Time varying fields and Maxwell's equations Chapter 9

Transcription

1 Tie varying fields and Maxwell's equatins hapter 9 Dr. Naser Abu-Zaid Page 9/7/202

2 FARADAY LAW OF ELETROMAGNETI INDUTION A tie varying agnetic field prduces (induces) a current in a clsed lp f wire. The tie varying agnetic field B (t) is said t induce an electrtive frce (ef) in the lp and this ef drives the current. Electrtive frce ef induced in a statinary clsed circuit is equal t the negative rate f increase f the agnetic flux linking the circuit. Lenz s law: The directin f the ef is such that the agnetic flux generated by the induced current ppses the change in the riginal flux. d ef ef E dl B ds Fingers indicate the directin f and the thub indicate directin f ds. hange in flux des nt equal zer in the fllwing cases: Dr. Naser Abu-Zaid Page 2 9/7/202

3 d 0 hanging B (t) linking statinary circuit. Either ving circuit in steady B, r ving B in statinary circuit binatin f bth changing B (t) and ving circuit If the clsed path cnsists f N-turn filaentary cnductr then d ef N The general fr f Farady s law is: Rewrite: E dl d d ef d ef B ds t d d B ds ds If the circuit is statinary then 0, s t E dl B ds B ds E dl B ds t Obtaining the pint fr using tkes There; B E ds ds t E dl E dl B ds ds B t t B E t Pint fr f Faradays Law, r Maxwells first equatin fr tie varying fields Dr. Naser Abu-Zaid Page 3 9/7/202

4 Exaple 6: uppse a tie varying agnetic field is defined in space (in a cylindrical crdinate syste as: B ˆ sin t az, B 0, 0 a) Deterine the induced electric field via Faradyay s law. b) hw that the electric field intensity satisfies the pint fr f Faraday s law. lutin: Dr. Naser Abu-Zaid Page 4 9/7/202

5 a) E E B ds 2 B d ef E dl aˆ B sin t, d sin t dd B ds 2 B cs t, But: 2 E dl E d 2E 0 : B E cs t, 2 b) B E t B cs t aˆ B cs t aˆ and Maxwell s equatin is satisfied. z z Dr. Naser Abu-Zaid Page 5 9/7/202

6 MOTIONAL EMF Mving nturs (Mtinal ef) nsider the fllwing device: Methd One: The agnetic flux linking is: B ds Byd d dy ef Bd Bdv Methd Tw: The frce n charge q ving at a velcity v in a agnetic field B is: F qv B The tinal electric field intensity is: F E v B q Then the tinal ef prduced by the ving cnductr is: ef E dl v B dl 0 v B dl va ˆ y Baˆ z Bdv and satisfies the RHR. d dxaˆ x Dr. Naser Abu-Zaid Page 6 9/7/202

7 If the circuit is ving in a changing agnetic field then the general fr f Faraday s law is: d d ef B ds B ds t v B dl Exaple 7: cnsider a wire lp that is rtating in the presence f a dc agnetic field given by B Baˆ y as shwn. The lp have a resistr R inserted in it, and rtates at a radian frequency f and lies in the xz plane at t 0. Deterine the current induced in the lp. Dr. Naser Abu-Zaid Page 7 9/7/202

8 APPLIATION OF FARADAY LAW Transfrers. Recent applicatins, includes Maglev and Witricity. Older applicatins include inductive heating. Transfrers turns N turns N 2 I I 2 V V 2 Fr an ideal Transfrer i i 2 v2 v N2 N N Reff R L N 2 Nn idealities ccur due t: Leakage flux Finite inductances Nn-zer winding resistances Hysteresis Eddy currents lss Usually apprxiate circuit fr transfrer is used. N N 2 2 Dr. Naser Abu-Zaid Page 8 9/7/202

9 re Flux Induced ef Ohic lsses and heating Lcal currents in the cnducting cre DIPLAEMENT URRENT AND MAXWELL EQUATION Fr static fields, Maxwell's equatins are given by: E 0 () H J (2) D v (3) B 0 (4) The first equatin is already dified fr tie varying fields. B E (') t The third and furth equatins reain the sae. What abut the secnd equatin? H J Taking the divergence f bth sides H 0 A cntradictin v J t J Dr. Naser Abu-Zaid Page 9 9/7/202

