1ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLecture 10Perfect Metals in Magnetism and InductanceIn this lecture you will learn:• Some more about the vector potential• Magnetic field boundary conditions• The behavior of perfect metals towards time-varying magnetic fields• Image currents and magnetic diffusion• InductanceECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityThe Vector Potential - Review0. =∇ ArIn electroquasistatics we had:0=×∇ErTherefore we could represent the E-field by the scalar potential:φ−∇=ErIn magnetoquasistatics we have:Therefore we can represent the B-field by the vector potential:()()0.. =∇=∇ HBorrµAHBorrr×∇==µA vector field can be specified (up to a constant) by specifying its curl and its divergenceOur definition of the vector potential is not yet unique – we have only specified its curlFor simplicity we fix the divergence of the vector potential to be zero:ArAr2ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMagnetic Flux and Vector Potential Line Integral - ReviewThe magnetic flux λ through a surface is the surface integral of the B-field through the surface∫∫=∫∫=adHadBorrrr..µλSince:AHBorrr×∇==µ()∫=∫∫×∇=∫∫=sdAadAadBrrrrrr...λWe get:B-fieldThe magnetic flux through a surface is equal to the line-integral of the vector potential along a closed contour bounding that surfacesdrA closed contourStoke’s TheoremECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityVector Potential of a Line-CurrentzIˆxyConsider an infinitely long line-current with current I in the +z-directionThe H-field has only a φ-componentrUsing Ampere’s Law:()IrH=πφ2Work in cylindrical co-ordinatesrIHπφ2=⇒But()()rIrrArrAAHozzooπµµµφ21−=∂∂⇒∂∂−=×∇=r()()⎟⎠⎞⎜⎝⎛=−rrIrArAooozzln2πµIf the current has only a z-component then the vector potential also only has a z-component which, by symmetry, is only a function of the distance from the line-currentIntegrating from roto r :JAorrµ−=∇23ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityVector Potential of a Line-Current DipolezIˆ+xyd+r−rConsider two infinitely long equal and opposite line-currents, as shownThe vector potential can be written as a sum using superposition:()⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛=+−−+rrIrrIrrIrAooooozln2 ln2ln2πµπµπµrThe final answer does not depend on the parameter rorrQuestion: where is the zero of the vector potential?Points for which r+equals r-have zero potential. These points constitute the entire y-z plane zIˆ−ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityH-Field of a Line-Current DipolexydSomething to Ponder UponPoisson equation:oερφ−=∇2zozJAµ−=∇2Vector Poisson equation (only the z-component relevant for this problem)Potential of a line-charge dipole:Vector potential of a line-current dipole:()⎟⎟⎠⎞⎜⎜⎝⎛=+−rrroln2επλφr()⎟⎟⎠⎞⎜⎜⎝⎛=+−rrIrAozln2πµrNotice the SimilaritiesH4ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMagnetic Field Boundary Conditions - I1Hoµ2Hoµ12HHooµµ=The normal component of the B-field at an interface is always continuousMaxwell equation:The net magnetic flux coming into a closed surface must equal the magnetic flux coming out of that closed surface (since there are no magnetic charges to generate or terminate magnetic field lines)0.. =∇=∇ HBorrµTherefore:ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMagnetic Field Boundary Conditions - IIKHH=−12The discontinuity of the parallel component of the H-field at an interface is related to the surface current density (units: Amps/m) at the interface 1H2HKThis follows from Ampere’s law:JHrr=×∇ or∫∫=∫adJsdHrrrr..The line integral of the magnetic field over a closed contour must equal the total current flowing through the contour1H2HKLKHHLKLHLH=−⇒=−12125ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityPerfect Metals and Magnetic Fields - IA perfect metal can have no time varying H-fields inside itNote: Recall that in magnetoquasistatics one can have time varying H-fields (as long as the time variation is slow enough to satisfy the quasistaticconditions)The argument goes in two steps as follows:• A time varying H-field implies an E-field (from the third equation of magnetoquasistatics)• Since a perfect metal cannot have any E-fields inside it (time varying or otherwise), a perfect metal cannot have any time varying H-fields inside it ()()ttrHtrEo∂∂−=×∇,, rrrrµ()0,. =∇ trHorrµ()()trJtrH ,, rrr=×∇ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityPerfect Metals and Magnetic Fields - IIAt the surface of a perfect metal there can be no component of a time varying H-field that is normal to the surfaceperfect metal()tHrThe argument goes as follows:• The normal component of the H-field is continuous across an interface• So if there is a normal component of a time varying H-field at the surface of a perfect metal there has to be a time varying H-field inside the perfect metal • Since there cannot be any time varying H-fields inside a perfect metal, there cannot be any normal component of a time varying H-field at the surface of a perfect metal6ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityPerfect Metals and Magnetic Fields - IIITime varying currents can only flow at the surface of a perfect metal but not inside itperfect metalThe argument goes as follows:• Time varying currents produce time varying H-fields• So if there are time varying currents inside a perfect metal, there will be time varying H-fields inside a perfect metal• Since there cannot be any time varying H-fields inside a perfect metal, there cannot be any time varying currents inside a perfect metal()tKrECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityCurrent Flow and Surface Current DensitySo how does a (time varying) current flow in perfect metal wires?Remember there cannot be any time varying currents inside a perfect metal……ayxConsider an infinitely long metal wire of radius acarrying a (time varying) current I in the +z-directionThe current flows entirely on the surface of the perfect metal wire in the form of a uniform surface current density K( t )K()()atItKπ2=ayxKφH()()()()() ()()rtIratKtHtKatHrπππφφ222==⇒=Can use Ampere’s law to calculate the H-field