Summary chapter 4. Electric field s can distort charge distributions in atoms and molecules by stretching and rotating:

Transcription

1 Summa chapte 4. In chapte 4 dielectics ae discussed. In thse mateials the electns ae nded t the atms mlecules and cannt am fee thugh the mateial: the electns in insulats ae n a tight leash and all the can d is mve a it within the atm mlecule. The cumulative effects f thse micscpic displacements accunts f the chaacteistic ehavi f dielectic mateials. Electic field s can distt chage distiutins in atms and mlecules stetching and tating: Althugh the cente f psitive chage and negative chage in atms and mst mlecules that eside in a e electic field cincides when placed in an electic field this is n lnge the case. This esults in plaiatin. We discussed atmic plaiailit, f a single atm a single mlecule. It elates the electic diple mment f the atm mlecule with the etenal electic field. p ae [a] Whee p is the electic diple mment. This is a vect pinting fm the negative t the psitive chage that has a magnitude equal t q*d, whee d is the distance etween the psitive and negative chage. We wte the vect p is chapte 3 as: p q Whee + ( - ) is the psitin vect f the psitive chage (psitin vect f the negative chage). We deived an equatin f f a hdgen atm: 3 4 a [] [] Whee a is the adius f the atm. Nte that the tpical plaiailit is much smalle than the atmic adius (check eample 4. and plem 4.). In additin we discussed mlecules that have a pemanent electic diple mment. Such diple mment iginates fm the diffeent electnegativit f the atms the mlecule cnsists ff. Electnegativit is a measue f the attactin f an atm f electns in a cvalent nd. A pemanent electic diple epeience a tque when placed in an unifm electic field. The tque aut the cente f the diple is given : N p E [3] If the field is nn-unifm the diple als epeiences a fce: F p E [4]

2 This epessin p E is the diectinal deivative f E in the diectin f p, s hw much des E change in the diectin f p, E in the diectin f p. We discussed this ne in chapte : p E p E p p E E p p E iˆ p E p E E kˆ p E p E ˆj [5] Nte that equatin [3] is als valid f a nn-unifm electic field as lng as ne is inteested in calculating the tque aut the cente f the diple. The tque aut an the pint is given : N [6] p E F As jects cnsists f a lt f pemanent induced electic diples we wuld like t use the electic diple densit. This quantit is called the plaiatin and is identified a capital P. Nte that this is a vect quantit. Its magnitude tells us the electic diple mments pe unit f vlume. Yu can cmpae it with the chage densit we intduced in chapte. Of cuse the chage densit is nt a vect ut a scala quantit. The field f a plaied ject is the summatin f the fields caused all electic diples the ject cnsists f. The electic ptential f a single diple was deived and discussed in chapte 3 (equatin 3.0): 4 scipt scipt p [7] S f a plaied ject: scipt P ' d ' 4 vlume scipt [8] Nte that fm equatin [7] t [8] we eplaced the electic diple mment, p, the pduct f the plaiatin and a vlume element, i.e. Pd. Equatin [7] cespnds t equatin 4.9 in the tet. As calculating equatin [8] is time cnsuming we simplified equatin [8] t: d ' 4 4 vlume scipt suface scipt da ' [9] Whee p is the und vlume chage densit and is the und suface chage densit. The ae given :

