Professor: Chungpin Liao (Hovering) 廖重賓（飛翔） cpliao@alum.mit cpliaq@nfu.tw

Electromagnetism - II （電磁學－II） Chapter 6. Polarization（偏極化）（due to the presence of ）（dielectrics (介電質)） Professor: Chungpin Liao (Hovering) 廖重賓（飛翔） cpliao@alum.mit.edu cpliaq@nfu.edu.tw EM -- Hovering

6.0 Introduction The previous chapters postulated surface charge densities (σ) as required by the B.C.’s obeyed by surfaces of conductors. Thus far, however, no consideration has been given in any detail to the physical laws which determine the occurrence and behavior of charge densities in matter（ρp& σsp）. Chap.6 ＆7 are for such physical laws. In the following, consider 2 possible pictures that could be used to explain why an object distorts an initially uniform E : EM -- Hovering

e-'s↓, ∵ Eint = 0 ∴ ρ = 0 remains sites are uncovered + o ρ>0 Conducting sphere. (metallic) On a macroscopic scale,ρ= 0 initially uniform Some e- 's are free to pass between the sphere and the lower electrode. Some positive sites are left behind and exposed at top. EM -- Hovering

E (initially uniform) induces dipoles. See next (*) ↑ & ↓similar to the case of metal sphere. + ― For ideal dielectric e- 's& are pairedon the atomic scale. + The sphere remains neutral. Polarizable & not conducting On a macroscopic scale,ρ= 0 EM -- Hovering

Because the charge of the orbiting e- 's is equal and opposite to the charge of the nuclei ( ), a neutral atom has no net charge. + （*）about formation of dipoles: via polarization. According to quantum theory, electrons orbiting the nuclei are not to be viewed as localized at any particular instant of time. It is more appropriate to think of the e- 's as 〝clouds〞 of charge surrounding the nuclei. An atom with no permanent dipoles has the further property that the center of the negative charge of the electron 〝clouds〞 coincides with the center of the positive charge of the nuclei. EM -- Hovering

In the presence of an E , the center of is pulled in the direction of E , while the center of is pushed in the opposite direction. + ― relative displacement (polarization) The two centers of charge no longer coincide. The particle (atom) acquired a dipole moment. EM -- Hovering

Charge density ρ Because charge accumulations occur via displacements of paired charges (polarization) as well as of charges that can move far away from their partners of opposite sign ( ie: free e- 's ), it is often convenient to distinguish these by separating the total charge density ρ into： ρ = ρu+ ρp paired ( bound charges ) (polarization) unpaired ( free charges) EM -- Hovering

Dipole (moment) p Dipole (moment) p polarization density P ( = Np ) permanent ( indep. of E ) modification of Maxwell’s laws Self-consistent description Generalized  dependent induced (depending on E ) simple constitutive laws P = P ( E ) ie: With a polarizable body placed into E , ρp is produced which in turn modifies E . EM -- Hovering

6.1 Polarization density (極化密度) －for non-conducting dielectrics Whether representative of atoms, molecules, groups of ordered atoms or molecules (domains), or even macroscopic particles, the dipoles are pictured as opposite charges ±q separated by a vector distanceddirected from the negative to the positive charge. An individual dipole has a dipole moment ( 偶極矩) P = qd | d | << any macroscopic dimension of interest. Consider now a medium consisting of N such polarized particles (dipoles) per unit volume. EM -- Hovering

＋ S V ＋ Q & A Q : What is the net charge q contained within an arbitrary volume Venclosed by a surface S？ A : if unpolarized : q= 0 if then polarized : q= net unneutralized charge left behind Since d  0 EM -- Hovering

(e.g. : ) S V For convenience, we’ll assume (without loss of generality) that the negative centers of charge are stationary and that only the positive centers are mobile during the polarization process. Consider the particles in the neighborhood of an element of area da on the surface S： All positive centers of charge now outside S within dV = d · da have left behind negative charge centers. V V can be a very small one enclosing the point of interest. EM -- Hovering

Gauss’ theorem polarization charge density Because there are Nd· da ( =NdV ) such negative centers of charge in dV, the net charge Q left behind in Vis ： -▽·P = ρp ∵ V ∴ arbitrary where P ≡ qNd = Np P = qd ( dipole moment ) EM -- Hovering

