MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.chapter 7electrodynamics-fields and waves488 Electrodynamics-Fields and WavesThe electromagnetic field laws, derived thus far from theempirically determined Coulomb-Lorentz forces, are correcton the time scales of our own physical experiences. However,just as Newton's force law must be corrected for materialspeeds approaching that of light, the field laws must be cor-rected when fast time variations are on the order of the time ittakes light to travel over the length of a system. Unlike theabstractness of relativistic mechanics, the complete elec-trodynamic equations describe a familiar phenomenon-propagation of electromagnetic waves. Throughout the restof this text, we will examine when appropriate the low-frequency limits to justify the past quasi-static assumptions.7-1 MAXWELL'S EQUATIONS7-1-1 Displacement Current Correction to Ampere's LawIn the historical development of electromagnetic fieldtheory through the nineteenth century, charge and its electricfield were studied separately from currents and theirmagnetic fields. Until Faraday showed that a time varyingmagnetic field generates an electric field, it was thought thatthe electric and magnetic fields were distinct and uncoupled.Faraday believed in the duality that a time varying electricfield should also generate a magnetic field, but he was notable to prove this supposition.It remained for James Clerk Maxwell to show that Fara-day's hypothesis was correct and that without this correctionAmpere's law and conservation of charge were inconsistent:VxH=JJ V Jf = 0 (1)for if we take the divergence of Ampere's law in (1), thecurrent density must have zero divergence because thedivergence of the curl of a vector is always zero. This resultcontradicts (2) if a time varying charge is present. Maxwell_Maxwell's Equations 489realized that adding the displacement current on the right-hand side of Ampere's law would satisfy charge conservation,because of Gauss's law relating D to pf (V D = pr).This simple correction has far-reaching consequences,because we will be able to show the existence of electro-magnetic waves that travel at the speed of light c, thus provingthat light is an electromagnetic wave. Because of thesignificance of Maxwell's correction, the complete set ofcoupled electromagnetic field laws are called Maxwell'sequations:Faraday's LawVxE= B E dl= -B *dS (3)at L d isAmpere's law with Maxwell's displacement current correctionVx H = Jf+D H -dl = Jr dS+d D dS (4)at dt eGauss's lawsV" D=pf > fs D.sdS= Pf dV (5)V B=0 B dS=0 (6)Conservation of chargeV" Jrf+L'=O JfdS+ v pfdV=O (7)As we have justified, (7) is derived from the divergence of (4)using (5).Note that (6) is not independent of (3) for if we take thedivergence of Faraday's law, V -B could at most be a time-independent function. Since we assume that at some point intime B = 0, this function must be zero.The symmetry in Maxwell's equations would be complete ifa magnetic charge density appeared on the right-hand side ofGauss's law in (6) with an associated magnetic current due tothe flow of magnetic charge appearing on the right-hand sideof (3). Thus far, no one has found a magnetic charge orcurrent, although many people are actively looking.Throughout this text we accept (3)-(7) keeping in mind that ifmagnetic charge is discovered, we must modify (3) and (6)and add an equation like (7) for conservation of magneticcharge.M = ýýý490 Electrodynamics--Fiedsand Waves7-1-2 Circuit Theory as a Quasi-static ApproximationCircuit theory assumes that the electric and magnetic fieldsare highly localized within the circuit elements. Although thedisplacement current is dominant within a capacitor, it isnegligible outside so that Ampere's law can neglect time vari-ations of D making the current divergence-free. Then weobtain Kirchoff's current law that the algebraic sum of allcurrents flowing into (or out of) a node is zero:V.J = 0=>JdS = E ik= (8)Similarly, time varying magnetic flux that is dominantwithin inductors and transformers is assumed negligibleoutside so that the electric field is curl free. We then haveKirchoff's voltage law that the algebraic sum of voltage drops(or rises) around any closed loop in a circuit is zero:VxE=O E=-VV* E dl=iO vA =0 (9)7-2 CONSERVATION OF ENERGY7-2-1 Poynting's TheoremWe expand the vector quantityV -(ExH) =H (VxE)-E .(VxH)= -H. B-_E D--E *Jr (1)at atwhere we change the curl terms using Faraday's andAmpere's laws.For linear homogeneous media, including free space, theconstitutive laws areD=eE, B=IAH (2)so that (1) can be rewritten asV. (ExH)+t(eE2 +AH') -E Jf (3)which is known as Poynting's theorem. We integrate (3) over aclosed volume, using the divergence theorem to convert theConservation of Energy 491first term to a surface integral:(ExH) -dS+ (E2+E + H) dV=- E JIdV (4)I V-(ExH)dVVWe recognize the time derivative in (4) as operating on theelectric and magnetic energy densities, which suggests theinterpretation of (4) asdWPou,+- = -Pa (5)where Po., is the total electromagnetic power flowing out ofthe volume with densityS = E x H watts/m2 [kg-s-3] (6)where S is called the Poynting vector, W is the electromag-netic stored energy, and Pd is the power dissipated orgenerated:Po.t= (ExH).dS= S dSW= [IeE + tH2] dV (7)Pd = E -JdVIf E and J, are in the same direction as in an Ohmic conduc-tor (E • Jr = oE2), then Pd is positive, representing power dis-sipation since the right-hand side of (5) is negative. A sourcethat supplies power to the volume has E and Jf in oppositedirections so that Pd is negative.7-2-2 A Lossy CapacitorPoynting's theorem offers a different and to some aparadoxical explanation of power flow to circuit elements.Consider the cylindrical lossy capacitor excited by a timevarying voltage source in Figure 7-1. The terminal currenthas both Ohmic and displacement current contributions:eAdv oAv dvv vA IS+ = C-+- C=-T R = (8)1 dT I dt R I 'AFrom a circuit theory point of view we would say that thepower flows
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