UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

LECTURE NOTES 9

ELECTROSTATIC FIELDS IN MATTER: DIELECTRIC MATERIALS & THEIR PROPERTIES

Polarization

We have previously discussed the electrostatic properties of conductors of electricity. Now we wish to discuss the opposite end of the spectrum – the electrostatic properties of insulators – non-conductors (very poor conductors) of electricity.

Suppose we have a block of insulator material, or even a gas or liquid – non-conducting – consisting of atoms, or atoms bound together as molecules (e.g. H2O, CO2, HCOOH, etc.) Solid materials that are insulators are things like wood, plastic, glass (amorphous SiO2), rubber, etc. All of these materials are very poor conductors of electricity – insulators. They are examples of dielectric materials, generically known as dielectrics.

An “ideal” dielectric material is one which contains no free charges. Since microscopically, the dielectric material consists of atoms, its macroscopic electrostatic dielectric properties arise from the collective (sum total) contributions of its microscopic constituents – at the atomic scale.

Each atom consists of a central, positive-charged “pointlike” nucleus 15 Rnucleus  few fermi's  1fm 10 m surrounded by “clouds” of electrons, bound to the nucleus. The atomic electrons are not free → typical radius of orbiting electrons is  few angstroms (1Å 1010 m ) ability to move / migrate!

Polarization of an Atom in an Externally-Applied Electric Field:

First, consider an atom (electrically neutral) in its ground state (lowest quantum energy level) such as the hydrogen atom (simplest case). The single electron orbiting the nucleus (a single proton) has a spherically–symmetric charge distribution (i.e. no  or  -dependence) in the  absence of any externally-applied electric field, i.e. Eext  0 :

The typical electric field strength “seen” by an electron orbiting the hydrogen nucleus (a single proton) due to the nuclear electric charge is:  19 e  111.610Qnucl  Ernucl  rˆˆ r 4r 212 4   8.85 10 10 2 o 10  10 e 11 which for rm1Å 10 gives a whopping Ernucl 1Å 1.44 10 Voltsm/ (very large!!!)

n.b. The nucleus “sees” this same electric field strength, due to the electron’s electric charge. This is a typical electric field strength internal to/in the vicinity of atoms (& molecules)!!!

Compare this internal electric field strength to the electric field strengths easily / routinely   lab 36 lab atom available in a laboratory setting of Eext  10 10 V/m. We realize that EEexternal internal i.e. 1036 10 10 11 Volts/m !!!

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

Because of this, when an atom (e.g. hydrogen) is placed in an external electric field, because Eext << Eint the atom “sees” the externally applied field as a perturbation to its internal electric field.

Neutral atom, no externally-applied electric field:

 Eext  0 : Typical Atomic Size: ratom ~ a few Ǻngstroms (1 Ǻ = 10−10 m)

ratom

Qnucl Spherically-symmetric electron “cloud” charge distribution (negative) e−

Point-like nucleus has positive charge Rnucl ~ few fm, i.e. ~ few x 10−15 m

A Neutral Atom in an Externally Applied Uniform / Constant Electric Field:

Suppose we place a neutral atom between the plates of parallel plate capacitor with  e.g. EExext o ˆ  1000 Volts/meter in gap-region of capacitor plates: Qehydrogen   nucl Atomic nucleus feels a net force of: FQEeExˆ nucl nucl ext o Electron “cloud” feels a net force of: FQEeEx  o ˆ  eee Thus we see that: FF  eExˆ (forces are equal & opposite – i.e. Newton’s 1st Law!) e nucl o

Qnucl   FeEx ˆ FeExxˆˆ  e o nucl o

Electron “Cloud”, w/ Electric Charge −e

2 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 The net effect of the externally-applied electric field EExext o ˆ is that the nucleus is displaced a  tiny amount (a small fraction of size of atom) from its Eext  0 position in the xˆ direction, and the centroid (i.e. center) of the electron “cloud” is displaced a tiny amount (n.b. the size / magnitude of the displacement of centroid of electron cloud is the same as that for nucleus) from  its Eext  0 position in the xˆ direction:

+ − + − + Centroid −   − + EExext o ˆ of e EExext o ˆ − + Cloud +Qnucl − + − xˆ d d + 2 2 − + d − + − + − + −  Displaced Electron Cloud Original Eext  0 Charge Distribution Atomic Configuration

- Obviously, because the electrons of the atom are bound to the nucleus, the mutual attraction of the atomic Coulomb force keeps the atom together – electron cloud is bound to nucleus. - Thus, because of the displacement of +Qnucl (= +e for hydrogen) in the xˆ -direction by an amount d/2 and the displacement of the centroid of the electron “cloud” charge distribution (Qe = −e for hydrogen) to the xˆ direction also by an amount d/2, caused by the application  of the externally-applied uniform / constant electric field EExext o ˆ we see that an electric   dipole moment p Qd Qdxˆ is induced (i.e. created) in the atom by the application of the  external electric field EExext o ˆ .

  36 11 Because EEext 10int 10 Volts/meter, the externally-applied field Eext is seen as a  (very) small perturbation on the internal electric field Eint ; hence a linear relationship exists   between the induced electric dipole moment p and the externally-applied electric field Eext :

  p   Eext ← n.b. vector relation

The constant of linear proportionality  is known as the atomic electric polarizability.