3 P P nˆ [0] S athe than have t sum up ve all cntiutins f the electic diples, we can sum up ve all und chages. In class we discussed the phsical intepetatin f und chages. The displacement f the chages in the mateial caused the plaiatin will nl esult in a net chage nea the suface f the mateial nea aeas whee the plaiatin is nt cnstant, i.e. has a nn-e divegence. Nte that thee ae tw was t detemine the electic field. One can fist detemine the electic ptential using equatin [9] and then calculate the electic field fm the electic ptential using: E O ne can use Culm s law and integate ve all chages. If n fee chages ae thee, this will e an integatin ve the and. S f this appach the plem cnvets t the plems we discussed in chapte. Of cuse thee is a thid methd t find E, ut that ne is nt ve eas and nt discussed in the tet. We culd stat ff with the electic field f a single diple, i.e. equatin 3.03, and then integate ve all diples f the ject. The math will e hendus S nmall we fist detemine the und chage using equatin [0], then use equatin [9] t find, and then use equatin [] t find E if asked f. F smmetic plems we might want t use Gauss law and diectl detemine E instead f equatins [9] and []. Nte that Gauss law f E takes int cnsideatin all chages, i.e. the und and the fee chages. As the electic field in the ject is ften nt cnstant, als the plaiatin in the ject vaies as a functin f the psitin. This makes it ften difficult t detemine the und chages. We theefe intduced a new paamete, the electic displacement D. This is als a vect quantit and is defined as: D E P [] The easn f this intductin ecmes clea nce we take the divegence f equatin [], i.e. D E P E Nte that the divegence f E is equatin t/, whee is the ttal chage densit i.e. und chage densit and fee chage densit (this is Gauss law in diffeential fm, i.e. equatin.4), s we can ewite equatin [3] nw t: D fee This Gauss law f the electic displacement. It sas that the divegence f D is equal t the fee chage densit. In integal fm this law ecmes: [] [3] [4]

4 suface D da Q fee enclsed [5] Nte that n the left side f equatins [4] and [5] ae n. F linea dielectic mateials, we can use equatin [4] and [5] t fist detemine the D. The advantage f such appach is that we d nt need t knw the und chage vlume and suface chage densit as nl the fee chages ae t e knwn. This simplifies the math cnsideale. Nte that f the capacit plems in chapte 4 we fist detemine D fm the fee chages and then use the fllwing equatin t detemine D fm E: D E P E E [6] e E E E e Nte that this equatin implied a lt f elatins. I wuld like u t memie thse elatins, and the shuld als e n u equatin sheet. E0 is called the pemittivit f fee space, is called the pemittivit f the mateial, is called the dielectic cnstant f the mateial (/ ), e is called the electic susceptiilit f the mateial and f linea dielectic mateials is defined : P E e [7] Cmpae equatin [7] with equatin [] and u have t cnclude that the electic susceptiilit, e, is elated t the atmic plaiailit,. Thee is a caveat thugh, as the E in equatin [] is the etenal applied field, and the E in equatin [7] is the ttal field felt the diple, i.e. the effect f the etenal electic field and the induced electic field caused the und chages. The latte is ften ppsite t the etenal electic field. S the elatin etween the electic susceptiilit and the atmic plaiailit is nt staight fwad and will e discussed in slid state. We discussed the unda cnditins f dielectic mateials: D E ave // ave D E elw // elw 0 fee [8] If thee is n fee chages, i.e. f =0, the pependicula cmpnent f D will e cntinuus and the paallel cmpnent f E will als e cntinuus acss the inteface. Ntice that the electic ptential, i.e. will als e cntinuus acss the inteface. The fist equatin can e deived fm equatin [4], i.e. Gauss law f D, and the nd equatin can e deived fm the fact that E is cul-less. The eneg f a dielectic sstem can e calculated fm the eneg densit equatin, i.e. : eneg densit D E [9] S the ttal eneg f the sstem

5 W D vlume Ed [0] Nte that the eneg densit is nt alwas cnstant thugh space and that ne might need t find sepaate epessins f the eneg densit inside and utside the mateial and then d the integal in tw steps. Equatin [0] includes the electstatic eneg sted in the fee chages, the electstatic eneg sted in the und chages, and the sping eneg sted in the und chages. The latte tw ae ppsite and equal in magnitude. At the end f chapte 4 we leaned hw t calculate fces fm the ttal eneg epessin. As the eneg is lweed when ne places a dielectic mateial inside a capacit, a fce is acting n the dielectic mateial, that sucks the dielectic mateial in etween the plates. We deived an epessin n hw t detemine the fce fm the capacitance and the electic ptential acss the capacit plates: F dc [] d We deived this epessin f the situatin that the electic chage n the capacit plates is cnstant. The epessin is hweve als valid f the case that the electic ptential diffeence acss the plates is kept cnstant.