Lesson 07 Electrostatics in Materials


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1 Electomagnetcs P71 Lesson 07 Electostatcs n Mateals Intoducton Mateals consst of atoms, whch ae made u of nucle (oste chages) and electons (negate chages). As a esult, the total electc feld must be modfed n the esence of mateals. We usually classfy mateals nto thee tyes: 1) Conductos: The electons n the outemost shells of the atoms ae ey loosely held and can feely mgate among atoms due to themal exctaton at oom temeatues. These electons ae shaed by all atoms. ) Delectcs: All electons ae confned wthn the nne shells of atoms, and can hadly mgate een by alyng a stong exctaton. 3) emconductos: Outemost electons ae modeately confned, whch may not mgate due to themal exctaton but become moable when alyng an extenal electc feld. Fg (a) Enegy leels and bands. (b) Band dagams of nsulato, semconducto, and metal. In tems of quantum theoy, the electons of an atom can only stay at dscete enegy leels (Fg. 71a, left). When a lage numbe of atoms aggegate n odeed manne (cystallne sold), each enegy leel slts nto densely saced leels (enegy band ). The allowed enegy bands can be oelaed o seaated by fobdden enegy ga (Fg. 71a, ght). Electc chaactestcs of mateals deend on the band stuctue and how they ae flled by the electons at the temeatue of 0 K (Fg. 71b):
2 Electomagnetcs P71) Conductos: Bands ae oelaed, o the conducton band s atally flled. ) Delectcs: Enegy ga s lage, and the conducton band s emty. 3) emconductos: Enegy ga s small, and the conducton band s emty. In ths lesson, we wll dscuss the feld behaos both nsde some mateal and on the nteface of dffeent mateals. 7.1 tatc Electc Feld n the Pesence of Conductos Chage and electc feld nsde a conducto Assume that some oste (o negate) chages ae ntoduced n the nteo of a conducto. Intally an electc feld wll be set u, whch wll ush the fee chages away fom one anothe by Coulomb s foce and modfy the electc feld dstbuton by tuns. Ths chagefeld nteacton ocess wll contnue untl: (1) all the chages each the conducto suface (no way to leae): ρ = 0, (7.1) and () the suface chages edstbute themseles such that no electc feld exsts nsde the conducto: E = 0 (7.) Eq. (7.) can be justfed by: 1) If E 0 somewhee nsde the conducto, the electc otental V ( ) s nonunfom. Ths wll esult n a contadcton that wok has to be done to moe fee chages. ) The fee chage dstbuton that leads to E = 0 actually coesonds to the lowest system enegy,.e., a state of equlbum (Lesson 9). Fo good conductos lke coe, the tme equed to each the equlbum s only n the ode of sec (Lesson 10).
3 Electomagnetcs P73 Aconducto nteface nce the chages on the conducto suface wll not be at est f thee s tangental electc feld comonent, the suface electc feld only has nomal comonent n the equlbum. goous bounday condtons (BCs) ae deed by alyng eq s (6.3), (6.4) wth esect to the dffeental contou abcda and the thn ll box (Fg. 7) acoss the aconducto nteface, esectely. Fg. 7. Dffeental contou and ll box used to dee BCs on the aconducto nteface (afte DKC). 1) Tangental BC: By eq s (6.4), (7.), E dl = E Δw + 0 Δw = 0, abcda Δh 0 t ) Nomal BC: BC: By eq s (6.3), (7.13), E = 0 (7.3) t E ds = E ρ Δ Δ =, s n Δh 0 ε 0 ρ s ε 0 E =, (7.4) n whee E n s n the nomal decton fom the conducto to the a. Examle 71: Consde a ont chage + located at the cente of a shecal conductng shell of fnte thckness (Fg. 73a). Fnd E, V nsde and outsde the shell.
4 Electomagnetcs P74 Fg (a) The geomety of a conductng shee. The coesondng (b) adal comonents of E (dashed), and (c) electc otental V. Ans: Because of the shecal symmety of the souce, we hae E a = E () eeywhee, and the Gaussan sufaces ae concentc shecal sufaces centeed at the ogn. < : E ds = E ( ) ( 4π ) =, E ( ) =. 3 ε 0 4 πε 0 () Fo < < o : By eq. (7.), E = 0, E ds = 0 (1) Fo fo any enclosed suface nsde the shell. Ths mles that fee chages of must be nduced on the nne suface of the shell. By the conseaton of chages and eq. (7.1), fee chages of + must be nduced on the oute suface of the shell. (3) Fo > o : The total chage enclosed by a Gaussan suface 1 s +, E ( ) =. 4 πε 0 The cue of E = E () s shown n Fg. 73b. One can dee the electc otental dstbuton V () by lne ntegal of E (Fg. 73c).
