Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5.

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1 PHY1 Electricity Topic 5 (Lectures 7 & 8) pcitors nd Dielectrics In this topic, we will cover: 1) pcitors nd pcitnce ) omintions of pcitors Series nd Prllel 3) The energy stored in cpcitor 4) Dielectrics Reding from Young & Freedmn: For this topic, red the introduction to chpter 4 nd sections 41 to 45 Introduction A cpcitor is mens of storing electric chrge We sw t the strt of the course tht n isolted conductor cn e chrged electrostticlly, so this could e considered s the most sic sort of cpcitor The chrges on the conductor tend to repel ech other, nd to reduce their energy they will lek wy to ground if ny slightly conductive pth is ville We cn store the chrge more esily if we cn reduce its energy, which mens reducing its potentil for given chrge This cn e chieved y rrnging tht the conductor crrying chrge is in close proximity to nother conductor crrying Although the chrges within ech conductor my e repelling ech other, they will e ttrcted y those in the other conductor nd their net energy will e reduced pcitors nd pcitnce onsider two seprted conducting pltes, connected to ttery The electromotive force of the ttery drives chrge from one terminl to the other One plte will therefore end up with positive chrge while the other will hve n equl negtive chrge The potentil difference etween the pltes will e equl to the voltge supplied y the ttery If the ttery is disconnected, the chrge will remin on the pltes, which will mintin the sme potentil difference As we will see, the mgnitude of the chrge stored on either plte is directly proportionl to the potentil difference etween the pltes We cn write this s = [1] where the constnt of proportionlity is known s the cpcitnce of the system, which is clled cpcitor The cpcitnce is thus the quntity of chrge stored per unit potentil difference etween the pltes The SI Unit of cpcitnce is the Frd (F), nd 1 Frd = 1 oulom / olt The Frd (like the oulom) is very lrge unit, nd most prcticl cpcitnces re mesured in microfrds (1 µf = 1 6 F) or picofrds (1 pf = 1 1 F) We shll shortly see tht cpcitors store electricl energy, s well s chrge Prllel Plte pcitor The simplest, nd most common, geometry for cpcitor consists of two flt metl pltes, close together ut seprted y lyer of insulting mteril When connected to potentil difference, they will crry opposite mounts of electric chrge, ± If the seprtion of the pltes is very smll compred with their lterl dimensions (s it usully is), we cn ignore fringe fields ner the ends, nd consider uniform electric field which only exists etween the pltes A E PHY1 Electricity 8

2 The ttrction etween the chrges mens tht they lie on the inner surfces of the pltes Assume the pltes hve n re A nd seprtion d Applying Guss s lw to the dotted surfce shown, which completely surrounds one plte, we hve EA = E = A (Note tht this is different from the cse of n isolted sheet of chrge considered previously, s here the electric field is ll on one side of the sheet, nd so only cuts one side of the Gussin ox ) We clculte the potentil difference moving from lower to upper plte from = dl = Ed E where the cncelling minus sign comes from the fct tht E nd the pth d re in opposite directions Thus Using our definition for from [1], we hve d = A = = A [] d Rerrnging this eqution lso provides us with n lterntive, nd more common, unit for which is F m 1 To store lrge mount of chrge t low potentil, ie to produce lrge cpcitnce, we oviously wnt lrge plte re A nd smll seprtion d This cn e mnufctured from two long strips of luminium foil, seprted y thin sheet of plstic film In prcticl cpcitor, this sndwich is covered y nother sheet of plstic nd then rolled up into cylinder As we will see lter, the plstic lyer my hve other enefits esides its insulting nture pcitors in Prllel nd Series When two or more cpcitors re connected together, they ehve like single cpcitor with modified cpcitnce It is useful to e le to clculte the effective cpcitnce of the comintion in terms of the individul cpcitnces pcitors in Prllel If two cpcitors re connected s shown, with one plte of ech connected to ground ( ) nd the other to voltge, then from [1] cpcitor 1 will crry chrge 1 = 1 nd similrly for The = = = where is the totl chrge is ( ) 1 1 effective cpcitnce of the comintion So = 1 For N cpcitors in prllel, = 1 N [3] (In prllel comintion, the totl cpcitnce is lwys greter thn ny individul cpcitnce) 1 PHY1 Electricity 9

3 pcitors in Series Two components re connected in series when they shre common terminl, s shown When connected to n externl potentil, chrge will flow onto the cpcitors s shown Note tht since there is no externl connection to the centre point of the circuit, the totl chrge on the lower plte of 1 nd the upper plte of must e zero Therefore oth cpcitors must crry the sme chrge, The voltge cross the circuit is just the sum of the voltges cross the individul cpcitors, = 1 1 But from [1], = 1 =, etc So nd therefore 1 1 = = = 1 1 or for N cpcitors in series, = [4] 1 N (In series comintion, the totl cpcitnce is lwys less thn the smllest individul cpcitnce) Energy Stored in pcitor We hve seen tht when chrge is t n electrosttic potentil, it possesses potentil energy tht is, the ility to do work A chrged cpcitor must therefore store energy, s well s chrge The work required to dd n infinitesiml chrge dq to cpcitor lredy crrying chrge q is q dw = dq = dq The totl work done to chrge cpcitor to chrge is therefore q W = dq = This is the electricl potentil energy U stored in the cpcitor Using eqution [1], this cn e expressed in vriety of wys = = = [5] U The Energy Density of n Electric Field We cn consider the energy stored in cpcitor (or other system of chrges) either s eing due to chrges held t potentils or s energy stored in n extended electric field Writing U 1 = nd Ed =, nd using [] = A, we hve d 1 A ( ) 1 = = d U Ed AdE But Ad is the volume etween the pltes, tht is the volume where the electric field exists The energy density, tht is the energy per unit volume, is therefore just u 1 = E [6] This expression is not only vlid for the prllel plte cpcitor, or indeed for ny other sort of cpcitor It is the generl expression for the energy density in n electric field PHY1 Electricity 3