Q & A ρp = - ▽·P negative charge centers left behind in V1 at r Out flow of positive charge centers from V at r Q : Why introducing P？ We seem only to replace the unknown ρp with another unknown P . A : It is easier to directly link P with E than ρp with E. EM -- Hovering

To see this： Recall that Qnet= 0 in V for the polarizable sphere after the application of E The P representation guarantees this is true. Suppose we find a Vcontaining the entire polarized body so that the S surface enclosing the volume V falls outside the body. P = 0 on S 0 For arbitrary S EM -- Hovering

P(r) can be a discontinuous function, e.g., in a dielectric material surrounded by free space, the polarization density P can fall from a finite value to zero at the interface. In such regions (i.e., interfaces), there can be a surface polarization charge density（ σsp）. Q & A Q： σsp = ? A： ∵ (†) EM -- Hovering

According to , P originates on polarization charge and terminates on charge. ρp = -▽·P + ― ▽·P = outflow ρp < 0 & | ρp|= ▽·P ∴ ρp = -▽·P Look at polarization surface charges（ σsp's ）due to uniform polarization of a right cylinder dielectric : EM -- Hovering

6.2 Laws and continuity conditions with polarization Modification of vacuum Maxwell’s equ’s. with P given ( simply ) without stating its cause self-consistently. With the paired (polarized) and unpaired (free) charge densities distinguished, Gauss’ E law becomes : free bound ▽·ε0E = ρu+ρp ─ (*) ∵ρp = -▽·P free ∴▽·(ε0E+P) = ρu ≡ D EM -- Hovering

At an interface, the Gauss’ E continuity condition (jump condition ) is obtained by integrating (*) over an incremental volume : Where, recall ： (†) ie：At a surface of discontinuity : EM -- Hovering

Polarization current density （Jp）and Ampere’s law Gauss’ E law is not the only one affected by polarization. If P = P(t) then the flow of charge across the surface S comprises an electrical current. Thus, we need to investigate charge conservation, and more generally, the effect of P(t) on Ampere’s law. According to def. of P (polarization density ), the process of polarization transfers an amount of chargedQ, dQ = P·da EM -- Hovering

in ∆t But ∴ Now, ∵ ━ recall polarization charge conservation ∴ EM -- Hovering

=0 = Next, the Ampere’s law: Recall that in the EQS approx., the magnetic field intensity H is not usually of interest, and so Ampere’s law is of secondary importance. But if H were to be determined, Jp would make a contribution. Besides, modification to the Ampere’s law will reveal the unpaired ( free ) charge conservation law, as we’ll see. Ampere’s law： EM -- Hovering

∵ conservation of polarization charge conservation of free charge Thus, the rate of charge transport in a material medium consists of a current density of unpaired charge Ju and polarization current density Jp, each obeying its own conservation law. EM -- Hovering

= ≡ ∴ Displacement flux density (D ) （位移通量密度） Primarily in dealing with E-dependent polarization phenomena, it is customary to define : ∴ if isotropic & linear We regard P as representing the material and E as a field quantity induced by the external sources and the sources within the material. This suggests that D be considered a 〝hybrid〞quantity. Gauss’ E law： EM -- Hovering

≡ = Continuity ( jump ) condition : ― (6.2.3) with becomes : i.e. ∴The divergence of D ( ▽·D = ρu) and the jump in normal D determine the unpaired charge densities. When in free space, D = -ε0E Ampere’s law can also be written as： EM -- Hovering

P = P(r,t) given, and follow. If ρu(r,t) is also given throughout the material, the total charge density in a Gauss’ E and jump condition are known. 6.3 Permanent polarization Usually, P = P(E). However, in some materials a permanent polarization is 〝frozen〞 into the material, meaning that P(r,t) is prescribed, indep. of E. Electrets ( 駐極體 ) used to make microphones and telephone speakers, are often modeled in this way. Thus, a description of permanent polarization problems follows from the same format as used in Chapters 4 & 5. EM -- Hovering