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 p Coulomb meters Coulombs2 SI units of atomic polarizability:    Eext Newtons// Coulomb Newtons m

The atomic polarizabilities  of atoms are often expressed in terms of  4 o which has SI 33 12 2 units of 4 o   meters , since  o 8.85 10 Farads/ m (  Coulombs / Newton  m )

The table below summarizes the (normalized) atomic polarizabilities 4 o of some of the lighter atoms, in units of 10−30 m3:

Atomic Nuclear Charge # e− in electron

Element Znucleus  4 o outer shell configuration ( ) denotes “Alkali” (single e− in H 1 0.667 “1” 1s closed Metal outer shell) shell

Noble (closed He 2 0.205 (closed shells) (1s2) Gas shells)

Alkali (single e− in Metal outer shell) Li 3 24.3 1 (1s2) 2s

Be 4 5.60 (1s2) (2s2)

C 6 1.76 (1s2) (2s2) 2p

Noble (closed Gas shells) Ne 10 0.396 (closed shells) (1s2) (2s2) (2p6)

Alkali (single e− in Na 11 24.1 1 (1s2) (2s2) (2p6) 3s Metal outer shell)

Noble (closed Ar 18 1.64 (closed (1s2)(2s2)(2p6)(3s2)(3p6) Gas shells) shells)

Alkali (single e− in K 19 43.4 1 (1s2)(2s2)(2p6)(3s2)(3p6)4s Metal outer shell)

Alkali (single e− in Cs 55 59.6 1 (1s2)(2s2)(2p6)(3s2)(3p6)(4s2) Metal outer shell) *(4p6)(4d10)(5s2)(5p6)6s

4 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

It can be seen that atomic polarizability  has dependence on outer-most shell/valence electron! (i.e. the least tightly bound electron to nucleus - due to screening effects of inner shell electrons).

It is (certainly) possible to obtain a theoretical relation - i.e. a theoretical “pre-diction” (technically, a post-diction) of the atomic polarizability,  for a given atom (or molecule)   relating how p (the induced electric dipole moment) depends on Eext (the applied external electric field). In order to do this, must “know” the atomic electron volume charge distribution /  electric charge density  r .

Griffiths Example 4.1, pages 161-62:

A crude model for atom, due to J.J. Thompson (c.a. ~ 1910) is his “plum pudding” model of the atom – i.e. a point nucleus of positive charge +Q surrounded by a uniformly charged spherical electron cloud of total charge –Q of radius a. This means assuming a constant volume charge density  for the atomic electrons orbiting the nucleus – i.e. one which is flat, out to a eatomic radius r = a (and zero after that) as shown in the figure below:

 eatomic 3Q   QQ  3 eatomic 4 a Vatom 4 3  a 3 Coulombs 3Q Coulombs 3 e  3 3 m atomic 4 a  m 

0 r = a r

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

This is a (crude, but simple) theoretical model of an actual atom (i.e. far from reality), but it works somewhat well - i.e. it is accurate to a factor of ~ 4.5 – c.f. with actual data – see below!

If the nucleus is displaced a relative distance d from the centroid of the atomic electron charge density distribution, then the electric field intensity that the nucleus “sees” at that point is:

 1 rˆ 3Q Ernucl    r d where rr  constant, for ra  . internal  2 eatomic 3 4 o v r 4 a

    r rr  dr  dxrˆ    (since here: ddx ˆ )

Then: r 222dx2  y  z

 The source point r moves around the interior of the spherically- symmetric atomic electron charge density “cloud” in carrying out the above volume integral.

One can see that explicitly doing this volume integral is a pain – it certainly can be done though! However, by use of the divergence theorem, we find there is an easier way to obtain what we  nucl  want Erinternal  from Gauss’ Law:    Q Erd  Er  dA encl vS  o So we take a fictitious, spherical Gaussian surface S of radius d < a centered on the centroid of the spherically-symmetric atomic electron charge density distribution as shown below:

Then: Qencl = charge enclosed by S, radius d

34Q 3 QVencl 3 d 43 a 3 d QQencl   a

 nucl nucl 2 Qencl ErdAEddinternal  internal 4 S  Area of o Gaussian Sphere, S' 3 d Q o  a Then: Erdnucl   rˆ ( rˆ direction is due to intrinsic symmetry of problem) internal 4 d 2

6 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

     Qd ˆ nucl ˆ @ Nucleus, i.e. rddx : Erdinternal  3 x 4 oa Now the force on the nucleus due to the centroid-displaced spherically-symmetric atomic electron charge distribution is:

Qd2 nucl nucl ˆ Qnucl = +Q: FrdQErdinternalnucl internal   3 x 4 oa

Since this is an electrostatics problem, this force on the nucleus must be precisely balanced by the force on it due to the externally applied field, i.e.

