Chapter 2 Maxwell s Equations in Integral Form

Transcription

1 9/3/9 - hapter Mawell s Equatios i Itegral Form. The lie itegral. The surface itegral.3 Farada s law. Ampere s circuital law.5 Gauss Laws.6 The Law of oservatio of harge.7 Applicatio to static fields - Mawell s Equatios Edl d dt d Dd dv V Electric field Magetic itesit flu desit Vm Wb m harge desit m 3 Hdl Jd d dt Magetic urret field itesit desit Am Am Displacemet flu desit m D d d -. The Lie Itegral Work doe i carrig a charge from A to i a electric field: E E Δl A Δl W A dw j j

2 9/3/9-3 cos dw qe l qej l j cos j qe l j j j j W q E l j A j j j j Defiitio: Voltage betwee A ad V WA A Ej lj q j - Defiitio: Voltage betwee A ad I the limit, V A A E dl = Lie itegral of E from A to. Defiitio (ote): electromotive force (emf) d = Lie itegral of E aroud E l the closed path. Notes: oservative ad Nocoservative fields oservative fields Lie itegral aroud a closed path is ero E.g. gravit, static E field No-coservative fields Lie itegral aroud a closed path is oero E.g., time-varig field -5

3 9/3/9-6 Eample A If E d l = L R the E dl A is idepedet of the path from A to (coservative field) Edl Edl Edl ARLA AR LA AR Edl Edl AR Edl = Edl AL AL -7 Eample: lie itegral F a a a, (,,3) alog the straight lie paths, F d l (,,) from (,, ) to (,, ), from (,, ) to (,, ) ad the from (,, ) to (,, 3). (,, 3) (,, ) (,, ) (,, ) -8 From (,, ) to (,, ), ; d d,, F, F dl,, From (,, ) to (,, ),, ; d d F a dl d a d a d a d a,, Fdl, Fdl,, 3

4 9/3/9-9 From (,, ) to (,, 3),, ; d d F a a a, dl d a (,,3), F dl d, d 6 (,,) (,,3) F dl 6 6 (,,) - I fact, F dl a a a d a d a d a d d d d,,3 F dl,,3,,3,, d,,,, 3 6, idepedet of the path. -. The urface Itegral Flu of a vector crossig a surface: a Flu = ()() Δ a Δ a Δ Flu = Flu ( cos ) cos a

5 9/3/9 - Normal aj j j α j Δ j I the limit, Flu j j j j j Flu, = d d = urface itegral of over. = urface itegral of over the closed surface. -3 Eample: D. (a) A a a, a a A, A d A d A d - (b), a a Aa a dd da A d d d A d d d 8 A d 8 5

6 9/3/9-5 (c) A a a, a a dd da A d d d A d dd A d -6 (d) From (c), A d d d A d dd ( ) d 3 A d 3 + = Review Questios What is the magetic flu crossig a ifiitesimal surface orieted parallel to the magetic flu desit vector?.8. For what orietatio of a ifiitesimal surface relative to the magetic flu desit vector is the magetic flu crossig the surface a maimum?.. Provide a phsical iterpretatio for the closed surface itegrals of a two vector fields of our choice. 6

7 9/3/9.3 Farada s Law * E dl d dt -8 d d * Note: this defiitio/statemet of Farada s law is cotroversial -9 E dl = Voltage aroud, the electromotive force (emf) aroud (but ot reall a force, also called electromotace) Vm m, or V. d = Magetic flu crossig, Wb m m, or Wb. d dt d = Negative of the time rate of chage of magetic flu crossig, Wb s, or V. - Importat osideratios () Right-had screw (R.H..) Rule The magetic flu crossig the surface is to be evaluated toward that side of a right-had screw advaces as it is tured i the sese of. 7

8 9/3/9 - () A surface bouded b The surface ca be a surface bouded b. For eample: R R O Q O Q P P This meas that, for a give, the values of magetic flu crossig all possible surfaces bouded b it is the same, or the magetic flu bouded b is uique. - (3) Imagiar cotour versus loop of wire There is a emf iduced aroud i either case b the settig up of a electric field. A loop of wire will result i a curret flowig i the wire. () Le s Law tates that the sese of the iduced emf is such that a curret it produces, if the closed path were a loop of wire, teds to oppose the chage i the magetic flu that produces it. -3 Thus the magetic flu produced b the iduced curret ad that is bouded b must be such that it opposes the chage i the magetic flu producig the iduced emf. (5) N-tur coil (5) N-tur coil For a N-tur coil, the iduced emf is N times that iduced i oe tur, sice the surface bouded b oe tur is bouded N times b the N-tur coil. Thus 8

9 9/3/9 - emf N d dt where is the magetic flu liked b oe tur. -5 D.5 si ta cos ta (a) d = si t d E d l si t dt cos V t emf dec emf > ic. emf < 3 t t Le s law is verified. 9

10 9/3/9-7 (b) d si t cos t si t E d l d si dt cos V t t -8 (c) d sit cos t si t E d l d si t dt cos t V -9 E. Motioal emf cocept l d v a coductig rails = a d = l = l vt coductig bar v t

11 9/3/9-3 E dl d d dt d l v t dt lv This ca be iterpreted as due to a electric field E F Q v a iduced i the movig bar, as viewed b a observer movig with the bar, sice l v l v a d a l E dl -3 where F Qv Qv a a Qv a is the magetic force o a charge Q i the bar. Hece, the emf is kow as motioal emf. Review Questios -3.. To fid the iduced emf aroud a plaar loop, is it ecessar to cosider the magetic flu crossig the plae surface bouded b the loop? Eplai..7. How would ou oriet a loop atea i order to receive maimum sigal from a icidet electromagetic wave which has its magetic field liearl polaried i the orth-south directio?