4 Time varying electromagnetic field

Transcription

1 ectrodynamcs and Optcs GFIT5 4 Tme varyng eectromagnetc fed 4.1 ectromagnetc Inducton Inducton due to moton of conductor onsder the Faraday s experment. The fgure shows a co of wre connected to a current measurng gavanometer. Near the co there s a bar magnet. When the co s hod statonary the gavanometer does not show current fow, but f we move the co ether toward or away from magnet the meter shows current n opposte drecton respectvey. movng the co S N the meter shows current We ca ths an nduced current, and the correspondng emf that has to be present to cause ths current s caed nduced emf. s we move the conductor n magnetc fed the charge carrers move wth the conductor and experence a force (Lorentz Force). Ths force s an extraneous force, t has not eectrostatc orgn: F F q( v B) v B q s the strength of the extraneous fed. The nduced eectromotve force (emf) aong the conductor s: B B B G ds v B ds ( ) If the movng conductor forms a oop, due to the nduced emf a current fows through t. onsder a straght conductor movng n homogeneous magnetc fed, perpendcuar to the pane drected nto the page. B ds v The vectors v, B and ds mutuay perpendcuar to each other. Due to the nduced eectrc fed the emf aong the conductor: + + ds v B ds vbds Bv B ( ) Bv. We used that v and B are perpendcuar, both are constant over the ength of the conductor, and ther vector product s parae to ds. 47

2 ectrodynamcs and Optcs GFIT5 Suppose the movng conductor sdes aong a statonary U shaped conductor as n fgure. The movng conductor has become a source of emf, charges moves wthn t from ower to hgher potenta and n the remander of the crcut charge moves from hgher to ower potenta. If the resstance of the sdng bar s r and the externa resstance s then: Bv. B r ds v I I U I < + r + r The devce above s caed near generator. The mechanca power needed to move the rod s converted nto eectrca power. If a drected cosed conductor oop moves n magnetc fed the nduce emf s: ( v B) ds, (Neumann Law) onsder the near generator agan, when the conductor moves toward the rght a dstance ds, the area encosed by the crcut s ncreased by d ds and the change n magnetc fux through the crcut s dφ B d B ds. B v ds The tme rate of change of fux s therefore: d Φ ds B Bv Faraday s Law of nducton states that the nduced emf n a crcut s equa to the negatve rate of change of the magnetc fux through t. Ths s caed fux-rue. dφ Ths s the aternatve form of the equaton for the emf n a movng conductor. It s often eases to appy the fux-rue nstead of Neumann s Law to obtan the nduce emf. The negatve sgn s due to Lenz s Law: The drecton of the nduced current s such that ts effect woud oppose the change n magnetc fux, whch gave rse to the current. emark: If we have a co of N turns and the fux vares the same rate through each turn, the nduced emf s are n seres and must be added: dφ N. 48

3 ectrodynamcs and Optcs GFIT The emf nduced n a rotatng co, generator onsder a conductng co of N turns each of area beng made to rotate wth anguar speed ω n a unform magnetc fed. B. N α n B + ω const. S t t be n B, where n s the norma of the co. s ω const., the ange α ω t. The fux through each turn s Φ Bcosα Bcosωt. ppy the Faraday s Law. The nduced emf n the co s: dφ N NBω( snω t). Introduce NBω snω t. NBω, t s caed maxmum vaue of the emf, snω t The output from a smpe generator s a snusoday varyng emf Inducton due to the change of fux-nkages a) mutua nducton ron core G prmary crcut secondary crcut s the current n the prmary crcut s vared wth the rheostat, the magnetc fed due to the current s vared aso. part of the magnetc fux through the secondary crcut s aso vared. It s found expermentay that an emf and nduced current appears n the secondary crcut. The secondary crcut s not movng n a magnetc fed so no motona emf s nduced n t. In such a stuaton no one porton of crcut can be consdered the source of emf, the entre crcut consttutes the source. The settng up of an nduced current sgnfes that the changes n the magnetc fed produce extraneous forces n the oop. The eectrc fed due to the extraneous forces causes the current carrers n a conductor to start movng and an nduced current s set up. In case of mutua 49