 nucl ˆ ˆ Finternal rdQEnucl external  rdQEx  nucl o Eexternal  Exo

    nucl  nucl  Frdinternal  Qnucl Frdexternal  xˆ

 nucl nucl  i.e. FrdFrdinternal external  0 manucl nucl0 a nucl 0  nucl nucl  FrdFrdexternal internal         Qd  Q ErdQnucl  Erdnucl  or: ErdErdnucl  xˆ nucl external  nucl internal  external internal  3 4 oa  But: p pQd    p  Erdexternal  3 4 oa

3 Turning this around: paE 4 o external

p 3 Theoretical “post-diction” for  using J.J. Thompson’s   pp4 oa  external “plum-pudding” model of an atom (ca ~ 1910)

A better theoretical model of the atom: use the Schroedinger Wave Equation (i.e. use quantum mechanics) which describes the wave-nature of electrons bound to the heavy / massive “point- like” nucleus. Electron wave function (“probability amplitude”) HE   Electron wave function: Appropriate Energy Eigenvalue(s)  rRrY,,  , Hamiltonian (bound state energy / Operator energy spectrum of electrons Spherical bound to nucleus) harmonic

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

2 Electron Probability Density: Pr,, * r ,,  r ,, r ,,

2 Electron Volume Charge Density:  rePrer,, ,,     ,,  eatomic    For an atomic electron in the lowest energy / ground state of e.g. the hydrogen atom: Lz orbital angular quantum # 2 1,0,0 rer,,  ,,   eatomic 1,0,0  nlm,, Ground state: n, l, m = 1, 0, 0 Principal quantum # Total orbital angular (=Radial quantum #) momentum L quantum #

1 For hydrogen atom in its ground state:  re,,  ra/ o (n.b. spherically symmetric!) 1,0,0 3  ao 2 o 4 o c c 10 Where: ao  Bohr radius22  20.53A 0.53 10 m emcmceeme

And where: h = Planck’s constant = 6.626 x 10-34 Joule-sec  h 1.054  1034 Joule  sec 2 e2 11 em fine structure "constant"  4 o c 137.036

(em = EM interactions’ dimensionless coupling strength)

And: cMeVfm197.327 - (1 MeV 10 6 electron volts, 110fmm 15 ) 2 Rest mass energy of electron: mce  0.511 MeV And: c = speed of light = 3 x 108 meters / sec

 For an atomic electron in the lowest energy / ground state of e.g. the hydrogen atom, the electron volume charge density is:

2 e 1,0,0 rer,, ,,  e2/rao eatomic 1,0,0  3  ao

  1,0,0 e 2/rao Qencl If we use e re,,  and Gauss’ Law EdA  for a Gaussian sphere of atomic 3 S  ao o radius r = d centered on the centroid of the atomic electron charge density distribution, the result from this more sophisticated / quantum-mechanically motivated model, for the atomic polarizability QM is:

p QM 3 See also Griffiths 3aam33303   0.112  10 QM o o o problem 4.2 Eext 44 0

Compare this to J.J. Thompson’s (relatively crude/simple) “plum-pudding” model of the atom, 3e with electron volume charge density   constant, and with aa , which gave: eatomic 3 o 4 ao

8 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

p 33303 pp  pp4 oaam o   o 0.149 10 Eext 4 0

They differ by ~ 30% from each other (for the hydrogen atom / simplest atom) - not bad for J.J.!! However, compare these two “post-dictions” e.g. to the actual experimental measurement for (supposedly atomic, i.e. not molecular) Hydrogen (from the above table):

 H2 expt 0.667 1030m 3 which is 4.5 → 6 larger than that predicted by either theory!!! 4 0 Eeek!!!

  Note that when Eext becomes comparable to, or larger than, Einternal then there no longer exists   a linear relationship between p and Eext . It becomes increasingly non-linear:

  i.e. p EEEHOTs23 ...... ' ext ext ext linear quadratic cubic

(See S.Errede’s UIUC Physics 406POM lecture notes on distortion for more details if interested.)

Electrostatic Molecular Polarizability

Molecules, consisting of groups of (two or more) atoms bound together (by the electromagnetic force / interaction), because of their intrinsic individual atomic structure, are often highly non-spherical in their geometrical shapes / configurations.

- An example of one such highly non-spherical molecule is the (linear) carbon dioxide (CO2) molecule:

O C O zˆ Axis of CO2 Molecule

Because of its shape (linear, and axially-symmetric) it should come as no surprise that:

Molecular polarizability Molecular polarizability    to CO22 molecule axis to CO molecule axis

CO2 CO2 i.e.  > 

2 2 CO2 40 C CO2 40 C  4.5 10 Nm/ >  2.0 10 Nm/

 CO2 4.5 or:  2.25 CO2  2.0

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 Thus for non-spherical molecules, the net induced electric dipole moment pmol may not be    parallel to Eext , i.e. pmol  molE ext .