Light in bulk matter – material aspect Optical concern : the response of dielectric ( i.e., nonconducting ) materials to EM Fields transparent dielectric : lens, prizm, air, plates, films, water. Phase speed of light in a homogeneous, sotropic dielectric (介電質) : Absolute index of refraction : , Since normally dielectrics of interest are nonmagnetic, Maxwell’s relation EM -- Hovering

∵ material hν atom : scatters light 1 2 Dispersion (色散) : In short, n = n (ω) ω<ω0 (or hν< hν0= Ei→j ) 1 absorb hν then re-emit photons (redirect light) only Non-resonant scattering (ground state) Matching Ei→j: excitation of atomic e-’s (quantum jump) absorption 2 dense medium collision Thermalization (randomization) Dissipative absorption EM -- Hovering

an atom e- e- e- + e- An r rule, the closer the frequency of the incident beam is to an atomic resonance, the more strongly will the interaction occur, and in dense materials, the more energy will be dissipatively (耗散) absorbed. Selective absorption Your color of skin, hair, eyes, clothing, leaves, fruits, etc. Why? (math) Dipole moment per unit volume : P = (ε-ε0)E For isotropic homogeneous and linear media : FE=qeE(t)=q0E0cosωt ( driving force on an e- ) EM -- Hovering

an atom e- e- e- + e- = = ω0 = natural freq. (resonant) of the bound e- dipole Medium= Now, let x(t) = x0cosωt (a guess) ≡ in-phase 180° out-of-phase EM -- Hovering

& For rarefied gas Now, since ∴We arrive at dispersion relation : In general, Works for rarefied gas (稀薄) EM -- Hovering

cf. In dense media : if ( little dissipation ) → & Colorless, transparent media have their ω0's outside the visible range. ( ω02 >> ω2 ) n ≈ const. Visible range EM -- Hovering

ie : free space Note that we still use the jump condition σsp = -n· (Pa-Pb) derived from ρp = -▽·P for permanent P . It should be recognized that once ρpis determined fromthe given P the methods of Chapters 4 & 5 ( EQS ) are directly applicable in obtaining E. EX. 6.3.1. A permanently polarized sphere uniformly polarized by P = P0 z → E = ? ( if no other field sources in the free space ) Step : P → →ψ→E ρp ρsp If R = 1 → unit sphere see next page EM -- Hovering

EM -- Hovering

free space uniformly polarized by P = P0 z σsp = ? ρp = ? in free space within the material ∵ ρp= 0&σsp = P0cosθ ∴ It is this surface charge density that gives rise to E . ( Abrupt changes of the normal component of P entail polarization surface charge densities(σsp’s ) ) EM -- Hovering

With ρu+ρp = 0 for both the free space and within the sphere, we have : ie : Laplace’s equ. To solve ψ, we now need the B.C.’s : Recall the EQS B.C.’s (P.151 ) : (††) ( Form Faraday’s ) actually not restricted to EQS (†) ( Form Gauss’s ) and ( p.90) ( 4.14) (*) h→0 (*) ∵(††) EM -- Hovering

∴ (††’) While (†) gives : ( †’) With (a)≡(o) outside (b)≡(i) inside sphere B.C.’s become : (at r = R) Solutions of in spherical coordinate (p.181) : We choose : 1 2 3 EM -- Hovering

Putting ψoψi in B.C.’s : ∴ ∴ = ∴ Q: What would be equivalent moment of the dipole at the origin giving the same result? cf. p.101 (4.4.10) dipole field volume EM -- Hovering

EX. 6.3.2. Fields due to volume polarization charge (ρp≠0 ) and B.C.’s plane parallel electrodes The problem is identical to that considered in Ex.5.6.1 (p.163) . see Fig.5.6.2 on p.165 EM -- Hovering

i EX. 6.3.3 An electret ( 駐極體 ) microphone －a device to pick up voice or sound a conducting grounded diaphragms (movable →h(t)) a thin sheet of permanently polarized material Given h(t), VR=? within electret ∵P0 & discontinuity air gap electret → uniform E. in (a) & (b) regions EM -- Hovering

