Lecture 16 Dielectrics: Dipole, Polarization

Transcription

1 Lecture 16 Dielectrics: Dipole, olarization Sections: 4.7, 5.7 Homework: See homework file

2 Electric Dipole and its Dipole Moment electric dipole: two point charges of equal charge but opposite polarity in close proximity electric dipole is characterized by its moment p (a vector that points from Q to Q!) p= Qd, C m Q E p= Qd Q d LECTURE 16 slide 2

3 Torque on Dipole in External Field dipole tends to rotate so that its moment aligns with the external field torque fulcrum is at dipole s center (dipole is bound) e 2 E d T = Q sinα Te = p E 2 F the dipole moment p determines the torque exerted on the dipole by an external electric field E E F Q Q p F F = = QE Fsinα T α e torque definition: T= R F dielectric polarization is the tendency of bound dipoles to orient along external electric fields LECTURE 16 slide 3

4 olarization in Dielectrics dielectrics have negligible DC conductivity and their charges are assumed immobile, i.e., they are bound to a fixed location polarization involves the orientation of microscopic bound dipoles so that their moments align more or less with the external E field there is a variety of mechanisms of polarization electronic polarization (atomic level) E ionic polarization (molecular level) p = 0 p orientational polarization occurs in materials that have permanently polarized microscopic sub-domains (electrets: quartz, polypropylene; polar liquids: water) LECTURE 16 slide 4

5 olarization Vector polarization vector is the dipole moment per unit volume = lim v 0 p i v, C/m dp= dv assume n (m 3 ) bound dipoles (atoms, molecules, sub-domains) per unit volume of averaged distance and orientation given by i 2 d av and dipole charge q d p i = lim = v 0 v N( q d ) v i d av number density of bound dipoles =n q d d av p av averaged dipole moment = np av What is the polarization vector in vacuum? LECTURE 16 slide 5

6 olarization Vector Example Atomic hydrogen contains n = atoms/m 3 at a certain temperature and pressure. When an electric field of 4 kv/m is applied, each dipole formed by an electron and the positive nucleus has an effective length of m. Find the atomic dipole moment p av and the polarization vector if the electric field is along a x. q e C LECTURE 16 slide 6

7 olarization Vector and Surface Bound Charge 1 specimen is homogeneous, isotropic, and L surrounded by vacuum (air) E ext is orthogonal to surface planes unit normals a n point inwards uncompensated bound charge exists on surface only ρ sb =Q sb /A ptotal Qsb L ρsba L = = ae = ae = ρsb ae = ρsban v v A L total dipole moment of specimen ext E ext = 0 Q sb a n a n A Qsb ext Eext = 0 is along E ext (isotropic material) and points from ρ sb = a, where = ρ n n n sb to ρ sb LECTURE 16 slide 7

8 olarization Vector and Surface Bound Charge Example 1 A slab of polarizable material has an area A and thickness L (see plot in pervious slide). The dipole number density is n. The slab is subjected to an external field E ext normal to A. The resulting average dipole moment in the material is p av, co-parallel with E ext. Express the surface bound charge density ρ sb on both sides of the slab in terms of p av and the polarization vector. LECTURE 16 slide 8

9 olarization Vector and Surface Bound Charge 2 assume E ext and are not orthogonal to the surface planes ptotal Qsb L ρsba L ρsb = = ae = ae = ae cosα = v v A L cosα boundary condition for the normal component ρsb an = ae an = ρ cosα E ext n = a n = ρsb ext = 0 holds regardless of surface orientation with respect to E outside the specimen = 0 compare with Da = ρ in EC n s sb A A LECTURE 16 slide 9 L A α a n α a n A A A ext = 0

10 olarization Vector and Surface Bound Charge Example 2 A slab of polarizable material has an area A and thickness L. The dipole number density is n. The slab is subjected to an external field E ext, which is at an angle α with respect to A (see plot in previous slide). The resulting average dipole moment in the material is p av, coparallel with E ext. Express the surface bound charge density ρ sb on both sides of the slab in terms of p av and the polarization vector. LECTURE 16 slide 10

11 olarization Vector and Surface Bound Charge 3 so far we have considered interface between air (non-polarizable medium) and dielectric (polarizable medium) now we consider the interface between two dielectrics ( ) (1) (2) (1) (2) sb = sb sb = ρ ρ ρ cosα (1) (2) ( ) = a ρ sb n1,2 general boundary condition for at dielectric interface E ext strongly polarizable dielectric #1dielectric #2 a n1,2 (1) ρ sb (2) weakly polarizable LECTURE 16 slide 11

12 olarization Vector and Volume Bound Charge make use of relation to surface charge inside volume = ρ ins n flux of is negative if bound charge inside surface s is positive s= ρ ins sb s take integral over s s s sb d s = Q [ v ] ins b d s = v ρ vb net bound charge inside s dv E ext a n L s s ins Q b a n Gauss Theorem = ρ vb Compare with D = ρv LECTURE 16 slide 12

13 Susceptibility and ermittivity 1 two types of charges: free charge (in conductors) and bound charge (in dielectrics) from microscopic point of view both types of charge exist in vacuum vacuum flux density relates to the total charge density = = ( ε0e) ρvt ρvf ρvb relate D (flux density) vector to free charge only at the same time = ρvb D = ρvf ( ε E) = D 0 D= ε 0 E LECTURE 16 slide 13

14 Susceptibility and ermittivity 2 polarization and E-field in isotropic materials: susceptibility relative permittivity = χe( ε0e) D= ε0 (1 χe) E ε r = 1 χe D= εε 0 re= εe anisotropic dielectrics (permittivity tensor) Dx εxx εxy εxz Ex D = ε ε ε E D = y yx yy yz y Dz εzx εzy ε zz Ez ε r εe nonlinear dielectrics [χ e = f(e), ferroelectric materials] What is the susceptibility of vacuum? LECTURE 16 slide 14

15 Susceptibility and ermittivity Example For the example of polarized hydrogen in sl. 6, calculate: (a) the susceptibility χ e,(b) the relative permittivity ε r, (c) the flux density vector D. LECTURE 16 slide 15

16 Dielectric Strength Homework We have implicitly assumed that a dielectric material preserves its properties regardless of the field strength. In practice, if the field magnitude exceeds certain critical value (the dielectric strength of the material), the polarization forces become too strong, the electrons break free from the atoms, accelerate through the material, thereby heating it, damaging its molecular structure, and ultimately causing even more electrons to break free. This avalanche process leads to arcing/sparking, which destroys the material (dielectric breakdown). The dielectric strength E ds is the strongest E-field that the material can sustain without breakdown. glass mica oil air MV/m 200 MV/m 12 MV/m 3 MV/m LECTURE 16 slide 16

17 You have learned: what a dipole is and how to calculate its potential and E field what torque an external E field exerts on a dipole of moment p what polarization is and how it is described by the polarization vector how polarization relates to surface and volume bound charge what susceptibility and permittivity mean what dielectric strength is LECTURE 16 slide 17