4 Dielectrics The cpcitnce of cpcitor could oviously e incresed if we could reduce the potentil t which given chrge is stored This could e chieved if the electric field ws reduced Is there some wy this cn e done, for given geometry of cpcitor? The nswer is tht it cn, if mteril known s dielectric is introduced into the gp etween the cpcitor pltes We hve lredy seen tht conductor contins free chrges, nd tht in n externl electric field these move until the net internl field reches zero (We could not fill the gp etween the cpcitor pltes with conductor, s this would llow chrge to escpe from one plte to the other, nd no chrge would e stored!) A dielectric is n insultor which, though neutrl overll, contins ound chrges which cnnot move throughout the ulk of the mteril ut which cn e displced slightly under the influence of n electric field Exmples might e simple toms, in which electrons move smll distnce one wy while the positive nucleus is displced slightly the other when field is pplied, or more complicted orgnic molecules with polr groups which re positive nd negtive while eing ound to the molecule In ech cse, the chrges move under the influence of the pplied field without leving their prent toms or molecules, until n internl restoring force lnces the force due to the externl field They therefore ecome microscopic dipoles, generting dipole moment in the direction of the pplied field In most mterils, the displcement, nd so the size of the dipole moment, is proportionl to the pplied electric field The net effect of this is tht within the dielectric, the mteril is neutrl ut the overll field is reduced, s the dipoles produce field which opposes the pplied field On the surfce of the dielectric, excess ound chrges exist, nd the mteril is sid to e polrised The digrm shows sl of dielectric, polrised y n externl field nd so reducing the electric field within its ulk The electric field inside the mteril is reduced y fctor k known s the dielectric constnt of the mteril This mens tht the potentil difference etween the two pltes of cpcitor filled with dielectric mteril is lso reduced y k From eqution [1], =, we therefore see tht the cpcitnce is incresed y the sme fctor For prllel plte cpcitor contining dielectric mteril, the cpcitnce is therefore ka = d [7] (The dimensionless constnt k is lso sometimes known s r, the reltive permittivity of the mteril) E PHY1 Electricity 31

5 Putting Wht You Hve Lernt Into Prctice uestion 51 A prllel plte cpcitor hs pltes with dimensions 3 cm y 4 cm, seprted y mm The pltes re connected cross 6 ttery Find () the cpcitnce; () the mgnitude of chrge on ech plte; (c) the energy stored in the cpcitor () The re of the pltes is m The cpcitnce is () The chrge stored is A d = = = 3 53pF 1 1 = = = 3 1 (c) The stored energy is U = = = J uestion 5 Wht is the cpcitnce of n isolted metllic sphere of rdius R = cm? The potentil of the sphere when crrying chrge is = 4 π R 1 11 = = 4π R = 4π = 1 F pcitnce is pf uestion 53 For the circuit shown opposite, find: () the effective totl cpcitnce; () the chrge nd potentil difference for ech individul cpcitor () We strt with the prllel su-circuit of nd 3, nd hve 3 = 3 = 4µ F This comintion is in series with the other two cpcitors, so the totl equivlent cpcitnce is given y 1 =6F =1F 3 =3F 4 =1F = = = = F () pcitors in series hve the sme chrge, nd this is lso the chrge on the overll equivlent cpcitnce = = = µ So 1 = 4 = 3 = 96µ PHY1 Electricity 3

6 Since nd 3 hve common potentil difference cross their terminls, the chrge they crry will e proportionl to their cpcitnce, so ¼ will e on, nd ¾ on 3 = 4µ ; = 7µ 3 We find the voltges from = 96 1 = = 16; = 96 1 = = = = = 4= 6 3 = = = (As check, = = 48 s required) uestion 54 ylindricl pcitor Wht is the cpcitnce of long cylindricl (coxil) cle of inner rdius, outer rdius nd length L s shown? How is the cpcitnce chnged if the insultion etween the conductors is plstic with dielectric constnt k? onsider chrge on the inner conductor, nd on the outer hoose s Gussin surfces cylinder of length L nd rdius r, etween the other cylinders From Guss s lw we hve da= E E π r L = E = π rl To find the potentil difference etween the conductors, we integrte over rdil pth from the outer to the inner conductor Since =, we hve dr = E dr = = ln = ln π rl π L π L πl = ln ( ) If n insultor of dielectric constnt k fills the spce etween the conductors, the cpcitnce is just incresed y fctor k uestion 55 Sphericl pcitor Wht is the cpcitnce of two concentric sphericl conducting shells of inner rdius nd outer rdius? We tke the sme pproch s in question 54, considering chrge on the inner conductor, nd on the outer sphere For Gussin surfce, we consider concentric sphere of rdius r, where < r < Applying Guss s lw yields 4π = = 4π r E r E PHY1 Electricity 33

7 Integrting from outer to inner conductor to find the potentil difference we get Then from dr = E dr = = = = 4π r 4π r 4π 4π =, we hve = 4π Prolems from Young & Freedmn for Topic 5: Try to do exercises 41 to 444 nd 147 to 475 The lter prolems re more chllenging (Numericl nswers to odd-numered questions re ville t the ck of the ook) PHY1 Electricity 34

Physics 1402: Lecture 7 Today s Agenda

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