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Lecture 7 Polarization Charge Dipoles and Dipole Moments

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Lecture 7 Polarization In this lecture you will learn: • Material Polarization • Mathematics of Polarization • Dielectric Permittivity • Conductors Vs Dielectrics • Appendix ECE 303 – Fall 2007 – Farhan Rana – Cornell University Charge Dipoles and Dipole Moments Consider a charge dipole: + q r d − q Dipole moment of the charge dipole is a vector p r such that: r pr = q d ECE 303 – Fall 2007 – Farhan Rana – Cornell University 1 Conductors in Electric Fields – A Review φ = V φ = 0 • Consider the problem when a conductive + - plate was placed inside an electric field + - + - + - • Conductors have “free charges” that are able + - to move around (mostly these “free charges” + - are electrons that are not attached to any + - particular atom) + - • Under the influence of external E-field these +- free charges move to completely screen out V the E-field from within the conducting material φ = V φ = 0 + - + - Conductors + - + - Sea of Free Electrons + - + - + - + - +++ + - + - + +ve nucleus + - + - +++ + - + - -ve sea of free + - + - +++ electrons +- +++ V σ ECE 303 – Fall 2007 – Farhan Rana – Cornell University Dielectrics in Electric Fields - Polarization Many materials do not have “free electrons” that can move around, but have electrons bound to atoms, as shown below Dielectric Dielectric in an E-field E + + + + +ve nucleus - + - + - + + + + -ve electron - + - + - + cloud + + + - + - + - + + + + - + - + - + + + + - + - + - + + + + - + - + - + In the presence of an E-field, the electron cloud in each atom distorts ALMOST INSTANTANEOUSLY so that each atom looks like a charge dipole ECE 303 – Fall 2007 – Farhan Rana – Cornell University 2 Dielectrics in Electric Fields – Polarization Vector r d r r - + p = Qd = dipole moment of each dipole -Q +Q r The polarization vector P is a vector such that: E r P = N pr - + - + - + v = NQd - + - + - + Where N is the number of charge dipoles per unit - + - + - + volume in the material - + - + - + r The units of P are: Coumlombs/m2 - + - + - + r The polarization vector P characterizes the - + - + - + polarization density of the medium under the influence of the electric field ECE 303 – Fall 2007 – Farhan Rana – Cornell University Dielectrics in Electric Fields – Electrical Susceptibility Naturally, one would expect the polarization of the material to be proportional to the strength of the electric field: r r P ∝ E r r ⇒ P = εo χe E The constant of proportionality χe is called the electrical susceptibility of the material E - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + ECE 303 – Fall 2007 – Farhan Rana – Cornell University 3 Material Polarization and Surface Charge Densities r P = P xˆ - + - + - + - + - + E - + - + - + - + - + σ = −P p σ p = +P - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + x • The stuff inside the box in on the average charge neutral (same number of positive and negative charges) • There is a net negative surface charge density on the left facet of the material as a result of material polarization • There is a net positive surface charge density on the right facet of the material as a result of material polarization ECE 303 – Fall 2007 – Farhan Rana – Cornell University Material Polarization and Surface Charge Densities d Area = A E - + - + - + - + - + How much charge is r in this volume?? - + - + - + - + - + P = P xˆ - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + Total interface negative charge due to dipoles in the volume Ad = - Q N A d If we divide the total interface charge by the area A we get the interface charge per unit area which would be the surface charge density σp QNAd ⇒ σ = − = −NQd = −P p A ECE 303 – Fall 2007 – Farhan Rana – Cornell University 4 Material Polarization and Volume