Rather, a more correct mathematical description (i.e. for the more general case), one that does   properly relate pmol to Eext is given by:

   pmol  molE ext

mol mol mol mol ext  Molecular Electrostatic px xxxyxz Ex  mol mol mol mol ext  Polarizability Tensor : p  E mol y yx yy yz y Rank-2 Tensor/3 3 Matrix pmol mol mol mol Eext zzxzyzz      z     pmol mol Eext

Now zˆ is the symmetry axis of molecule. It is always possible to choose the principal axes  mol mol mol xˆˆ,,yzˆ of the molecule such that off-diagonal elements of mol vanish (i.e.  xyyzxz,,, etc.)

mol mol mol For the linear, axially-symmetric CO2 molecule,  zz   and  xxyy.

mol mol ext mol ext mol ext mol ext mol ext pExxxxxyEyxz  EEEzxxx x

mol mol ext mol ext mol ext mol ext mol ext Writing things out completely: pyyx  EExyyyyzEEEzyyy y

mol mol ext mol ext mol ext mol ext mol ext pzzx  Exzy  EEEEyzzzzzz z

   pmol  molE ext

Alignment (Polarization) of Polar Molecules in an Externally-Applied Electrostatic Field

A neutral atom has (and many types of neutral molecules have) no intrinsic, permanent    electric dipole moment – i.e. if Eext = 0 then patom (or pmol ) = 0. Such molecules are known as   non-polar molecules. However, when an external electric field is applied E  0 , then p  ext atom (or pmol ) ≠ 0. Such electric dipole moments are known as induced electric dipole moments.

Some molecules, such as the all-important water molecule (H2O) actually do have permanent   electric dipole moments (i.e. even when Eext = 0, pmol ≠ 0)! The non-vanishing permanent electric dipole moment for some molecules arises because of how outer-shell atomic electrons are (non-democratically) shared amongst the individual atoms making up the molecule.

For example, in the water molecule (H2O) the outer-shell (most loosely-bound) atomic electrons tend to cluster preferentially around the oxygen atom, and since the H2O molecule is not axially symmetric (the two hydrogen atoms form an opening angle between them of 105o with oxygen atom at the vertex) leaving net negative charge at oxygen vertex and net positive charge at hydrogen atom end.

10 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 p H2O H+ H+

  = 105o p 6.1 1030 Coulomb  meters HO2 (very large!!!)

− O one reason why H2O is very good solvent!

Molecules with permanent electric dipole moments are known as polar molecules. (e.g. water, benzene, toluene, . . . )

If polar molecules, with permanent electric dipole moments are placed in a uniform applied  external field Eext , two things can/will happen: 1.) Permanent electric dipole moments tend to align / line-up with the external field:   permanent pmol E ext  2.) The external applied electric field E can / does induce a (non-permanent) electric dipole  ext  induced  moment: pmol  molE ext

     Thus: ptotalppp permanent induced permanent  E mol mol mol mol mol ext   pinduced  mol induced permanent Usually, for typical values of Eext  1000 Volts/meter, thus ppmol mol for polar  atomic molecules with permanent electric dipole moments. Why?? Because EEinternal ext .

So to “zeroth order” when a molecule with a permanent electric dipole moment is placed in an  external electric field Eext :  total permanent induced permanent atomic ppmol mol because ppmol mol if EEinternal ext .

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

What happens when polar molecules (and/or pure physical electric dipoles) are placed in a  uniform external electric field Eext ?

 +Q E  Exˆ ext o FQEQEx ext  o ˆ  d/2 dp,, Qdr d  d/2  r   FQEQEx  ˆ −Q  ext o  Eext Ex o ˆ

  Force on +Q(@ r ): Fr QErext    QEx o ˆ   Force on –Q(@ r ): Fr QErext    QEx o ˆ    Net force on polar molecular dipole: FFFQEQEnet ext ext 0  →  No net force on polar molecular dipole!!! (for uniform / constant EExext o ˆ )

 However,  a net torque on the polar molecular dipole: NrF

The net torque acting on the polar molecular dipole is:    NNNrFrFnet  Take torques about the midpoint  of dipole  dd FF  22  dd11  QEext  QE ext Q d  E ext  Q d  E ext p 2222     Qd Eext but p  Qd    p  Qd  EExext o ˆ   Thus: NpEnet ext    And: NpEpEnet ext sin o sin EExext o ˆ  ˆ ˆ n.b.: Nnet points in direction dx (Use right-hand rule for -product!!!)

12 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

    The torque Nnet acting on the electric dipole (with pure electric dipole moment p  Qd )    due to uniform / constant electric field Eext Ex o ˆ acts such that p lines up parallel to Eext .   The polar molecule is free to rotate, and will do so until p  EExext o ˆ , as shown below:

 E-field-aligned polar molecule / electric dipole moment p :

 + Eext − + −     + Eext p  Qd Eext −

+ − +  Exo ˆ − + −  + Eext −

   n.b. when polar molecule / electric dipole moment p is aligned with E , note that the     ext net torque on the dipole, Nnet vanishes: NQdEnet ext but when dE ext the cross product    vanishes! NpEnet0sin o  because 0 when p  Exext o ˆ.

 We have shown (P435 Lect. Notes 8) that the potential energy U p of a pure electric dipole   p in an external electric field Eext is:

  WPEpE.. ext pE ext cos

Work/Potential Energy vs. Angle Θ of Electric Dipole In External Electric Field:

+pEext

W(Θ)=P.E. (Θ) 0 (Joules)

−pEext Θ 0 90o 180o (π/2) (π)

  p p     Eext p Eext Eext      NpEext 0,  ext p  Eext p anti- Eext

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

What happens to polar molecules (and / or pure physical electric dipoles)  when they are placed in a non-uniform externally-applied electric field Eext ?

