lecture4 local field ph611

8/7/2019 Lecture4 Local Field PH611

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Local Field and Clausius - Mosotti

Equation"Particles", i.e. atoms or molecules in a liquid or solid arebasking in electrical fields - the external field that weapply from the outside is not necessarily all they "see" interms of fields. ± First , of course, there is a tremendous electrical field

inside any atom. ± Second , we have fields between atoms, quite evident

for ionic crystals, but also possible for other cases of bonding.

± All these fields average to zero ±macroscopically

± What is microscopic view?



Here, we are looking the effect of externalfield on atoms and molecules.what an atom "sees" as local electricalfield or the local field E loc to be the field

felt by one particle (mostly an atom) of thematerial .we may express E

loc as a superposition of

the external fieldE

ex and some fieldE

m atintroduced by the surrounding material.E

loc = E ex + E

mat



In ter n al/ local field:

When dielectric material is placed in the externalelectric field, it is polarized creating electricdipoles. Each dipole sets electric field in the

vicinity. Hence the net electric field at any pointwithin the dielectric material is given by:

The su m of exter n al field a n d the field due toall dipoles surrou n di ng that poi n t. This n etfield is called ³i n ter n al or local field or Lore n tzfield´



Lore n tz m odel .Decompose the total field into f our components .

E mat

1 st component µ E n ear ¶ : due to atoms or ions inside sphere.

2nd component µEL¶ : due to cut out part µsphere ¶ ( Lorentz)3 rd component µ EP¶ : due to surface of the material, area charge,macroscopic p olarization .4 th component µEext ¶:External electric field

E loc

= E ex

+ E pol

+ E L

+ E near



How large are those fields?F ield f or standard geometry (material) in a capacitor is de f ined as (standard

Problem in E lectrostatics) E = N P / oWh ere N is called Depolarization F actor (pure number)

Shape Axis N Sphere any 1 /3

T hin slab n o rmal 1T hin slab in plane 0 Cylinder L o ngitudinal 0 Cylinder T ransverse ½

Hence

4 th c o mp o nent E ext 3 rd c o mp o nent E

p = ± P/ I0 (for µthin slab ¶, -ve si gn showsdirectio n)2 nd c o mp o nent E L = P/3 I0 (for µsphere¶ )1 st c o mp o nent E n ear = 0 for isotropic materials (cubic) , which iseasy to imagine.



HenceE

loc = E ext ± P /I0 + P /3 Io +0

If E 0 = E ext ± P /I0 , is the h o m o gene o us f ield averaged over thewhole volu m e of the ho m o g e n eous m aterial. (m acroscopicapproach )T hen Local field finally becomes; Lore n tz Relatio n

E loc =E

0 + P /3 Io



In ter n al/ local field------

Let us consider a dipole with charges +q and ±qseparated by a small distance dx. T he dipolemoment is given by: = qdx

Now consider a point P at a distance r from thecentre of dipole.

T hen the electric field for the system can be resolveinto two components.

(i) E r = 2 cos / 4 0 r 3(ii) T he tangential / transverse component perpendicular to the radial

component is given by

E = sin / 4 0 r 3



P

E r E

=q.dx

V(r, ) = (1/4 0)( cos /r 2)

E

-q +q

dxConsider a dielectric material placed in external electric field of strength Eand also assuming an array of equidistant dipoles within the dielectricmaterial, which are aligned in the field direction as shown in the fig:



Let us find the local field at µX¶ due to all dipoles in thearray.

T he field at X due to dipole A is given by;

EXA = E r +E [here r = a & = 0 0]

Er

= 2 cos0 0 / 4 0 a 3 = 2 / 4

0 a 3

& E = 0

Hence EXA = / 2 0 a 3



(1)



(2)



(3)

! Er



¼½»

¬«

¼½

»

¬

«!

!

2

13

2

13

)3

1(

00

0

O

O I

I

IIE

IE

N or

N

N

r

r

r

.213 0 ¹ º ¸©

ª¨!O O I E

N

T he last equation is known as Clausius-Mosotti formula. It isalso known as Lorentz-Lorentz equation in context of optics.

T he remarkable fact is that the atomic polarizability which isa microscopic quantity can be measured at ease bymeasuring a macroscopic quantity (dielectric constant) byusing a capacitor.



0

2

20

113 2 2

or,

1. L orentz- L orentz Equation

3 2

r

r

N

N n

n

I E O I O I

EI

¨ ¸¨ ¸! ! © ¹© ¹ ª º ª º

¨ ¸! © ¹

ª º

Clausius-Mosotti equation in alternativeforms



If the material consists of different types of moleculesthen

± Where N i is the no of molecules per unit volume and i is thepolarizability of i th kind of molecule.

0

1 12 3

r

i iir

N

I EI I ! §

T his equation can also be written as

Where, N A is the Avogadro number , M= molecular weight, V= density and P M = Molar polarization .

0 0

1

2 3 3r A

M r

N M MN P

I EE VI V I I

! ! !

lecture4 local field ph611

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