Charge Densities More generally, one can write a volume polarization volume charge density due to material polarization as: r ρp = −∇ .P (A formal proof is given in the Appendix) There will be a net non-zero volume charge density inside a material if the material polarization is varying in space In 1D situations: ∂P ρ ()x = − x p ∂x ECE 303 – Fall 2007 – Farhan Rana – Cornell University Material Polarization and Charge Densities E σ p = −P σ p = +P r P = Pxˆ x x=0 x=L Px P 0 L x r ρp = −∇ .P ∂P ρ = − x p ∂x Pδ (x − L) 0 L x ∂P − Pδ ()x ρ = − x = −P δ ()x + P δ (x − L ) p ∂x ECE 303 – Fall 2007 – Farhan Rana – Cornell University 5 Mathematics of Polarization – The “D” Field Gauss’ Law states: r ∇ .εo E = ρ But charge densities could be of two types: 1) Paired charge density ρp (due to material polarization) 2) Unpaired charge density ρu (due to everything else – the usual stuff) So: r r r ρ = −∇ .P ∇ .εo E = ρu + ρp = ρu − ∇ .P Using: p r r ⇒ ∇ .()εo E + P = ρu If one defines the D-field inside materials as: r r r D = εo E + P Then inside materials Gauss’ Law becomes: r ∇ .D = ρu ECE 303 – Fall 2007 – Farhan Rana – Cornell University Mathematics of Polarization – Dielectric Permittivity If one defines the D-field as: r r r r Then: ∇ .D = ρ D = εo E + P u r r r r r r Note that: D = εo E + P = εo E + εo χe E = εo (1+ χe )E If one defines the dielectric permittivity of a material as: ε = εo (1+ χe ) then one can write the D-field inside materials as: r r D = ε E Inside materials the D-field obeys the Gauss’ Law: r r ∇ .D = ∇ .ε E = ρu ECE 303 – Fall 2007 – Farhan Rana – Cornell University 6 Mathematics of Polarization – Polarization Current Density So far we have looked at the current density due to the motion of free unpaired charges: r r E Ju = σ E - + - + - + The motion of paired charges also results in + + + a current density: - - - r r - + - + - + r ∂P ∂ (εo χe E ) Jp = = ∂t ∂t - + - + - + r Jp is called the polarization current density r r r r ∂ (ε E ) Ampere’s Law is correctly given by: ∇ × H = J + J + o u p ∂t r r r ∂ (ε E) Which can also be written as: ∇ × H = J + u ∂t ECE 303 – Fall 2007 – Farhan Rana – Cornell University Mathematics of Polarization – Modified Maxwell’s Equations Putting it all together …. One can forget all about material polarization, and polarization charge densities, and polarization current densities, as long as one uses the dielectric permittivity ε instead of the free space permittivity εo Maxwell’s equations in dielectric materials take the form: r r ∇. ε E = ρ ∇. (εo E ) = ρu + ρp ( ) u r r ∇. µo H = 0 Here the charge density ∇. µo H = 0 OR ρu is the unpaired charge density r r r ∂ µ H r ∂ µoH ∇ × E = − o ∇ × E = − ∂t ∂t r r r r r ∂ ()ε E r r ∂ ()ε E ∇ × H = J + J + o ∇ × H = Ju + u p ∂t ∂t Here the current density is due to the unpaired charges ECE 303 – Fall 2007 – Farhan Rana – Cornell University 7 Dielectric Permittivity – Boundary Conditions - I σu + σ p How to relate the electric fields on both sides of E E the dielectric interface ?? 1 2 Gauss’ Law in the presence of dielectric material is: ε1 ε2 r r ∇. (D) = ∇. (ε E ) = ρu r r or ∫∫ D . da =∫∫∫ ρu dV σu + σ p Draw a Gaussian surface at the interface: D D (D2 − D1)A = σu A 1 2 ⇒ D2 − D1 = σu A ε1 ε2 or ε2 E2 − ε1 E1 = σ u ECE 303 – Fall 2007 – Farhan Rana – Cornell University Dielectric Permittivity – Boundary Conditions - II σu + σ p How else can one relate the electric fields on both sides of the dielectric interface ?? E E Gauss’ Law in the presence of dielectric material is 1 2 also: r A ∇. (εo E ) = ρu + ρp ε1 ε2 Draw a Gaussian surface at the interface: εo (E2 − E1) A = (σu + σ p ) A or εo (E2 − E1) = σu + σ p σu + σ p The parallel component of the E-field at a material interface is always continuous at the interface (no E1 E2 change here) ε1 ε2 (E2 − E1) = 0 ECE 303 – Fall 2007 – Farhan Rana – Cornell University 8 Dielectrics Vs Conductors