Lecture 12 Dielectrics: Dipole, Polarization

Lecture 12 Dielectrics: Dipole, Polarization Sections: 4.7, 5.7 Homework: See homework file

LECTURE 12

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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 12

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Field of a Dipole: Far-field Assumption distance from dipole to observation point is much larger than dipole z

P(r ,  ,  )

R



Q

x

r

( d / 2) cos 

d

actual

R

y



z

R

Q



Q

rd far-field approximation

x

r

( d / 2) cos 

d

P(r ,  ,  )

R y



Q

LECTURE 12

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Field of a Dipole – 2 • the far-field assumption leads to d d R  r  cos  , R  r  cos  2 2 • using superposition

Q  1 1  V ( P)     4  R R   1 1 Q   V ( P)   4 r  d cos  r  d cos   2 2 

  Q d cos    4 2  d    2   r   cos     2    0 p  ar Qd cos  V ( P)   2 4 r 4 r 2 LECTURE 12

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Field of Dipole – 3 • dipole potential decreases with distance as 1/r2 • dipole potential is zero in the azimuthal plane θ = 90° • E field obtained from V (using SCS)

Qd E  V  2 cos  a r  sin  a  3 4 r • E field of dipole decreases with distance as 1/r3

LECTURE 12

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The charges of a dipole are located at the points D1 (0,1,0) and D2 (0,−1,0). (a) What is the direction of the dipole moment if the charge at D2 is negative? (b) Along which axis is the potential the strongest for a given r ≠ 0?

(c) How is the E vector oriented with respect to the dipole axis for observation points in the xz plane?

LECTURE 12

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Torque on Dipole in External Field

  F  QE F  F  F sin    Te Q   Q p



   E   • torque fulcrum is at dipole’s  • dipole tends to rotate so that its moment aligns with the external field

center (dipole is bound) torque definition: T  R F   d | Te | 2   Q | E |   sin    Te  p  E    2  F 

the dipole moment p determines the torque exerted on the dipole by an external electric field E dielectric polarization is the tendency of bound dipoles to orient along external electric fields LECTURE 12

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Polarization 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 E  (atomic level)    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 12

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Polarization Vector polarization vector is the dipole moment per unit volume p  i i , P  lim v

v  0

C/m 2

dp  Pdv

• assume n (m−3) bound dipoles (atoms, molecules, sub-domains) per unit volume of averaged distance and orientation given by dav and dipole charge qd

p N ( qd dav )  i i P  lim  v 0

v

v

 P  n q d d av pav

number density of bound dipoles

LECTURE 12

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Polarization Vector and Surface Bound Charge – 1 L specimen is homogeneous, isotropic, and       surrounded by vacuum (air)     











Eext

is orthogonal to surface planes

unit normals an point inwards

Eext



























 









an an P

A



Eext

Qsb Qsb uncompensated bound charge exists on surface only → ρsb =Qsb/A p |Q | L |  A | L P  total  sb a E  sb a E |  sb | a E    sba n v v A L total dipole moment of specimen

P is along Eext (isotropic material) and points from –ρsb to +ρsb P  Pna n , where Pn    sb LECTURE 12

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Polarization Vector and Surface Bound Charge – 2 assume Eext and P are not orthogonal to the surface planes

p total | Qsb | L |  sb A | L |  sb | P  aE  aE  aE cos  v v A  L • boundary condition for the normal P component ext |  sb | E P  an  a E  a n    sb cos 

Pn    sb

A cos   A

L

       A      A       a n               a      n   A        A

P

holds regardless of surface orientation with respect to E outside the specimen P = 0 compare with D  a n   s in PEC LECTURE 12

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Polarization 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

 sb   sb(1)   sb(2)  | P (1) |  | P (2) | cos 

general boundary condition for P at dielectric interface

  sb   P (1)  P (2)   a n

#1 dielectric #2 Eext dielectric             

      

      

      

      

             n 

a

P (1)  sb P (2)

strongly polarizable

LECTURE 12

     



 

weakly polarizable slide

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Polarization Vector and Volume Bound Charge make use of relation to surface charge inside volume ins | | Pn ||  sb flux of P is negative if bound charge inside surface s is positive ins s  P  s    sb

Eext

L

    P          an 

s

s  an P 

Qbins

take integral over s ins P  d s   Q b  

net bound charge inside s

s

  P  ds    vb dv s[ v ]

Gauss Theorem

v

   P   vb LECTURE 12

Compare with   D  v slide

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Susceptibility and Permittivity – 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

  ( 0 E)   vf   vb • relate D (flux density) vector to free charge only   D  vf • at the same time   P    vb    ( 0 E)    D    P

 D   0E  P LECTURE 12

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Susceptibility and Permittivity – 2 • polarization P and E-field in isotropic materials: susceptibility

P   e ( 0 E)

 D   0 (1   e ) E  r

• relative permittivity

 r  1  e

D   0 r E   E

• anisotropic dielectrics (permittivity tensor)  Dx   xx  D     y   yx  Dz   zx

 xy  xz   Ex      yy  yz  E y  D  εE    zy  zz   Ez 

• nonlinear dielectrics [χe = f(E), ferroelectric materials] LECTURE 12

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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 Eds is the strongest E-field that the material can sustain without breakdown. glass

25-40 MV/m

mica

200 MV/m

oil

12 MV/m

air

3 MV/m LECTURE 12

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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 P how polarization P relates to surface and volume bound charge what susceptibility and permittivity mean what dielectric strength is

LECTURE 12

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