibium: but E inside! What is V? V EL vbl How about moving B (o changing B)? 36

17 Non-Coulomb Wok Non-Coulomb foce dives e against F E F m Non-Coulomb wok: L F NC evbl What is emf of this battey? Non-Coulomb wok pe unit chage: F NC L e V vbl emf vbl motional emf Ba may have some esistance int : V emf int I 37

18 Changing Magnetic Field Solenoid: inside B NI d outside B Constant cuent: thee will be no foces on chages outside (B=, E=) What if cuent is not constant in time? Let B incease in time E~1/ E~dB/dt Non-Coulomb E NC! 38

19 Diving Cuent by Changing B E NC causes cuent to un along the ing What is the suface chage distibution? Positive chage buildup on the oute suface. What is emf and I? emf E dl E NC 2 2 E NC 2 R NC 2 I Ring has esistance, R 39

20 Quantitative Relationship Between B and emf Can obseve expeimentally: I=emf/R E NC ~emf 1. E NC ~db/dt 2. E NC ~ coss-section of a solenoid emf d dt 2 B 1 1 4

21 Magnetic Flux emf d dt 2 B 1 1 B magnetic flux mag on the aea encicled by the cicuit Magnetic flux on a small aea A: B nˆ A B A Definition of magnetic flux: B nda ˆ B da mag 41

22 Faaday s Law emf d mag dt Faaday s law cannot be deived fom the othe fundamental pinciples we have studied Fomal vesion of Faaday s law: d E ˆ NC dl B nda dt Sign: given by ight hand ule Michael Faaday ( ) 42

23 Including Coulomb Electic Field d E ˆ NC dl B nda dt Can we use total E in Faaday s law? Etotal dl ENC EC E dl NC dl E C dl = d E dl B nda ˆ dt 43