φ = 0 Conductors: φ = V + - + - + - Free unpaired charges move to completely + - + - + - + - + screen the E-field on time scales longer than - + - + - + the dielectric relaxation time - + - + - + - + - + - + - + - + - + Dielectrics: - + - + - + - Paired charges originating due to material polarization partially screen the E-field +- almost instantaneously V φ = V φ = 0 When σu = 0: + - + - + - + - ε2 E2 − ε1 E1 = 0 + - + - + - + - also: + - + - + - + - εo (E2 − E1) = σ p + - + - The discontinuity of the normal component of the + - + - E-field is due to the paired charges at the interface +- (even when σu = 0) V ECE 303 – Fall 2007 – Farhan Rana – Cornell University Conductors or Dielectrics Some materials are conductors and some are dielectrics …………. Dielectrics Conductors Tightly Bound Electrons Sea of Free Electrons + + + + +ve nucleus +++ + +ve nucleus + + + -ve electron +++ -ve sea of free cloud electrons + + + +++ + + + +++ + + + +++ + + + +++ ε σ ECE 303 – Fall 2007 – Farhan Rana – Cornell University 9 Conductors and Dielectrics ……. but many important materials are conductors as well as dielectrics Tightly bound core electrons and a sea of free electrons • Most conductors like gold, copper, + +ve nucleus and silver, and semiconductors like + + + Silicon, are both conductors and + + + -ve electron dielectrics cloud + + + • They have a sea of free electrons -ve sea of free electrons that results in a finite value of + + + conductivity and they also have tightly bound core electrons that + + + result in a value for the dielectric permittivity + + + ε σ ε Dielectric relaxation time for these materials = τ = d σ ECE 303 – Fall 2007 – Farhan Rana – Cornell University Appendix: Polarization Charge Density - I The expression relating the polarization charge density to the divergence of the polarization vector, r ρp = −∇ .P can be proved more formally as shown below: The potential of an isolated dipole sitting at the origin and pointing in the z-direction is: pr cos()θ r r + q rr p = qd φ()= 2 θ 4πεo r r d − q More generally, the potential of a dipole sitting at position r’ and pointing in an arbitrary direction is: r r + q 1 pr .
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  • A Beautiful Approach: Transformational Optics

    A Beautiful Approach: Transformational Optics

    A beautiful approach: “Transformational optics” [ several precursors, but generalized & popularized by Ward & Pendry (1996) ] warp a ray of light …by warping space(?) [ figure: J. Pendry ] Euclidean x coordinates transformed x'(x) coordinates amazing Solutions of ordinary Euclidean Maxwell equations in x' fact: = transformed solutions from x if x' uses transformed materials ε' and μ' Maxwell’s Equations constants: ε0, μ0 = vacuum permittivity/permeability = 1 –1/2 c = vacuum speed of light = (ε0 μ0 ) = 1 ! " B = 0 Gauss: constitutive ! " D = # relations: James Clerk Maxwell #D E = D – P 1864 Ampere: ! " H = + J #t H = B – M $B Faraday: ! " E = # $t electromagnetic fields: sources: J = current density E = electric field ρ = charge density D = displacement field H = magnetic field / induction material response to fields: B = magnetic field / flux density P = polarization density M = magnetization density Constitutive relations for macroscopic linear materials P = χe E ⇒ D = (1+χe) E = ε E M = χm H B = (1+χm) H = μ H where ε = 1+χe = electric permittivity or dielectric constant µ = 1+χm = magnetic permeability εµ = (refractive index)2 Transformation-mimicking materials [ Ward & Pendry (1996) ] E(x), H(x) J–TE(x(x')), J–TH(x(x')) [ figure: J. Pendry ] Euclidean x coordinates transformed x'(x) coordinates J"JT JµJT ε(x), μ(x) " ! = , µ ! = (linear materials) det J det J J = Jacobian (Jij = ∂xi’/∂xj) (isotropic, nonmagnetic [μ=1], homogeneous materials ⇒ anisotropic, magnetic, inhomogeneous materials) an elementary derivation [ Kottke (2008) ] consider×