 1 Qs e.g. Erext  ˆ  polar molecule in E-field associated with a point charge, −Qs : 4 o r

+Q  1 Qs  Erext  ˆ r 4 o r    −Qs p  Qd θ Eext @ dipole  (local to dipole)  r

–Q

  Force on +Q: Fr QErext   Force on –Q: Fr QErext    But here: Erext Er ext !!!     Net Force on dipole: FFrFrQErQErnet   ext ext     Fnet QErEr ext ext     FQEnet ext where EErErext ext  ext    For polar molecules drr few Ǻngstroms – typical atomic distance scale!!!

  Thus, since d is so small, here we may use: EdEext ext

  E extEd ext  See Griffiths d xx   ext ext EEdyy equation 1.35 d   ext ext EEdzz

   FQEQdEnet ext   ext  pE  ext

    nd FpEnet ext n.b. Fmanet dipole dipole (Newton’s 2 Law of Motion)

  If external field E is non-uniform, a net force exists on an electric dipole / polar molecule ext   which is proportional to the spatial gradient of the externally applied electric field Eext which causes the dipole / polar molecule to accelerate!

14 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

30 Suppose we have a polar liquid – e.g. water (H2O) with pp6.1 10 coulomb-meters mol H2 O permanent electric dipole moment for each / associated with each water molecule.

Why don’t polar water molecules spontaneously align with each other, thereby minimizing their overall energy?? (e.g. like magnetic dipoles (atomic) inside a permanent magnet)

Consider a universe in which there are only two such polar molecules (and nothing else). Then the overall/total potential energy (by the Principle of Linear Superposition) is:

 WPEUUpErpErpp12.. pp 12  pp12 121 212 

   E -field of dipole2 E -field of dipole1 rr21     @ dipole1 r1 @ dipole2 rr21 () Define: rr12 r

  11 E -field of dipole2 (@ dipole1): Er3 prrˆˆ p 21 3  211 2 4 o r   11 E -field of dipole1 (@ dipole2): Er3 prrˆˆ p 12 3  122 1 4 o r

 p1   Er21@  rr1 

   rrr21  Er12@  p2

 pp12 pp 12  WPEpErpEr . . 121 212  11  11     3p rprppˆˆ  3 prprpp  ˆ  ˆ  3321 11 12  12 22 12 44oorr  

    Now: rr12 r and p12ppp 21

11       WPE.. 3 prprppˆˆ  3 prprpp  ˆˆ  3 21  1212    12 4 o r Then: 21   3 prˆˆ pr pp  3 12  12 4 o r    Notice that: WPE/.. occurs when p is anti-parallel to p , i.e. when p p min min 1  2  21 and when: p12,pr to ˆ (or anti -  to rˆ ) then → pr1ˆ 0 and pr2 ˆ 0.

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 pp pp 11  WPE12.. 12  pp when we have: p  Qd min min 3 12 11 2 o r

  r r

  p22 Qd So let’s put in some actual numbers: 12 2  8.85 10 C F o Nm 2 m

A typical intermolecular separation distance for the water (H2O) molecule is r ~ 10Ǻ = 10 x 10−10 m = 10−9 m = 1 nm H22OHO p p p 6.1 1030 Coulomb  meters 12HO2

2 pp12 pp 12 11 30 22 W P. E .12 3  6.1  10  6.69  10 Joules min min 2  8.85 10 109  Very small energy!!!

Now liquid water (room temperature T  300K) also has thermal energy associated with it.

Boltzmann Kinetic Theory: – get a contribution to thermal energy of ½ kBT for a.) each kinetic degree of freedom that are operative at that b.) each rotational degree of freedom absolute temperature T. c.) each vibrational degree of freedom

-23 kB = Boltzmann’s Constant = 1.38 x 10 Joules/K

For water (H2O) @ T = 300K: a.) kinetic: vvv,,  3 kT xyz2 B b.) rotational: I , IkT  2   2 B c.) vibrational: none  0 (i.e. no quantum vibrations of water molecule H2O no H-O-H vibrational excitations @ T = 300K)

HO 5 Thus, for a single H2O molecule: WkT2  . thermal2 B Since we’re considering two H2O molecules (here), then (using superposition principle):

HO22 HO 23 20 Wthermal TOT2 W thermal 5 k B T 5  1.38 10 300 2.07 10 Joules

HO2 20 pp12 22 Wthermal  TOT2.07 10 Joules Wmin 6.69 10 Joules

HO212 pp Ratio: WTOTWthermal min  30.9 Sum: WTOT  WHO212 W pp 2.07 1020 J 6.69 10 22 J 2.00 10  20 Joules HO2 thermal min

16 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

Thus, we see that the internal energy of water (@ the microscopic level) is dominated by thermal

pp12 energy at T  300K (room temperature). In order for the dipole-dipole term Wmin to actually dominate, the water temperature would have to be TKW9.7 for pp12 to dominate W HO 2 0 . HO2  min TOT

So, typically (but not always) for polar materials (gases / liquids / solids), macroscopic amounts 23 of individual molecules N A 6 10 molecules/mol the thermal energies overwhelm the dipole-dipole interaction energies at room temperature.