Formally, we have just solved Laplace’s equ. in each of the bulk regions. ( However, expressing things in E is easier here. ) B.C.1 : x 1 at x = d B.C.2 : recall 33 2 EM -- Hovering

cf. , 1 2 The figure shows the situation where ie: In the air gap, the field due to σsu on the electrodes reinforces that due to σsp . While in the electret, it opposes the downward directed field due to σsp. Next, ρu = ? (0 < x < h) → i → υ To calculate i＝ EM -- Hovering

∴ integrate over S (Gaussian surface) 0E ( 0) from i = condEwire is neglected On bottom electrode ― (&) energy from sound wave (＆) : ∴ a υ(t)-differential equ. of time-varying coefficients EM -- Hovering

Not only difficult to solve but also υ(t) cannot be a good replica of h(t), as required for a good microphone, if all terms are of equal importance. Can be remedied if the deflection h1(t) is only a perturbation around an equilibrium value h0 . i.e.:h1(t) << h0 Linearization expansion : h(t) = h0+h1 h(t) = h0 υ(t) =υ0 Equilibrium see if they can exist (＆) : =0 =0 Note: v0 vR vcap EM -- Hovering

where h0  C0 =0 Q: C0 =q0/v0 < , but v0 =0?  C0 = q0/vcap, vcap = d P0/0 Linearization(perturbation) around the above equilibrium (＆) → ≡ (†) driver 1 2 We could solve (†) for its response (υ1(t)）to a sinusoidal drive EM -- Hovering

Alternatively, however, the freq. response can be determined at two limits, with more physical insight : ωh1 low freq. : << : 1 2 high freq. : >> : 1 2 Usually, sound pressure wave → h(t) → υ(t) , instead of having h(t) as given. ∴ Dynamical equations of the diaphragm have to be treated in general, to give h(t). Q: Weird microscope: h(r,t), P0(E) , and? EM -- Hovering

6.4. Constitutive laws of polarization → i.e., P = P(E) =? macro macro Dipole formation, or orientation of dipolar particles, usually depends on the local E in which the particles are situated. This local microscopic E is not necessarily equal to the microscopic E field. Yet certain relationships between the macroscopic quantities E and P (called constitutive laws ) can be established without a knowledge of the relations between the local microscopic fields and the macroscopic E fields. One special constitutive law : permanent polarization P = P0(r) indep. of E. Much commonly, P in media depend on E. EM -- Hovering

Isotropic media : → no preferred orientation in the absence of E. If we assume that P in an isotropic medium depends on the instantaneous field ( E ) and not on its past history, then P = P(E) and P//E Indeed, if P//E , then a preferred direction different from that of E would need to exist in the medium, which contradicts the isotropy assump. An 〝electrically nonlinear〞medium for which P saturates〞for large valves of E. | P | | E | If the medium is electrically linear, in addition to being isotropic, then a linear relationship exists between E and P : χe : dielectric susceptibility EM -- Hovering

All isotropic media behave linearly and obey P = ε0χe E if the applied E is sufficiently small. An isotropic medium cannot have a term in the Taylor expansion of P independent of E : i.e., =0 linearity nonlinearity Displacement flux density D ≡ε0E+P ∴ for linear, isotropic media (& no memory ) : D ≡ ε0E+ ε0χe E = ε0 (1+ χe)E = εE = ε0εrE relative dielectric coefficient ∴εr = 1+χe Typical values of χecan be found in Table 6.4.1 (P.221) EM -- Hovering

Index of refraction μr →1 ≈ There are materials in which P depends not only on the current E , but also on the sequence of preceding states as well ( hysteresis 遲滯 ) Transducers ：P ( E, temp., strain ) usingP pyroelectric piezoelectric EM -- Hovering

Recall：( linear media ) Gauss’ E law : Faraday’s law : 6.5 Fields in the presence of electrically linear dielectrics － self-consistent treatment of P & E. ( P ←→E ) interact In this and the next section, the polarization density P is induced by E , which itself is modified by P. ∵ D ≡ε0E+P =ε0E+ ε0χe E = ε0 (1+ χe )E = ε0εrE = εE ∴This self-consistent approach can be reduced to solving E (or D ) with the whole 〝feedback〞 from P given in ε EM -- Hovering

Professor: Chungpin Liao (Hovering) 廖重賓（飛翔） cpliao@alum.mit cpliaq@nfu.tw