Time Varying Fields & Maxwell Equations

University:
Washington State University
Course:
Physics 101 | General Physics
Academic year:

2022
Views:

198

Pages:

5
Author:

Janor

Time Varying Fields & Maxwell EquationsFaraday's Law : "The is time varying magnetic flux produces anelectromagnetic force (EMF) This cmt producesCurrent in closed path. Mathematicaly.EMF = - d(NO) = -Nd 0(volt)dtdtWhere NO = Total magnetic flux.N = Number of turns.For Single turn (or) Loop N = 1So,EMF = - d- sign is due to Lenz's lawdtwhich states that the induced emfopposes changing magnetic fluxoNote " d$ " is not Zero in the following cases.dtStCase 1A non moving loop area is present in the timevarying maxwell's field in this case produces emt iscalled as transformers emf.y=oaycurrentLoop1towardsIndused Emt.Loop(2)you-isI(t)y=dcurrentdirt awayfrom you EMF = - d oct )-/dtp(t) =//B(t) . dsIf B(t) is uniform thenp(t) = B (t) II dsQ(t) = Blt). dsl() = /B(+) / 151 COS ODs -22in)cmt = - d [ B(t) area COS OBS]dtIf B is normal (90) to the Surface then is is alsonormal to the Surface and B, S are in Same directionSo, OBS = 0 by definition.Emf = (E E ll-4[ V = - $ E.dl Emt is opposite of v]Some Important points -F= Q( VeX3)E Ve x BQE = 3 C X B- 5 Case 2 : "A moving Loop area is present static magnetic fieldin this case cmt is called motional emf.Put 5th eq" in 4th( Emt ) motional pc Ve x B ).dlIf Ve, / B are uniform then Emf = VeX B JdlEmt = =(VexB )lEmt = VeBlBIt loop rotated with "w = 2rf " rotational frequency thenangle between 3, 3 changes according frequency tothis case.(Emf ) motional=- B (area) d (cos wt)dtCase 3 " A moving loop area is present in one time varing magneticfield then total Emf is Sum of transformer &motional Emf."Emt = - do(t) + plv X B ). dldt Observations :1Faraday's Law Emt = -do-/dtEmf = Q E. dl-2Q = 11B. ds-3Put (2) 3 3in/of E.dl = d B.dsdtSE. de = ll (-dB dt ) ds-4By stokes theorem :PE.dl = ds-5Put5in4() ds =11-dB dt ) dsCompare both SidesV X E = JB-It6Note1. Ampere's Circuital law$ 4.dl = I/By Stokes theorem.) ds -2By definitionI = ((J.ds) -3Put2&3in1Jf( ).ds = Jj(J.ds)By comparing both Sides.VXH = 4 Eq" "4 Signifies that Magnetic field is Rotational.Eq" 4" is applicable for Static field. PFor time varying field maxwell added dD to eq" 4atV X H = J + IDat -5= T E+ EdEat,VXH =JWTE+Cd&Jt = Jc + JdtotalCurrentdensity.Maxwell Equations.FormForm 2S. No.LawDifferential (or) PointIntegral1.Fraday's LawVXE = - OBSE.dl = dB at dsat2.Ampere's circuital law= J +DBHidl = It SSofGaus's Law of Electric->3.V.D = Pv$ D.ds = QfieldGaus's Law of->4.V.B =0p IM ds = 0Magnetic field# ## For free spaceVXB = le.J# If A is magnetic potential vector then B = T x A#J=VXH=>X (UVXA ) = le ( )