No net macroscopic alignment of permanent microscopic electric dipoles occurs at T  300K. When one places either non-polar or polar materials (at T  300K) in externally applied electric  field Eext , nevertheless, there is a net, partial macroscopic alignment of either induced microscopic electric dipole moments (for non-polar materials) or permanent electric dipole moments (for polar materials).

A“snapshot” of a dielectric material: (with either induced atomic / molecular electric dipole moments or permanent electric dipole moments at T  300K)

  No external applied field ext 0 With external applied field ext 0

Random orientation(s) of induced / permanent Partial alignment of induced / permanent electric dipole electric dipole moments at microscopic scale - moments at microscopic scale – atomic or molecular at atomic or molecular at T  300K. T  300K – when remove Eext , alignment rapidly disappears.

Both pictures change from one instant to the next: - Thermal fluctuations (fluctuations in thermal energy density) - Quanta of thermal energy (EM energy – virtual photons) “traded” between constituents (atoms / molecules) at microscopic scale.

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

  Electric Dipole Moment Per Unit Volume of Material (a.k.a. Electric Polarization) r

For a single atom (or molecule or physical dipole), we have seen that the electric dipole moment    of the charge distribution is given by: pr r  r  d v

For atoms / molecules, the integration volume v is associated with the space “occupied” by that atom / molecule, but it essentially can be over all space.

For a macroscopic sample of matter (gas, liquid or solid consisting of many, many atoms, 23 molecules (i.e. a sample containing N A ~10 molecules) it makes sense that we can define a quantity / parameter known as the electric dipole moment per unit volume (also known as the   electric polarization) r :

Electric Dipole Moment Per Unit Volume (a.k.a. Electric Polarization):

NN  pr Qdr   iimol iii mol r ii11  Volume Element dd Volume Element    Where the infinitesimal volume element d is centered on the vector r , and the ri vectors associated with each atom/molecule are contained within the infinitesimal volume element d. 1 N Thus: rr  i . N i1

  Note that the physical interpretation of r is that it is the macroscopic average of the microscopically-summed-over electric dipole moment per unit volume. Important!!!

  What are the S.I. units of the electric polarization, r ? From above, we see that they are:

Q  length Q  Coulombs / meter2 length 3 length2

  Thus, r has the same S.I. units as surface charge density,  !

Note that if macroscopic dielectric / insulating material has no net electric charge associated with N it, i.e. dQ r Qii  r 0 in each of the infinitesimal volume elements, d centered on the i1  vectorr , then the monopole moment M 0 of the charge distribution associated with the entire  dielectric material, integrating over the whole volume: MQ dQr   rd 0. 0 TOT vv

18 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 Thus, any potential Vr that arises due to (microscopic, and thus macroscopic) polarization of the dielectric material will be due to the next most important term in the multipole expansion of the potential, namely the (macroscopic) electric dipole moment associated with that material!

  We already know the potential Vr (and corresponding Er Vr ) associated dipole  dipole dipole with a single, microscopic pure electric dipole moment p is:

  1 prˆ Vrdipole  2 4 o r

 The electric dipole moment pr is located  at the source point r

For a macroscopic dielectric material, an infinitesimal volume element d  has a  net electric dipole moment pr associated with it of:

     pr prrd  (i.e. r ) d 

   The infinitesimal contribution to the potential dVdipole  r due to this prrd  is thus:    11prrrˆˆ r dV r d  dipole  22 44oorr   Then integrating this expression over the volume v of the dielectric material, we have:   1 r rˆ Macroscopic potential due to Vr dVr   d   dipolevv dipole 2 a polarized dielectric material 4 o r Now:  11   n.b.  refers to differentiation with     r rr   respect to source coordinates only!!! rˆ rrˆˆ  2 2 r rr    r rr  3 3 r rr 

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

   1 rˆ rrˆˆ   1 r rˆ Thus, using  we can write Vr  d  as:  2 2 dipole v 2 rrrr  4 o r

11   Vr r d  dipole v  4 o r

{n.b. In practice, it is often extremely difficult to explicitly carry out this integration, except for a   few cases with very simple geometry and / or simple polarization(s), e.g. r = constant.}

  Note that in general that r (= electric dipole moment per unit volume / a.k.a. electric polarization) is never known a-priori (i.e. before the fact), but is instead inferred a-posteriori (i.e. after the fact) from known, physically measurable quantities / parameters.

How is this accomplished??? integrand in above integral    Note that: fAf A A  f Then: Af fAf    A 11   Let: f  and: A  r rr  Then:   111r 11    Vrdipole  r d   d       rd   444vvv rrr '     ooo use the divergence theorem- convert to surface integral 11 11   Vrdipole   rndAˆ    rd   44Sv rr   '   oo    r  r Thus: B B  11 11   rdA +  rd  Sv BB'  44oorr  

= Potential due to (bound) = Potential due to (bound) surface charge density volume charge density     B rrn  ˆ B rr   (SI units: Coulombs/m2) (SI units: Coulombs/m3)

nˆ = Outward-pointing unit normal vector on bounding surface S

 The bound surface charge density  B rrn  ˆ and bound volume charge density   B rr   are called bound charge densities because these charges cannot move – they are certainly not free charges – because they are bound to atoms / molecules.