5 Electomagnetcs P tatc Electc Feld n the Pesence of Delectcs Concet of nduced doles A delectc molecule could be nonola (e.g., H, CH 4 ) o ola (e.g., H O, HCl, NH 3, O 3 ), deendng on whethe thee s nonzeo electc dole moment [eq. (6.17)]. Howee, a delectc bulk made u of a lage numbe of andomly oented molecules (ola o nonola) tycally has no macoscoc dole moment n the absence of extenal electc feld. When an electc feld s aled, the bound chages of the delectcs cannot feely mgate to the suface. Instead, (1) each nonola molecule s olazed fo the ostely chaged nucle and negately chaged electons ae slghtly dslaced n ooste dectons (dstoton of electon cloud). () Most of the nddual dole moments (ectos) ae algned n the same decton due to the electc toque. In ethe case, a macoscoc dole moment emeges. Fg A coss secton of a olazed delectc medum (afte DKC). Although an electc dole s neutal n ensemble, t odes nonzeo otental and electc feld [eq s (6.18), (6.16)], whch wll modfy the total feld nsde and outsde delectcs. Polazaton ecto and equalent chage denstes To analyze the effect of nduced doles, we defne a (mcoscoc) olazaton ecto P as
6 Electomagnetcs P76 the olume densty of electc dole moment: k P lm (7.5) Δ 0 Δ whee k denotes the kth dole moment nsde a dffeental olume Δ. If the olazaton ecto P s nhomogeneous (.e., aes wth oston) somewhee, thee must exst net (bound) olazaton chage at that oston. Ths henomenon can be llustated n two cases. Fg (a) The model to deduce the olazaton suface chage densty. (b) Net olazaton olume chage exsts whee the olazaton ecto P s nhomogeneous. 1) The olazaton ecto P s dscontnuous on the adelectc nteface, whee thee must exst net olazaton chage (Fg. 75a). To quanttately model ths henomenon, consde a dffeental aalleleed V bounded by a closed suface b, whee the dole moment of each molecule s = qd. The to suface of has a unt outwad b nomal ecto a n a n and an aea of Δ. The effecte heght of the aalleleed V s d. The net olazaton chage wthn V s: Δ = nq ( d a ) Δ = ( P a ) Δ n whee n (1/m 3 ) s the numbe densty of molecules. The coesondng olazaton n,
7 Electomagnetcs P77 suface chage densty Δ Δ s: s P a ρ = n (C/m ) (7.6) ) Consde two doles wth dffeent olazaton ectos P 1, P n the nteo of a dffeental olume V bounded by a closed suface (Fg. 75b). Although each nddual dole s electcally neutal, some net chage can exst on the nteface due to ncomlete cancellaton of the olazaton chage between adjacent doles. To mantan the electc neutalty, must be equal n magntude but ooste n sgn to the total suface olazaton chage s1 + s. Let ρ eesent the olazaton olume chage densty, ( P an ) ds = P ds = ( P) V ρ d. By eq s (7.6), (5.4), = d = V. The equement of ( + ) ( P) s1 s s1 + s = esults n: ρ = (C/m 3 ) (7.7) <Comment> 1) Eq. (7.6) can be egaded as a secal case of eq. (7.7) on the adelectc nteface, whee the degence of the olazaton ecto P s nfnte. ) Equalent olazaton chage denstes ρ s, ρ can be used n collaboaton wth eq s (6.10), (6.15) to ealuate the electc feld and otental contbuted by olazed delectcs. Electc flux densty In the esence of delectc mateals, the total electc feld would be ceated by both the fee and olazaton chages. Fundamental ostulate eq. (6.1) s thus modfed as:
8 Electomagnetcs P78 ρ + ρ E =. By eq. (7.7), we hae: D = ρ, (7.8) D ε 0 ε E + P (C/m ), (7.9) = 0 whee the electc flux densty D chaactezes the contbuton fom the fee chages. The ntegal fom of eq. (7.8) (Gauss s law) becomes: D ds = (7.10) Fo lnea, homogeneous, and sotoc delectcs, the olazaton ecto s ootonal to the electc feld (by the elastc sng model): P = ε 0 χee, (7.11) whee the electc suscetblty χ e s a dmensonless quantty ndeendent of magntude (lnea), oston (homogeneous), and decton (sotoc) of E. By eq s (7.9), (7.11), D = εe, (7.1) whee the emttty of the medum ε s defned as: ( 1+ χe ) ε 0 ε = (7.13) <Comment> The stategy s usng a sngle constant ε to elace the tedous nduced doles, olazaton ecto, and equalent olazaton chages n detemnng the total electc feld. Examle 7: Consde a aallellate caacto. (1) D descbes the suface densty of fee chages on the conductng lates. () P descbes the suface densty of olazaton chages.