4 ectrodynamcs and Optcs GFIT5 nducton one crcut acted as a source of magnetc fed and emf was nduced n a separate ndependent crcut nkng some of the fux. b) sef nducton G Due to experences f we dsconnect the source from the co and make short-crcut, the ammeter shows a decreasng current. Whenever a current s present n any crcut, ths current sets up magnetc fed that nks wth the same crcut and vares when the current vares. So there s an nduced emf n t resutng from the varaton n ts own magnetc fed. Such an emf s caed sef nduced emf. In the two prevous cases, we must concude that the nduced current n the oop s caused by an nduced eectrc fed whch s assocated wth the changng magnetc fed. Ths fed s caed non-eectrostatc fed and denote t by n. So the nduced emf n ths case s the ne ntegra of n around the oop, and accordng to experences the Faraday s Law of nducton n case of changng fux: d U o Φ Faraday s Law, ntegrated form: d n ds B d g In a varyng magnetc fed an emf s nduced n any cosed crcut and s equa to the negatve of the tme rate of change of the magnetc fux through the crcut. If an eectrostatc fed, produced by eectrc charges s aso present n the same regon, t s aways conservatve and so ts ne ntegra around any cosed path s aways zero. Hence n the next equaton s the tota eectrc fed nductng both eectrostatc and non eectrostatc contrbutons: d ds B d g To obtan the dfferenta form of Faraday s Law of nducton, et s transform the eft hand sde n accordance wth Stokes s theorem: ds d g ( ) Snce the oop and the surface are statonary the operaton of tme dfferentaton and ntegraton over the surface can have ther paces exchanged; or: d B B d d t The Faraday s Law: B ( ) d d t 5

5 ectrodynamcs and Optcs GFIT5 B + d, t Due to the arbtrary chosen surface the ntegrand must be zero: B +, t The dfferenta form of Faraday s Law of nducton: B t The negatve tme rate of change of magnetc fed at a pont equas to the cur of the fed. The cur of the vector s not zero, consequenty the nduced eectrc fed s non conservatve fed. It s mportant to remark that the tme-varyng magnetc fed causes eectrc fed to appear n space regardess of whether or not there s a wre oop n ths space. The presence of a oop ony makes t possbe to detect the exstence of an eectrc fed at the correspondng ponts of space as a resut of a current beng nduced n the oop. ectrc fed s set up not ony by charges but tme-varyng magnetc fed as we. The fed set up by the charges have sources so the fed nes begns and termnate at charges, and f they are at rest or n statonary moton ths fed s conservatve. The fed set up by tmevaryng magnetc fed have no sources, the fed nes are cosed curves, and non conservatve Sef Inductance of a Long Soenod The magnetc fed and nducton nsde a ong soenod s: NI NI H, B μ Denote the fux of one turn by Φ m. The fux through each turn: NI Φ m B d μ The fux of the co s: N Φ NΦ m μ I The fux of a ong soenod s proportona to the current. The proportonaty factor s caed sef nductance, denoted by L: Φ LI, μ N L L s numercay equa to the fux nkage of a crcut when unt current fows through t. The coeffcent L depends ony on the shape of the conductor, so the geometry and medum nsde t. Unt of L s: Vs [ L] 1 1henry 1H If the current varyng wth tme then Φ () t LI() t and so the sef nduced emf s: 51

6 ectrodynamcs and Optcs GFIT5 dφ di U L LI In case of a soenod generay the number of turns s a great number and as L s proportona to N the sef nductance s so great that we can consder t as the sef nductance of the whoe crcut Mutua nductance of two cosey wound cos onsder the next fgure: I1 N1 N ( t) The current I 1 n co 1 sets up magnetc fed as: NI 1 1() t B1 μ The fux through one turn: N 1 Φ () 1 μ I 1 t In case of cosey wound cos ths s aso the fux of one turn through the second co, so the whoe fux of co s: NN 1 Φ () 1 NΦ 1 μ I1 t. The tota fux through the second co s proportona to the current of the frst co. The proportonaty factor s caed the mutua nductance of the two cos and denoted by L 1 or M: Φ 1 L1I1 nd the mutua nductance: NN 1 L1 M μ The nduced emf n the co : dφ1 di1 di1 U1 L1 M The dea of mutua nductance s used n the transformer. 4. The generazaton of the oop theorem for a snge oop onsder the next snge oop. Denote the resstance of the whoe crcut by, the capactance by, the nductance of the co by L, (at the same tme ths s the nductance of the whoe crcut), and the apped emf by. 5