20 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

   Then, using these two relations,  B rrn  ˆ and B rr  , we see that:

11 11   Vrdipole   rdA  rd  Sv '  44oorr    11rr   BB dA + d Sv '  44oorr  

 Physically what this formula / above expression says is that the electric potential Vrdipole     and hence E-field Erdipole Vr dipole  due to the macroscopic electric polarization r of dielectric material is formally / mathematically equivalent to that of a potential associated with a  bound surface charge density  B rrn  ˆ plus a contribution to the potential associated   with a bound volume charge density B rr  

So instead of having to integrate over contributions from infinitesimal-sized point-like atomic / molecular dipoles contained within the volume v of the dielectric material, we simply find /    determine the bound charge densities  B and B , and then calculate Vr and Er the same exact way(s) we have already done e.g. for the free charge densities  f and  f !!!

 Consider an infinitesimal, single volume element dr  inside e.g. a uniformly polarized   block of dielectric material - i.e. a dielectric with uniform / constant polarization rx o ˆ e.g. a block of dielectric material (such as plastic) inserted between the plates of a parallel plate capacitor:

3 n.b: The number density of molecules is nmol = # molecules/unit volume (i.e. # molecules/m ).

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 21 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

Suppose the mean charge displacement, i.e. of molecules, going from an  unpolarized state “ ” to a polarized state, e.g. “ ” is an average distance of ddx ˆ . d

Then the (average/mean) amount of charge displaced in a volume element d  is (see above figure on previous page) is:

   = Infinitesimal surface area element of one dQ nmol Qd  n mol Qd dS where: dS nˆ dS   side of the infinitesimal volume element d 3  nQdndSmol  ˆ and: nmol = # molecules/m dQ = the amount of charge displaced in the polarization process, crossing an area element dS.

Now we re-arrange / rewrite this a bit:

  Since: p  Qd = average/mean electric dipole moment per molecule in the volume element d .

  Then: dQ nmol Qd nˆˆ dS n mol p n  dS

    But: nprmol   r {Since the electric polarization, r is defined as the (macroscopic) electric dipole moment per unit volume!!!}   i.e.: rnprmol  where nmol = number density of molecules (# molecules/unit volume)  and pr = electric dipole moment per molecule, in volume element d    3 2 {Check units/dimensions of rnprmol  = (#/m * Coulomb-m = Coulombs/m – Yes!!!)}

Thus we see that:   dQ nmol Qd r ndSˆˆˆ n mol p  r  ndS  r  ndS dQ B = bound charge QB on the surface area dSof one side of the infinitesimal volume element d .

  If we now integrate dQB  r nˆ dS (= amount of charge displaced in the polarization process) over all 6 surfaces of the infinitesimal volume element d  (for simplicity, assume d  to be a cube of side dS), and for again for simplicity’s sake, let us assume that the electric   polarization, r has negligible variation (i.e. is a constant) over the infinitesimal volume   element, d such that e.g. rx o ˆ .  net    QB (of dr @ )  rndSrndSrndS ˆˆˆ123          Then:  r nˆˆˆ456 dS   r n dS   r n dS

where: nnxˆˆˆ14  and nnyˆˆˆ25  and nnzˆˆ36 ˆ

22 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

  Then if: rx ˆ , and o  net    QB (of dr @ )  rndSrndSrndS ˆˆˆ123           r nˆˆˆ456 dS   r n dS   r n dS

With nnxˆˆˆ14  and nnyˆˆˆ25  and nnzˆˆ36 ˆ , then: net QQQQQQQB 123456

 odS xˆˆ x  x ˆˆ y  xzˆ ˆ  xxˆˆ xy ˆˆ xzˆ ˆ  0 Thus we see {here} that the net charge displacement across the entire surface S enclosing the net infinitesimal volume element, d  is QB  0 , because QQ41 and QQQQ23560.   (Note that this is true only for the case of uniform/constant polarization,  throughout d net i.e. in general, QB  0 for arbitrary polarization  throughout d .)

    If rx o ˆ , then (obviously) the bound volume charge density B rr   0   because: Boorrxx  ˆˆ   0.

   n.b. This result also holds for macroscopic volumes: QrdA B S  i.e. there will be no bound volume charge density, B r  0 if the electric polarization     (electric dipole moment per unit volume) r is uniform / constant, e.g. rx o ˆ throughout the macroscopic dielectric medium.

  If r is not uniform in the macroscopic volume v ,  then: Q r dA   r  nˆ dA 0 B SS

 Note that if dQ r  r nˆ dS 0 exists on the entire/total surface dSassociated with a volume element d  then this is also = to the net charge that flows out of (or into) the infinitesimal volume element d  across/through the total surface area element dS when the dielectric material is polarized.

 Then a net charge –dQB must remain in the infinitesimal volume d , so then:

     rd Q    r  dA       r  d    rr   vSvBB  B

Thus, we see that  BB and  are (bound / induced / polarization) surface and volume charge densities, respectively.

