MF LCTROSTATIC FILDS IN MATRIAL MDIA 3/4/07 LCTURS
Materials media may be classified in terms f their cnductivity σ (S/m) as: Cnductrs The cnductivity usually depends n temperature and frequency A material with high cnductivity (σ >>) is a metal A material with lw cnductivity (σ <<) is insulatr A material with cnductivities lies in between is a semi-cnductr If a material exhibits infinite cnductivity at T=0 k it is a supercnductr Nn cnductrs (insulatrs dielectrics ) A dielectric material (in which D = ε applies) is Hmgeneus if ε des nt change frm pint t pint Istrpic f ε des nt change with directin Linear if ε des nt change with the applied field 3/4/07 LCTURS
-Plarizatinand electric flux density Unlike the free charges in vacuum which prduce an electrstatic field, the dielectric medium des nt have any free charges In the absence f an external electric field The electrns frm a symmetrical clud arund the nucleus with the center cincide with the nucleus( the electrns still bunded in mtin) When an external electric field is applied Since the electrn can nt mve freely, the center f electrn clud will mve away frm the nucleus (plarizatin) The plarized atm r mlecule is represented by an electric diple + - - - - - - - - The displacements are limited by strng - restring frce set up by the charge cnfiguratin in the mlecule This plarizatin gives rise t an electric fields in ppsite directin t the applied field
The state f plarizatin is described by the vectr f plarizatin P [Cul/m ]. It is defined as the electric diple mment per unit vlume as; V Dielectric V where, v p i... P( r) lim V 0 v p V i [Cul/m Is the vectr summatin f the diple mments cunted ver the vlume V ] ext p i r 0 Nte: fr unifrmly plarized dielectric, P is cnstant ( P P â), while in the general media, it is functin f psitin ( P P ( r ) â) 3/4/07 4
Unifrm plarizatin i.e., the material is equivalent t a free space (ε= ε ) with surface and vlume plarized charges (r bunded charges). P C/m 3 v b where ext ext Nnunifrm plarizatin - - ρ vb ρ vb 0 + + - - 5
Let us have a medium in space cntaining bth free and bunded charges The electric field intensity due t bth types can be calculated as: p. ( and b are the free and bunded charges respectively [Cul/m 3 ] ) Since b. P. (. P) r. ( P) The mixed quantity P is a definitin fr the electric flux density D P [C ul/m ]. D D & General differential frm f Gauss s law V. D dv dv Ntes: V S D. ds dv V Q en General integral frm f Gauss s law The surce f D is the true free charge due t. D while the plarizatin vectr is stirring up D The surce f is the ttal charges (true free charge + plarizatin charge b ) due t. ( p ) /. P results frm the impressed field n the dielectric but it is a functin f the ttal field ttal, the impressed ne and the field results frm plarized charges
e is a dimensinless quantity called electric susceptibility It measures hw susceptible r sensitive a given dielectric is t electric field P D e e ) ( r D = ( + e ) is called permittivity f the dielectric r e P ) ( is the relative permittivity D P ) ( Nw we can define the electric flux density D and its relatin t the electric field intensity 3 /.... m C p vp
Basic equatins f electrstatic field in material media: Differential frm Integral frm. D D. ds dv s 0.d v 0 Summary: c D. D P ( ) P e vp. p... C / m 3 8
s D. ds s ds s Lim[ D nˆ S D nˆ.. S f ( h)] ss h 0 0 (as h 0) nˆ. (D - D ) s r D n - D n s P b -Bundary cnditins Surfaces that dividing the space int regins f different permittivity and/r carrying a surface distributin f free charges cause the lines f r D t be discntinuus and hence we call them bundaries nˆ nˆ D Gauss law: s Applying the equatin: Lim [P.nˆ S P.nˆ S h 0 ( Lim [ h ] b sb h 0 S h nˆ ) nˆ D. r f ( h )]. (P - P ) 0 (as h 0) sb r The equatin:. d 0 h 0 c Lim [.ˆ.ˆ P ds s sb. S Pn - P n h c f ( h )] 0 0 (as h 0) n ( - ) 0 r ˆ v dv b sb nˆ nˆ sb S h nˆ P P
Special case ( s = 0): Fr s = 0 D n = D n n = n cs = cs.() s = 0 Als, we have = sin = sin.( ) qn. () qn. () gives, tan tan
ε r =.5 =4 ε r =.5
3-Capacitance It is an idealized circuit element representing the electrstatic energy stred in the system and is characterized by culmbvltage law V Q V =(/C)Q Any tw cnducting bdies, regardless f their shapes and sizes, when separated by an insulating (dielectric ) medium, frm a capacitr C=Q/V C Q V s B A D. ds. d [ Farad ]
xample: parallel-plate capacitr Frm the definitin f vltage Taking int accunt that the capacity is given by 3
xample: capacitance f caxial line The electric field is described fr a < r < b by: The vltage V between inner and uter cnductrs is The capacity is than given by 4
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