7 ectrodynamcs and Optcs GFIT5 g g 1 g L Set up a cosed curve g aong the crcut shown n the fgure, and appy the Faraday s Law of nducton: dφ U dφ ds g ds ds L di + g1 g The next term s the potenta dfference across the capactor: ds ds g Due to the dfferenta form of Ohm s Law: ρ j + ρ j, di ρ jds ds+ L g1 g1 The next term s just the emf of the crcut by defnton: ds In case of thn wres: j ds I and j : ds ρ j ds I ρ I, g g 1 g 1 1 s the resstance of the whoe crcut. di I + L The generazaton of the oop theorem: LI+ I+ The charge of the capactor s varyng due to the current. The transferred charge n tme s: d d I, that s I, and I The dfferenta equaton for the charge: L+ + 53

8 ectrodynamcs and Optcs GFIT5 Ths s a non-homogeneous near dfferenta equaton of the second order wth constant coeffcents for the (t) functon nergy n an nductor, magnetc energy densty onsder a sera L crcut shown on fgure, and appy the generazed oop equaton: L di L I + Mutpyng ths equaton by I, we have: di LI I I + To mantan a current n a crcut, energy must be supped. The energy requred per unt tme, n other words the power s I, the power of the seat of emf. The term I s the energy spent n movng the eectrons through the crysta attce of the conductor and s transferred to di the ons that make up the attce, so the power dsspated n the resstor. The ast term LI s then nterpreted as the energy requred per unt tme to bud up the magnetc fed n space. Therefore the rate of ncrease of the magnetc energy s the frst term, and can be wrtten as: di d 1 LI LI That s the magnetc energy: 1 Wm LI + constant Due to agreement, f I the energy of the magnetc fed s zero: W m constant The energy due to magnetc fed: 1 Wm LI onsder now a ong soenod: NI H Φ LI, and H I N fnay Φ B N H 1 Wm L I I Φ I BN BH N The voume of the co and the voume of the space of the magnetc fed s: V 1 Wm B HV The magnetc energy densty s defned as: Wm 1 wm B H : V though ths expresson has been justfed for the magnetc energy densty n a very speca case, a more detaed anayss woud ndcate that the resut s competey genera. 54

9 ectrodynamcs and Optcs GFIT5 If B s the magnetc nducton and H s the magnetc fed strength at a pont then the energy stored n an eementary voume dv s: 1 dwm B H dv, and n a fnte voume: 1 Wm wmdv B H dv V V 4.3 Forced eectrca oscatons n a sera L crcut onsder the next eectrca crcut: L The generazed oop equaton s: LI+ I+ The generazed oop equaton for the charge: L+ + Let s suppose that an aternatng emf s apped as: cosω t s the maxmum vaue of the apped emf, ω s the cycc frequency. L+ + cos ωt The oop equaton concdes wth the dfferenta equaton of forced mechanca oscatons. mx + κ x + Dx F cosω t The correspondence between the forced oscaton and the eectrca crcut s x m L κ 1 D D 1 ω ω m L κ α α m L F We know that the genera souton of a non homogeneous equaton equas the sum of the genera souton of the correspondng homogeneous equaton and a parta souton of the non homogeneous equaton: 55

10 ectrodynamcs and Optcs GFIT5 nh.gen. hom.gen. + nh.part. s the genera souton of the homogeneous equaton contans an exponenta decreasng term: hom.gen. t L e, after suffcent tme eapses, becomes very sma and t may be dsregarded. Ths s caed transent process. So the statonary souton of the non-homogeneous equaton s a parta souton of ths non homogeneous equaton. We ook for ths souton and denote t by. Set up a hepng equaton, mutpy t by the compex unt and add t to the orgna oop equaton: L+ + cos ω t L + + snω t, 1 L ( + ) ( cosω t+ snω t) Use the uer-reaton: ϕ e cosϕ + snϕ Introduce the compex charge: + ', Use the compex emf: 1 ωt L+ + e t e ω 1 L+ + We sha try to fnd the parta souton n the next form: t e ω s the compex amptude of the charge. Take the frst and second dervatves of the compex charge: ωt ωe ω I, The frst dervatve of the compex charge s the compex current I. ( ) ωt ω ω e Insertng nto the dfferenta equaton: Lω + ω + Instead of the dfferenta equaton we have a smpe compex agebrac equaton for. 1 1 ω + ωl+ ω Introduce the compex mpedance as: 56

