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 7 electrodynamicsfields and waves 488 Electrodynamics Fieldsand Waves The electromagnetic field laws derived thus far from the empirically determined Coulomb Lorentz forces are correct on the time scales of our own physical experiences However just as Newton s force law must be corrected for material speeds approaching that of light the field laws must be corrected when fast time variations are on the order of the time it takes light to travel over the length of a system Unlike the abstractness of relativistic mechanics the complete electrodynamic equations describe a familiar phenomenonpropagation of electromagnetic waves Throughout the rest of this text we will examine when appropriate the lowfrequency limits to justify the past quasi static assumptions 7 1 7 1 1 MAXWELL S EQUATIONS Displacement Current Correction to Ampere s Law In the historical development of electromagnetic field theory through the nineteenth century charge and its electric field were studied separately from currents and their magnetic fields Until Faraday showed that a time varying magnetic field generates an electric field it was thought that the electric and magnetic fields were distinct and uncoupled Faraday believed in the duality that a time varying electric field should also generate a magnetic field but he was not able to prove this supposition It remained for James Clerk Maxwell to show that Faraday s hypothesis was correct and that without this correction Ampere 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 the current density must have zero divergence because the divergence of the curl of a vector is always zero This result contradicts 2 if a time varying charge is present Maxwell Maxwell s Equations 489 realized that adding the displacement current on the righthand 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 electromagnetic waves that travel at the speed of light c thus proving that light is an electromagnetic wave Because of the significance of Maxwell s correction the complete set of coupled electromagnetic field laws are called Maxwell s equations Faraday s Law VxE B at L E dl d is 3 B dS Ampere s law with Maxwell s displacement current correction Vx H Jf D at H dl Jr dS d dte D dS 4 Gauss s laws V D pf fs D sdS V B 0 Pf dV 5 B dS 0 6 Conservation of charge V 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 the divergence of Faraday s law V B could at most be a timeindependent function Since we assume that at some point in time B 0 this function must be zero The symmetry in Maxwell s equations would be complete if a magnetic charge density appeared on the right hand side of Gauss s law in 6 with an associated magnetic current due to the flow of magnetic charge appearing on the right hand side of 3 Thus far no one has found a magnetic charge or current although many people are actively looking Throughout this text we accept 3 7 keeping in mind that if magnetic charge is discovered we must modify 3 and 6 and add an equation like 7 for conservation of magnetic charge M 490 7 1 2 Electrodynamics Fiedsand Waves Circuit Theory as a Quasi static Approximation Circuit theory assumes that the electric and magnetic fields are highly localized within the circuit elements Although the displacement current is dominant within a capacitor it is negligible outside so that Ampere s law can neglect time variations of D making the current divergence free Then we obtain Kirchoff s current law that the algebraic sum of all currents flowing into or out of a node is zero V J 0 JdS E ik 8 Similarly time varying magnetic flux that is dominant within inductors and transformers is assumed negligible outside so that the electric field is curl free We then have Kirchoff s voltage law that the algebraic sum of voltage drops or rises around any closed loop in a circuit is zero VxE O 7 2 7 2 1 E VV E dl iO vA 0 9 CONSERVATION OF ENERGY Poynting s Theorem We expand the vector quantity V ExH H VxE E VxH H B E at D E Jr at 1 where we change the curl terms using Faraday s and Ampere s laws For linear homogeneous media including free space the constitutive laws are D eE B IAH 2 so that 1 can be rewritten as V ExH t eE 2 AH E Jf 3 which is known as Poynting s theorem We integrate 3 over a closed volume using the divergence theorem to convert the 491 Conservation of Energy first term to a surface integral E ExH dS 2 E H dV E JIdV 4 I V ExH dV V We recognize the time derivative in 4 as operating on the electric and magnetic energy densities which suggests the interpretation of 4 as dW Pou Pa 5 where Po is the total electromagnetic power flowing out of the volume with density S E x H watts m 2 kg s 3 6 where S is called the Poynting vector W is the electromagnetic stored energy and Pd is the power dissipated or generated Po t ExH dS W IeE tH Pd E JdV 2 S dS 7 dV If E and J are in the same direction as in an Ohmic conductor E Jr oE 2 then Pd is positive representing power dissipation since the right hand side of 5 is negative A source that supplies power to the volume has E and Jf in opposite directions so that Pd is negative 7 2 2 A Lossy Capacitor Poynting s theorem offers a different and to some a paradoxical explanation of power flow to circuit elements Consider the cylindrical lossy capacitor excited by a time varying voltage source in Figure 7 1 The terminal current has both Ohmic and displacement current contributions eAdv S 1 dT oAv I dvv vA C dt R C T I A R I 8 From a circuit theory point of view we would say that the power flows from the terminal wires being dissipated in the M 492 Electrodynamics Fieldsand Waves 4 ra 2 I rc Figure 7 1 The power delivered to a lossy cylindrical capacitor vi ispartly dissipated by the
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