Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06EE243 Advanced Electromagnetic TheoryLec # 7: Dielectric Materials• Geometry for Homework Problem 3.1• Multipoles• Interaction of Multipoles• Boundary Value Problems for Dielectrics• Relationship of dielectric constant to moleculesReading: Jackson Ch 4Copyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Geometry HW 3.1• Positive line source induces negative line sources on cylinders that can be approximated as being at their centers• Find the potential at the observation point• See if reciprocity holds when source and observation are interchangedxy(0,0)(1,0)(2,0)(0,0.5)+λs1Observation−λ1−λ2Copyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Overview• Conductivity is produced by free carriers (holes and electrons) moving to surfaces.• But charges (electrons) bound to nuclei (protons) can be displaced slightly by E.• This creates dipole and possibly higher spherical expansion eigenfunction moments.• They in turn create– Surface charge; reduction in internal field; internal lattice fields; stored energy; force on dielectricsCopyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Dielectrics• Charge produces E field• E field produces Polarization and surface charge• E field inside dielectric is loweredcharge 1 = q1Surface charges due to polarization PDielectric Object++++---PCopyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Eigenfunctions in Cylindrical Coordinates• x and y become ρ and φ• Azimuthal = sin (mφ) and cos (mφ)• Radial = Bessel functions J (oscillatory like)• Hankel functions and Modified Bessel functions I and K (exp. like)• Boundary Conditions– Wedge: Eigenfunction values of m (could be fractional)– Cylinders: Zeros of Bessel functionsJackson pp. 112Copyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Eigenfunctions in Spherical Coordinates• x, y, z become ρ, φ and θ• Azimuthal uniform = sinusoidal φ• Or azumuthal and θ combine on sphere surface as spherical harmonics()()()φφθθπφθφθφθφθφθθπφθilmlmlmlilmlmimmllmeYdgYAYAgePmlmllYcossin815),(),(),(),(),(cos!!412),(21*0−=Ω==+−+=∫∑∑∞=−=ExampleJackson pp. 108Copyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Spherical Harmonic Expansion• Representation outside the charge distribution• Take eigenfunction moments over distribution()()()'3'2''',,530''*''0010)(3...2141)(),(||)(41)(),(12441)(xdxrxxQrxxQrxprqxdxrYqxxxxxrYqlxijjijijijiijllmlmlilmllmlmρδπερφθρρπεφθππε∫∑∫∫∑∑−=⎥⎦⎤⎢⎣⎡++⋅+=ΦΩ=−=Φ+=Φ∞=−=+Copyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Results Spherical Harmonic Expansion• Negative gradient gives Eρ, Eθand Eφ•Energy• Dipole case• For dipole the average over a spherical volume is the value at the center of the sphere ()||4)(300xxpnpnxE−−⋅=πε()...061)0()0( +∂∂−⋅−Φ=∑∑jiijijxEQEpqW3210212112||4))((3xxpnpnppW−⋅⋅−⋅=πεCopyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Ponderable Media• Apply Maxwell’s equations locally and then average to get macroscopic ME.• Curl E = 0 added up over volume remains the same => - Gradient of Potenetial• Media free charges and dipole moments contribute to potential P = sum pMOLECULE• Add up charge and dipole contributions• Integrate dipole term by parts to move differential operator from 1/r to P.• Obtain Ponderable = having serious weightCopyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Ponderable Media: Results• Div P is like charge• Displacement• Boundary Conditions– D normal discontinuous– E tangential continuous0)()(][12112211200=×−=⋅−=⋅∇=⋅∇+=⋅∇−=⋅∇nEEnDDEDPEDPEσερρερεCopyright 2006 Regents of University of California11EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Image Charge Geometry-aaqq’’q’q’’ is necessary to create the fields for z < 0.Copyright 2006 Regents of University of California12EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06BVP Dielectrics: Image in Half-Space• Charge q at z = a in material with ε1above half space of material with ε2• Postulate two image charges– q’ at z = -a for fields z > 0– q’’ at z = a for fields z < 0• Match D normal and E tangential•Find()2/322121202112121''1212'12)(2ρεεεεεεπσεεεεεεε+⎟⎟⎠⎞⎜⎜⎝⎛+−=⋅−−=⎟⎟⎠⎞⎜⎜⎝⎛+=⎟⎟⎠⎞⎜⎜⎝⎛+−−=aaqnPPqqqqPOLCopyright 2006 Regents of University of California13EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Integral Representations & EquationsDielectric Objectρ(x’)Surface Charge PatchesObservation Point outside moves to surfaceBoundary Integration pointVolume integration pointsObservation Point inside moves to surfaceadnxxGxnxxGxdxxGxxSV′⎥⎦⎤⎢⎣⎡′′′Φ−′Φ′+′′′=Φ∫∫δδδδπρπε),()(),(41),()(41)(31adnxxGxnxxGxdxxGxxSV′⎥⎦⎤⎢⎣⎡′′′Φ−′Φ′+′′′=Φ∫∫δδδδπρπε),()(),(41),()(41)(32ε1ε2PCopyright 2006 Regents of University of California14EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Integral Representations for Dielectrics• Separate representations inside and outside homogeneous dielectrics– Use a free space Green’s function for the media in which the observation point is immersed– Will require knowledge of both the potential and its normal derivative on the boundaryCopyright 2006 Regents of University of California15EE 210 Applied EM Fall 2006, Neureuther Lecture #07 Ver 09/17/06Integral Equations for Dielectrics• Use Integral Representations • Form integral equations by– Requiring potential to be continuous and D normal discontinuous by div P.• Solve integral equations– Every point has two unknowns (potential and Div P) and two