11 ectrodynamcs and Optcs GFIT5 Z L ω +, or Z + Lω ω ω The equaton we have obtaned s caed the compex Ohm s Law: IZ, Introduce the nductve reactance: X L I. Z Lω, and the capactve reactance The compex mpedance may be represented on a compex pane: Im X 1 ω X L X 1 Z + Lω + ( XL X) ω The absoute vaue of the compex mpedance s caed the rea mpedance or smpy mpedance. X L X ϕ Z e 1 Z + L + XL X ( ) ω ω The so caed trgonometrc form of the compex mpedance s: Z Ze ϕ ϕ s the argument: 1 Lω ω tgϕ, or cos ϕ Z The compex current: ωt e ( ωt ϕ ) I e ϕ Z Ze Z Intate the maxmum vaue of the rea current: I Z ( t ) Ie ω ϕ I ppy the uer-reaton for the compex current: ( ωt ϕ) I Ie Icos( ωt ϕ) + Isn ( ωt ϕ) The rea part of the current and so the souton s: I ( t ) I cos ( ωt ϕ ), where I 1 + Lω ω 57

12 ectrodynamcs and Optcs GFIT5 If ϕ > then the current ags behnd the apped emf wth a phase ag of ϕ. If ϕ < then the current eads the emf wth a phase of ead of ϕ Instantaneous votages across the dfferent crcut eement 1. esstor: ( ω ϕ) ( ω ϕ) U I Ie t U t e ( t ) U U e ω ϕ, where U I The rea part of the votage across the resstor: U( t) U cos ( ωt ϕ ) We have shown that the potenta dfference between the termnas of a resstor s n phase wth the current.. apactor: π π ω ϕ ω ϕ ( ωt ϕ) π I 1 t t U e I e X I e U e ω ω The rea part of the votage across the capactor: () cos π U t U ωt ϕ, where 1 X, and U IX ω π 1 We used the formua, that e. π The votage across a capactor ags the current by. 3. Inductor π π π ωt ϕ+ ωt ϕ+ ( ω ϕ) ( ω ϕ) t t L ω ω L L U LI LIe L Ie e X Ie U e The rea part of the votage across the nductor: () cos π UL t UL ωt ϕ+, where X L Lω, and UL IX π We used the formua, that e. The votage across the nductor eads the current by π. L t the anayss of aternatng-current crcuts we often appy the rotatng vector dagrams. In such dagrams the nstantaneous vaue of a quantty that vares snusoday wth tme s represented by the projecton onto the horzonta (rea) axs. The ength of the compex vector corresponds to the maxmum vaue of the quantty and rotates counter-cockwse wth constant anguar speed ω. These rotatng vectors are caed phasors and the dagram contanng them phasor dagrams. It s caed compex phase dagram aso. 58

13 ectrodynamcs and Optcs GFIT5 Im U L + ω constant I e U U U L + U + U The compex descrpton cannot be used for the transent process and when the apped emf s not snusoda functon The effectve vaue of varyng current To characterze an aternatng current we use the concepts of effectve or root-mean-square vaue. The effectve vaue of an aternatng current s that steady current whch woud do the same work on the same resstor durng a tme T (tme of perod) as the aternatng current. I rms I I snωt Determne the effectve vaue of snusoday varyng current: W I T, () rms The two works are equa to each other that s: T rms T W I t () I T I t 1 T rms T () I I t Ths s the root-mean-square vaue of an aternatng current: 1 T Irms I () t T Ths s the square root of the average vaue of the square of the current or votage. If I I snωt, then: T T T I I ωt I sn ω ( 1 cos ) ω ω 1 sn Irms I t t t T T T I I rms, smar way rms 59

14 ectrodynamcs and Optcs GFIT Power n an L seres crcut When a source wth an nstantaneous emf ( t) suppes an nstantaneous current I () t, to a crcut, the nstantaneous power t suppes s: I P() t () t I() t ( ) cosωt I cos ωt ϕ cosωt cos( ωt ϕ) ppy the next trgonometrca expressons, and add the equatons: cos( α + β) cosαcos β snαsn β cos( α β) cosαcos β + snαsn β cos( α + β) + cos( α β) cosαcos β If α ωt, and β ωt ϕ, then α + β ωt ϕ, and α β ϕ. The nstantaneous power s: I P() t cos( t ) + cos ω ϕ ϕ The tme average of ths nstantaneous power s: I I P cosϕ cosϕ rmsirms cosϕ The tme average of the power s denoted by over bar, and cosϕ s caed power factor. 6