1ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLecture 11Faraday’s Law and Electromagnetic InductionandElectromagnetic Energy and Power FlowIn this lecture you will learn:• More about Faraday’s Law and Electromagnetic Induction• The Non-uniqueness of Voltages in Magnetoquasistatics• Electromagnetic Energy and Power Flow• Electromagnetic Energy Stored in Capacitors and Inductors• Appendix (some proofs)ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityFaraday’s Law RevisitedadHtadBtsdEorrrrrr...∫∫∂∂−=∫∫∂∂−=∫µA closed contourBFaraday’s Law: The line integral of E-field over a closed contour is equal to –ve of the time rate of change of the magnetic flux that goes through any arbitrary surface that is bounded by the closed contourImportant Note: In electroquasistatics the line integral of E-field over a closed contour was always zeroIn magnetoquasistatics this is NOT the case()∫∫=∇−= 0.. sdsdErrrφ2ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityElectromagnetic Induction and Kirchoff’s Voltage LawNow consider a circuit through which the magnetic flux is changing with time (Kirchoff’s voltage law is violated)++--dtdadBtsdEλ−=∫∫∂∂−=∫⇒rrrr..dtdVRIRIsdEλ−=−+=∫⇒21.rr1R2RI()()dtdRRRRVIλ21211+−+=⇒V+-()tλ++--1R2RIV+-Kirchoff’s voltage law comes from the electroquasistaticapproximation:0. =∫sdErr0.21=−+=∫⇒ VRIRIsdErr()21RRVI+=⇒ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLenz Law++--Suppose an induced current I is flowing through the wire:()dtdRRIλ211+−=⇒1R2RIThe induced current in the wire produces its own magnetic fieldLenz Law is just an easy way to remember in which direction the induced current flowsThe law states that the induced current will flow in a direction such that its own magnetic field opposes the time variation of the magnetic field that produced it Example: Suppose the magnetic flux through the wire loop shown above wasincreasing with time (so that dλ/dt > 0). Lenz would tell us that the induced current would flow in the clockwise direction so that its own magnetic field would oppose the increasing magnetic flux through the loop In the equation above this fact comes out from the negative sign on the right hand side()tλ3ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityNon-Uniqueness of Voltages in Magnetoquasistatics - I++--()dtdRRIsdEλ−=+=∫21.rr1R2RI2V1VQuestion:What is the voltages difference V2-V1?One may be tempted to write …….112221VRIVVRIV=−=−0012=−⇒=⇒VVI….. which cannot be correct since we know that:We have:What went wrong? Our usual concepts of circuit theory and potentials which are based on conservative E-fields are not valid when time varying magnetic fields are present()dtdRRIλ211+−=()tλECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityNon-Uniqueness of Voltages in Magnetoquasistatics - II++--1R2RI2V1VLets do some real experimental measurements and see what we get for V2-V1+-VC1C2Faraday’s Law for contour C1:voltmeterdtdRIRItadBsdEλ−=+⇒∂∫∫∂−=∫21..rrrrFaraday’s Law for contour C2:11210..IRVVVRIVtadBsdE=−=⇒=−⇒∂∫∫∂−=∫rrrr⎟⎠⎞⎜⎝⎛+−=−=⇒dtdRRRVVVλ21112This value for V2-V1would actually be measured experimentally if the measurement is done as shown above()tλ4ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityNon-Uniqueness of Voltages in Magnetoquasistatics - III++--1R2RI2V1VLets do another real experimental measurement and see what we get for V2-V1+-VC1C3Faraday’s Law for contour C1:voltmeterdtdRIRItadBsdEλ−=+⇒∂∫∫∂−=∫21..rrrrFaraday’s Law for contour C3:21220..IRVVVRIVtadBsdE−=−=⇒=−−⇒∂∫∫∂−=∫rrrr⎟⎠⎞⎜⎝⎛+=−=⇒dtdRRRVVVλ21212This value for V2-V1would actually be measured experimentally if the measurement is done as shown aboveLesson: The result of a voltage measurement depends on how exactly you do the measurement when time varying magnetic fields are present ()tλECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityInductors and Faraday’s Law - IConsider an inductor with a time varying current:+_()tIInductance = LCThe magnetic flux is given by:()()tILt =λConsider Faraday’s law over the closed contour C:adHtadBtsdEorrrrrr...∫∫∂∂−=∫∫∂∂−=∫µ()tV()()()dttIdLdttdtV==⇒λλ5ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityInductors and Faraday’s Law - II+_()tIInductance = LC’Consider Faraday’s law over the new closed contour C’:()()()2.21....verticalverticaltVsdEdttdsdEtVadHtadBtsdEo=∫⇒−=∫+−∫∫∂∂−=∫∫∂∂−=∫rrrrrrrrrrλµ()tVλQuestion: is there electric field anywhere else besides at the terminals?? There must be E-field inside the loop !!This E-field is not very easy to determine analytically.ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityThe Current-Charge Continuity Equation in Electromagnetism - IStart from:tEJH∂∂+=×∇rrrεTake divergence on both sides:()()tEJH∂∇∂+∇=×∇∇rrrε...0()tEJ∂∇∂+∇=⇒rrε..0Use Gauss’ law:ρε=∇ Er.To get:tJ∂∂=∇−ρr.Current-charge continuity equationWhat does this equation mean??6ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySuppose the current density is spatially varying as shown belowx∆xxx∆+()xJx()xxJx∆+The difference in current density at x and x+∆x must result in piling up of charges in the infinitesimal strip …….() ( )()txtxtxxJtxJxx∂∆∂=∆+−,,,ρ()()()txtxxtxJtxxJxx∂∆∂=∆−∆+−⇒,,,ρ() ()ttxxtxJx∂∂=∂∂−⇒,,ρNow generalize to 3D and get the desired result:()()ttrtrJ∂∂=∇−,,.rrrρThe Current-Charge Continuity Equation in Electromagnetism - IIECE 303 – Fall 2007 – Farhan Rana – Cornell University• A continuity equation of the form:()()ttrtrJ∂∂=∇−,,.rrrρestablishes a relation between the flow rate of a quantity (in the present case the flow rate of charge density) and the time rate of change of the quantity• In the next few slides we will try to find a power-energy continuity equation for the electromagnetic field• A continuity equation is in fact a “conservation law”For example, the current-charge continuity equation expresses charge conservation. It says that charge cannot just disappear. In the integral form it becomes:() ()()dttdQdVtrtadtrJ =∫∫∫∂∂=∫∫− ,.,rrrrρIf there is a net