Section 2 3 TRƯỜNG ĐIỆN TỪ

No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]

Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India 2.3 Faraday’s Law 2.3-3 Faraday’s Law d C E • dl  – dt S B • dS B S C dS 2.3-4 C E • dl = Voltage around C, also known as electromotive force (emf) around C (but not really a force), V m  m, or V S B • dS = Magnetic flux crossing S, 2 Wb m  m   , or Wb d –  B • dS = Time rate of decrease of dt S magnetic flux crossing S, Wb s, or V 2.3-5 Important Considerations (1) C Right-hand screw (R.H.S.) Rule The magnetic flux crossing the surface S is to be evaluated toward that side of S a right-hand screw advances as it is turned in the sense of C 2.3-6 (2) by Any surface S bounded by C The surface S can be any surface bounded C For example: z R z R C O x C Q O y P x Q y P This means that, for a given C, the values of magnetic flux crossing all possible surfaces bounded by it is the same, or the magnetic flux bounded by C is unique 2.3-7 (3) Imaginary contour C versus loop of wire There is an emf induced around C in either case by the setting up of an electric field A loop of wire will result in a current flowing in the wire (4) Lenz’s Law States that the sense of the induced emf is such that any current it produces, if the closed path were a loop of wire, tends to oppose the change in the magnetic flux that produces it 2.3-8 Thus the magnetic flux produced by the induced current and that is bounded by C must be such that it opposes the change in the magnetic flux producing the induced emf (5) N-turn coil For an N-turn coil, the induced emf is N times that induced in one turn, since the surface bounded by one turn is bounded N times by the N-turndcoil Thus emf – N dt 2.3-9 where  is the magnetic flux linked by one turn  D2.5 B B0 sin t ax  cos t a y B  d S = B S  sin t d C E  d l  dt B0 sin t    B0 cos t V z C x y 2.3-10  B0 dec  2 –B0 3 t inc emf B0 –B0 emf <  2 3 emf > Lenz’s law is verified t 2.3-11 (b) S B • dS 1  B0 sin t – B0 cos t 2     B0 sin t –   C E • dl z C x d 1   – B0 sin t –  dt     B0   – cos t – V   y 2.3-12 (c) z S B • dS  B0 sin t  B0 cos t    B0 sin t     E • dl C C x d     – B0 sin t   dt       – B0 cos t  V   y 2.3-13 Motional emf concept B C l S x z B  d S = B ly B0l  y0  v0t  v0 ay conducting rails y B B0az S dS conducting bar y  y  v 0t 2.3-14 d C E • dl  – dt S B • dS d   B0l  y0  v0t  dt  B0lv0 This can be interpreted as due to an electric field F E  v0 B0 a x Q induced in the moving bar, as viewed by an observer moving with the bar, since l v0 B0 l  x0 v0 B0 a x • dx a x l  x0 E • dl 2.3-15 where F Qv B Qv0 a y  B0 a z Qv0 B0 a x is the magnetic force on a charge Q in the bar Hence, the emf is known as motional emf