10 Add an arbitrary ter H J G Repeating the previus prcess H 0 J G v D J G t t D t And Aper's law is dified t be D H J t Integrating ver an pen surface enclsed by and using tke's there: Illustratin: H dl J ds D ds t An electrically charging capacitr with an iaginary cylindrical surface surrunding the lefthand plate. Right-hand surface R lies in the space between the plates and left-hand surface L lies t the left f the left plate. N cnductin current enters cylinder surface R, while current I leaves thrugh surface L. nsistency f Apère's law requires a displaceent current I D = I t flw acrss surface R. Taken fr wikipedia Dr. Naser Abu-Zaid Page 0 9/7/202

11 Exaple 8: cpare the cnductin and displaceent current densities in cpper 7,, 5.80 at a frequency f MHz. Repeat fr Tefln, which has 8 2.,, 30 at MHz. Assue lutin: Fr cpper : Fr Tefln MAXWELL EQUATION AND BOUNDARY ONDITION Fr tie varying fields, Maxwell's equatins are given by: Pint Fr Integral Fr ignificance B d Faraday's Law E t E dl B ds D J t H H dl J ds D v D ds Q B 0 B ds 0 D ds t Aper's Law Gauss's Law fr electrstatics N islated agnetic charge (Gauss's Law fr agnetstatics) urces: harge and urrent Density. Auxiliary equatins (cnstitutive relatins): Dr. Naser Abu-Zaid Page 9/7/202

12 D E B H The cnductin current density: J E The bundary cnditins fr physical edia, E E D t t2 H H t t2 n D n 2 B B n n2 K 0 s Oh's Law: ; ONDITION FOR A PERFET ONDUTOR E 0 inside a perfect cnductr Faraday's law: ; B E t B t 0 H 0 inside a perfect cnductr Aper's law: J 0 inside a perfect cnductr, if regin II is a perfect cnductr then E t 0 Ht K Ht K aˆ r n D utward nral n B n 0 s Dr. Naser Abu-Zaid Page 2 9/7/202

13 Exaple 9: hw that the fllwing vectr fields in free space satisfy all f Maxwell's equatins. E E cs t k z aˆ H x Ex Where, E, k are cnstants. x cs t k z x aˆ y lutin: a) Faraday s law =2 fr Faraday s law t be satisfied, this iplies; b) Aper s law =2 fr Aper s law t be satisfied, this iplies; c) Gauss law d) The ther Gauss law (atisfied) (atisfied) Dr. Naser Abu-Zaid Page 3 9/7/202

14 Exaple 0: At the interface between tw regins as shwn in the figure, find the agnetic field intensity vectr at x 0 if: H aˆ aˆ x 0 x y z Rein,, z y x Rein2, 2 2, 2 lutin: THE RETARDED POTENTIAL Tie-varying ptentials, usually called retarded ptentials. Reeber that the scalar electric ptential ay be expressed in ters f a static charge distributin; The vectr agnetic ptential ay be fund fr a current distributin which is cnstant with tie; Dr. Naser Abu-Zaid Page 4 9/7/202

15 The differential equatins satisfied by ; and fr, Having fund the gradient, and, the fundaental fields are then siply btained by using Or the curl, Tie-varying ptentials (defined in a way) which are cnsistent with tie varying Maxwell s equatins; leads t: Later, we will find that any electragnetic disturbance will travel at a finite velcity f thrugh any hgeneus ediu described by and. In the case f free space, this velcity turns ut t be the velcity f light, apprxiately It is lgical, then, t suspect that the ptential at any pint is due nt t the value f the charge density at se distant pint at the sae instant, but t its value at se previus tie, because the effect prpagates at a finite velcity. Thus; Dr. Naser Abu-Zaid Page 5 9/7/202

16 Where indicates that every t appearing in the expressin fr has been replaced by a retarded tie, Thus, if the charge density thrughut space were given by Then Where is the distance between the differential eleent f charge being cnsidered and the pint at which the ptential is t be deterined. The retarded vectr agnetic ptential is given by uary: Use the distributin f and t deterine and by applying: Electric and agnetic fields are then btained by applying: Dr. Naser Abu-Zaid Page 6 9/7/202