Maxwell's Equations Applied Electromagnetic Waves 3

Material Properties In a material medium: D = εE (ε = permittivity) B = µH ( µ = permeability ) ε = ε 0ε r εr = relative permittivity µ = µ0 µr µr = relative permeability Note: The fields E and B are the physical fields, meaning they exert a force on a charged particle that can be measured. The other two fields are defined. 27 Material Properties (cont.) Where does permittivity come from? V0 + - + + + + - Ex - - − + - - - + - + - + + - + Water D ≡ ε0 E + P 1 P≡ ∆V ∑p ∆V i Molecule: pi = p  i p = qd 28 Material Properties (cont.) D ≡ ε0 E + P Linear material: P = ε 0 χe E The term χe is called the “electric susceptibility.” Note: χe > 0 for most materials so = D ε 0 E + ε 0 χe E = ε 0 (1 + χ e ) E Define: ε r ≡ 1 + χe Then D = ε 0ε r E 29 Material Properties (cont.) Teflon Water Styrofoam Quartz εr εr εr εr = 2.2 = 81 (a very polar molecule, fairly free to rotate) = 1.03 =5 Note: εr > 1 for most materials: ε r ≡ 1 + χe , χe > 0 30 Material Properties (cont.) Where does permeability come from? B Iron Because of electron spin, atoms tend to acts as little current loops, and hence as electromagnetics, or bar magnets. When a magnetic field is applied, the little atomic magnets tend to line up. H ≡ 1 M ≡ ∆V ∑m ∆V i 1 µ0 B−M Electron: i mi = nˆi ( iA) a nˆi A = π a2 31 Material Properties (cont.) = B µ0 H + µ0 M The term χm is called the “magnetic susceptibility.” M = χm H Linear material: Note: χm > 0 for most materials so = B µ0 H + µ0 χ m H = µ0 (1 + χ m ) H Define: µ= r (1 + χ m ) Then B = µ0 µ r H 32 Material Properties (cont.) Material Relative Permeability µr Vacuum 1 Air 1.0000004 Water 0.999992 Copper 0.999994 Aluminum 1.00002 Silver 0.99998 Nickel 600 Iron 5000 Carbon Steel 100 Transformer Steel 2000 Mumetal 50,000 Supermalloy 1,000,000 Note: Values can often vary depending on purity and processing. http://en.wikipedia.org/wiki/Permeability_(electromagnetism) 33 Lorenz Force Law The fields E and B are the two physical fields, since they exert a force on a particle (the Lorenz force law). The D and H fields are the defined fields. Lorenz force law: F= q (E + v × B ) This experimental law gives us the force on a particle with charge q moving with a velocity vector v. 34 Terminology Properties of ε or µ Variation Independent of Dependent on Space Homogenous Inhomogeneous Frequency Non-dispersive Dispersive Time Stationary Time-varying Field strength Linear Non-linear Direction of E or H Isotropic Anisotropic 35 Isotropic Materials Isotropic: This means that ε and μ are scalar quantities, which means that D || E (and B || H ) D =εE B = µH y y B D E H x x 36 Anisotropic Materials Here ε (or μ) is a tensor (can be written as a matrix) Example: “biaxial medium” Dx = ε x E x   Dy = ε y E y  Dz = ε z E z  or  D x  ε x 0 0   E x  D  =  0 ε  E  0 y  y   y  Dz   0 0 ε z   E z  D= ε ⋅ E This results in E and D NOT being in the same direction. 37 Anisotropic Materials (cont.) Practical example: uniaxial substrate material  Dx  ε h 0 0   Ex  D  =  0 ε  E  0 h  y   y  Dz   0 0 ε v   Ez  There are two different permittivity values, a horizontal one and a vertical one. Teflon substrate Fibers (horizontal) 38 Anisotropic Materials (cont.) RT/duroid® 5870/5880/5880LZ High Frequency Laminates This column indicates that εv is being measured. 39