Local Field and Cla usius - Mosotti Equation "Pa rti cle s", i.e . atoms or mo lec ule s in a li quid or so li d are basking in electrica l fields - the ext ernal field that we apply from the outside is not necessarily all they "see" in terms 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 ofbonding. ± All these fie lds average to zero ± macroscopically ± What is microscopic v iew ?
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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
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
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
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.
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
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:
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.