9 Electomagnetcs P79 (3) ε E 0 descbes the suface densty of total chage o uncomensated fee chage. Fg Physcal meanngs of electc flux densty D, olazaton ecto P, and electc feld ntensty E llustated n the examle of a aallellate caacto (afte C. C. u). Examle 73: A ont chage + at the cente of a shecal delectc shell of emttty ε ε 0 (Fg. 77a), whee χe ε = 1 + s the elate emttty. Fnd D, E, V, P, and ρ s. Fg (a) The geomety of a delectc shee. The coesondng (b) adal comonents of D (sold), ε E (dashed), P (dashdot), and (c) electc otental 0 V.
10 Electomagnetcs P710 Ans: (1) By shecal symmety, D a = D (). By eq. (7.10) and the fact that delectc mateals contbute to no fee chage: D ds = D ( ) ( 4π ) = D ( ) =, fo all > 0. 4 π () By shecal symmety, E a = E (). By eq. (7.1),, E ( ) =, whee 4 πε ε ε 0, fo < < ε = ε 0, othewse 0. (3) V () s deed by ntegaton of E (). (4) By eq. (7.9), P = D ε E = a P ( ), whee P ( ) = D ( ) ε 0E ( ), 0 1 ( 1 ε ) P ( ) = 4π 0, othewse, fo < < 0. (5) By eq (7.6), the suface olazaton chage densty becomes: ρ s a = 1 1 ( 1 ε ) ( a ) = ( 1 ε ) ( 1 1 ε ) π 4 4π o ( > 0), fo = o 4π ( < 0), fo = ; The total olazaton suface chage s: s = ρ = s 1 1 ( 1 ε ) ( 4π ) = ( 1 ε ) 4π 1 ( 1 ε ) ( > 0), fo = o ( < 0), fo = ; They ceate a adally nwad E to educe the total electc feld n the delectcs. 7.3 Geneal Bounday Condtons fo Electc Felds Deaton As n aconducto nteface, we aly the ntegal foms of the two fundamental ostulates on the dffeental contou abcda and the thn ll box wth Δh 0 acoss the nteface of
11 Electomagnetcs P711 two delectc meda (Fg. 78) to dee the BCs fo the tangental and nomal comonents of the electc feld. Fg Dffeental contou and ll box used to dee geneal BCs (afte DKC). E dl = E Δw + E Δw, 1) Tangental BC: By eq. (6.4), ( ) ( ) 0 abcda E t Et ) Nomal BC: BC: By eq. (7.10), D ds = a 1 t t = 1 = (7.14), ( D a + D a )( Δ ) = ρ Δ 1 n ( D1 D ) = ρs n n1 s, (7.15) Eq. (7.15) can also be wtten as: D ρ 1 n D n = s, whee n D ( = 1, ) denotes comonent of medum 1). D n the decton of a n (unt nomal ecto dected fom medum to <Comment> 1) Eq s (7.3), (7.4) ae secal cases of eq s (7.14), (7.15), whee { E D } 0, = ae used. ) Only fee suface chage densty ρ s counts n eq. (7.15). If the two ntefacng meda ae both delectcs, ρ s = 0, D1 n = Dn. 3) Eq s (7.14), (7.15) eman ald een the felds ae tmeayng (Lesson 14, o Ch 7 of the textbook). BCs exlan why most otcal comonents (wth mateal ntefaces) ae olazaton deendent.
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