Lecture 5 pcitnce h. 5 rtoon - pcitnce definition nd exmples. Opening Demo - Dischrge cpcitor Wrm-up prolem Physlet Topics pcitnce Prllel Plte pcitor Dielectrics nd induced dipoles oxil cle, oncentric spheres, Isolted sphere Two side y side spheres Energy density Grphicl integrtion omintion of cpcitnce Demos Super DG Electrometer oltmeter irculr prllel plte cpcitor ylindricl cpcitor oncentric sphericl cpcitor Dielectric Sl sliding into demo Show how to clirte electroscope
Definition of cpcitnce pcitnce A cpcitor is useful device in electricl circuits tht llows us to store chrge nd electricl energy in controllle wy. The simplest to understnd consists of two prllel conducting pltes of re A seprted y nrrow ir gp d. If chrge + is plced on one plte, nd - on the other, the potentil difference etween them is, nd then the cpcitnce is defined s. The SI unit is, which is clled the Frd, nmed fter the fmous nd cretive scientist Michel Frdy from the erly 800 s. Applictions Rdio tuner circuit uses vrile cpcitor Blocks D voltges in c circuits Act s switches in computer circuits Triggers the flsh ul in cmer onverts A to D in filter circuit
Prllel Plte pcitor 3
Electric Field of Prllel Plte pcitor Gussin surfce d + + + + + + + + + + + E - - - - - - - - - - - + q - q A Are of plte A EA q! 0 E! 0 q A q! 0EA Ed qd! 0 A q q! 0A d qd!0 A f! i!! E " dˆr oulom/olt Frd Ed f # i Integrte from - chrge to + chrge so tht! E! dˆr "Edr f! i + " Edr Ed +! 4
Show Demo Model, clculte its cpcitnce, nd show how to chrge it up with ttery. irculr prllel plte cpcitor r d! 0A d r r 0 cm 0.m A!r!(.m) A.03 m d mm.00 m!.03m oulom (0 ) }Frd Nm.00m olt 3 " 0!0 F 300pF p pico 0-5
Demo ontinued Demonstrte. As d increses, voltge increses.. As d increses, cpcitnce decreses. 3. As d increses, E 0 nd q re constnt. 6
Dielectrics A dielectric is ny mteril tht is not conductor, ut polrizes well. Even though they don t conduct they re electriclly ctive Exmples. Stressed plstic or piezo-electric crystl will produce sprk. When you put dielectric in uniform electric field (like in etween the pltes of cpcitor), dipole moment is induced on the molecules throughout the volume. This produces volume polriztion tht is just the sum of the effects of ll the dipole moments. If we put it in etween the pltes of cpcitor, the surfce chrge densities due to the dipoles ct to reduce the electric field in the cpcitor. 7
Permnent dipoles Induced dipoles _ ++ _ E 0 the pplied field E the field due to induced dipoles E E 0 - E 8
Dielectrics The mount tht the field is reduced defines the dielectric constnt " from the formul E E 0! without he dielectric., where E is the new field nd E 0 is the old field Since the electric field is reduced nd hence the voltge difference is reduced (since E d ), the cpcitnce is incresed.! ' 0 $ % " &! # where " is typiclly etween 6 with wter equl to 80. Show demo dielectric sl sliding in etween pltes. Wtch how cpcitnce nd voltge chnge. Also show luminum sl. 0 9
d q E 0 d " E 0! 0! q A E 0 q! 0A qd! 0A! 0A d E E 0! E 0! d q!q 0 0!! 0 0
Find the cpcitnce of ordinry piece of coxil cle (T cle) f! i! f # i! E " dˆr! E. dˆr Edr cos80!edr Integrte from to or - to + Er k! r rˆ!! E.! " dˆr " Edr k# " dr +k# lnr r! k" ln! L k 4!"0 # ir is higher thn
cpcitnce of coxil cle cont. So, ln!"0l!" ln 0!" L ln L 0!" ln L Now if 0.5mm nd.0mm, then 43 L 6 " 0 ln4 pf m 0 L 6 " 0.38!! pf 86 L m $ 0 (for ir) And if ", then 0.5 mm.0 mm " % For "
Model of coxil cle for clcultion of cpcitnce Outer metl rid Signl wire - to + 3
pcitnce of two concentric sphericl shells dr -q +q Integrtion pth!! E! # " dˆr + # Edr s! E. dˆr Eds cos80!edr E! + " Edr + " kq dr +kq" dr kq r kq(! ) r q / 4!" 0 #! ) kq( r Let get very lrge. Then 4!" 0 for n isolted sphere 4