Synonyms of each other here

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 23 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

More on the Physical Interpretation of the Bound Charges:

We have seen that  BB and  represent real, physical accumulations of electric charges on the surfaces and/or in the volume of dielectric materials, respectively.

However, the primary distinction associated with these two physical quantities is that QdA   and Qd    are associated with bound charges – i.e. charges that are bound BBS BBv to atoms / molecules – they are not free surface / volume charge densities.

c.f. bound charges  BB and  with free charges  free and  free (which can move freely on bounding surfaces ( S ) and volumes ( v )) .

Consider a fully-polarized, very long dielectric rod of radius R with uniform polarization o zˆ ║ to the zˆ -axis of the polarized dielectric rod:  R o zˆ R zˆ QB− +QB+

d L   p  QdzB ˆ Carefully cut out a very thin disk  o zˆ e.g. of thickness d  10Ǻ = 10−9 m d nzˆ ˆ nzˆ ˆ zˆ 2 1  2 The thin disk has cross-sectional area A = πR : QB− QB+ QB− QB+  d We can replace the polarization,  by bound surface charges QB− and QB+!   Thus, the thin disk has (macroscopic) electric dipole moment p QdBB Qdz ˆ

However, the macroscopic electric dipole moment of the disk is also:   22 p Volume of disk  *(Ad )  R d oB R d zˆˆ  Q d z  (Since the macroscopic polarization  = macroscopic electric dipole moment per unit volume!!!) 22 Therefore: oBBooR dQd  or : Q  R  A Q  Or:   B  QAnAzzAA ˆ ˆˆ where: nzˆ ˆ B A BB1 o o 1  QB  B   QAnAzzAABB ˆ2 oˆˆ o and: nzˆ2 ˆ A  but:  B nˆ QQ Thus we see that: QQ  A and:  BB   BBo BBAA o

24 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

If we instead make an oblique cut in the long, polarized rod: Aoblique  A end cos ˆ n  θ zzˆˆ, o

Aoblique  A cross-sectional area A = R2 end cos  oblique  Bend nznˆˆ  oˆ  ocos Qoblique oblique A oblique cosA A same result! Bend Bendend   ocos o

If the polarization is non-uniform, then  accumulations of bound charge within the polarized dielectric material as well as on the surface of the dielectric material.

    Diverging Polarization r :  r R rnˆˆ  rr B rR rR   R 2 e.g. rro ˆ P R QAB B sphere4 R B  r   B rr  0 We know by charge conservation that: 2 QQBB 4 R B    Qrdrd     BB vv

Griffiths Example 4.2:

Determine the electric field associated with uniformly polarized sphere of radius R  o zˆ constant zˆ nrˆˆ  o zˆ  yˆ

xˆ Here:   Volume bound charge density: Borrz  ˆ 0   Surface bound charge density:  rR rnˆˆ  zrˆ cos zr  cos BoorR

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 25 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

In Griffiths Example 3.9 we obtained for   k cos on a sphere of radius R that for k o :

 o  rrRcos  3 o Vr   R3  o cos rR  2  3 o r

Now zr cos , and inside the sphere rR , the electric field intensity is:    o ErRVrRinside   zˆ for rR  33oo

Note that outside the sphere rR , the potential VrR is identical to that of a point electric dipole at the origin!  44 p Volume of sphere RRz33   ˆ 33o Electric dipole moment per unit volume

  R3314 Rpr 1 ˆ o ˆ Vr R22cos  o  zrˆ   2 3434oorr o r  Then use ErRVrRoutside   to obtain electric dipole field intensity outside the sphere (see P435 Lecture Notes 8).

  1 Lines of ErRoutside  and Ezinside  o ˆ : zˆ 3 o

 Note the discontinuity in the normal component of E @ rR !!

This is due to the existence of surface bound charge density B on surface of the sphere!!

26 Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2008 Lecture Notes 9 Prof. Steven Errede

 The Polarization Current Density JB

Suppose we have an initially un-polarized dielectric, which we then place in an external electric   field Eext – but one which varies extremely slowly with time. Then the (induced) polarization  in the dielectric will also vary extremely slowly with time, simply tracking the externally-applied electric field.

Using electric charge conservation (i.e. the using the empirical fact that electric charge cannot be created, nor can it be destroyed), we obtain the so-called continuity equation for bound charge in a dielectric, which we simply give here (for now – we will discuss in more detail, in the future):

   JBBrdA    rd  Svt

  Using the divergence theorem and B rr   we obtain:      rt, J,BBrtd  r  d  rtd ,   d   vvvvtt  t

The above volume integral equation must be valid for arbitrary volumes, v (e.g. for ≈ a few molecules in the dielectric → entire dielectric). Therefore the integrands in the volume integrals must be equal to each other at each/every point in space, and at each/every instant in time:

   rt, Continuity equation for J,rt   B  t bound electric charge

2 2 SI units of Polarization Current Density, JB t = Coulombs/m -sec = Amperes/m

(1 Ampere of current  1 Coulomb / second of charge passing through a  imaginary plane)

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 27 2005 - 2008. All rights reserved.