Sphericl cpcitor or sphere Recll our fvorite exmple for E nd is sphericl symmetry R k The potentil of chrged sphere is with 0 t r &. R The cpcitnce is k R R k 0 Where is the other plte (conducting shell)? 4!" R It s t infinity where it elongs, since tht s where the electric lines of flux terminte. k 0 0 nd R in meters we hve R 0 0 0!0 R(m) 0! R(cm) R( cm) pf Demo: Show how you mesured cpcitnce of electroscope Erth: (6x0 8 cm)pf 600 µf Mrle: pf Bsketll: 5 pf You: 30 pf 5
pcitnce of one chrged conducting sphere of rdius reltive to nother oppositely chrged sphere of rdius d d 4!$ 0 (+m+m +m 3 +m 4 +..) m /d d >>! If d gets very lrge, then 0 pf d 0 cm 0 cm m 0.5 0-0 (.) (+.5 +.5 +.5.) 0-0 (.)(/(-m)) 0. x 0-0 F 0.0 nf 0 pf 6
Electric Potentil Energy of pcitor As we egin chrging cpcitor, there is initilly no potentil difference etween the pltes. As we remove chrge from one plte nd put it on the other, there is lmost no energy cost. As it chrges up, this chnges. At some point during the chrging, we hve chrge q on the positive plte. The potentil difference etween the pltes is As we trnsfer n mount dq of positive chrge from the negtive plte to the positive one, its potentil energy increses y n mount du. du dq q dq. The totl potentil energy increse is q q U! dq Also U 0 using q 7
Grphicl interprettion of integrtion q/c U N! i dq " qi qi q q/c Are under the tringle U! dq 0 du dq where q q! dq 0 0 q Are under the tringle is the vlue of the integrl Are of the tringle is lso! h Are ()(h) ()( )! 0 q dq 8
Where is the energy stored in cpcitor? Find energy density for prllel plte cpcitor. When we chrge cpcitor we re creting n electric field. We cn think of the work done s the energy needed to crete tht electric field. For the prllel plte cpcitor the field is constnt throughout, so we cn evlute it in terms of electric field E esily. Use U (/) E " #! 0 "!! A nd Solve for $AE, ES nd sustitute in U (! AE)( ES)! E ( ) AS U " E! AS volume occupied y E We re now including dielectric effects: $ Electrosttic energy density generl result for ll "! E geometries. ES To get totl energy you need to integrte over volume. 9
How much energy is stored in the Erth s tmospheric electric field? (Order of mgnitude estimte) tmosphere h Erth R R 6x0 6 m 0 km olume 4! R E 00 0 m U " 0E! olume h olume "! U U 6 4 8 3 4 (6! 0 ) (! 0 ) 8.6 0 m " 8 )(0 )(8.6 0 3! m Nm m (0 4.3! 0 This energy is renewed dily y the sun. Is this lot? The totl solr influx is 00 Wtts/m Usun U 6 6 00 # 3.4(6! 0 ) "! 0 J s! 0 Usun # " 0! 0 Only n infinitesiml frction gets converted to electricity. J J dy ) World consumes out 0 8 J/dy. This is /000 of the solr flux. 0
Prllel omintion of pcitors Typicl electric circuits hve severl cpcitors in them. How do they comine for simple rrngements? Let us consider two in prllel. We wish to find one equivlent cpcitor to replce nd. Let s cll it. The importnt thing to note is tht the voltge cross ech is the sme nd equivlent to. Also note wht is the totl chrge stored y the cpcitors?. + + ( + ) +! +
Series omintion of pcitors Wht is the equivlent cpcitor? oltge cross ech cpcitor does not hve to e the sme. The chrges on ech plte hve to e equl nd opposite in sign y chrge conservtion. The totl voltge cross ech pir is: ) ( ) ( + + + So + Therefore, +
Smple prolem 0 µf 5.0 µf 3 4.0 µf ) Find the equivlent cpcitnce of the entire comintion. nd re in series. +! + 0! 5 50 3.3µF 0 + 5 5 nd 3 re in prllel. eq + 3 3.3 + 4.0 7.3µ F 3
Smple prolem (continued) 0 µf 5.0 µf 3 4.0 µf ) If 00 volts, wht is the chrge 3 on 3? / # 6 3 3 4.0" 0! 00 3 4.0" 0! 4 ouloms c) Wht is the totl energy stored in the circuit? U! 6 4 eq! " 7.3 " 0! U 3.6" 0 J F " 0 3.6 " 0 J 4