guns in TV tubes. These devices play important role in time varying voltages and currents, in generating and receiving electromagnetic

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1 Lecture 8 DILTRI RORTIS OF MTRILS Lecture 8 - apactor system of charges charge of the sngle plate: total charge of the system = potental fference between plates: S W.H.Freeman & co V r r S // r S V V V lectrcal capactance S V S V B B unform fel capactor, chargng, plates, In general capactor s a set of two conuctors of arbtrary shape solate from surrounngs. We call them plates. apactor s a evce that stores energy n electrostatc fel. It can be slowly charge an mae to release ts energy raply, gvng great power values. apactor can also be use to prouce electrc fel, for example n electron guns n TV tubes. These evces play mportant role n tme varyng voltages an currents, n generatng an recevng electromagnetc waves. We say that capactor s charge f ts plates carry equal an opposte charges +q, - q respectvely. The net charge of capactor s zero. hargng the capactors takes place when we connect ts plates to opposte termnals of the battery. Due to ts potental fference the battery transfers equal an opposte charges to the plates. The potental fference of a battery appears across the plates. The amount of charge gathere on the plates epens on the magntue (absolute value) of potental fference an vce versa - the potental fference between the plates epens on the charge place on them. The electrcal capactance of the capactor s the proportonalty constant between voltage across, an charge on the system. It s constant for the gven system of plates an epens exclusvely on ts constructon an materal between plates.

2 Lecture 8 - apactor V F V fara F nf pf capactance: confguraton -sze an shape of the plates materal fllng the capactor parameters: capactance F rate voltage V polarty apactor n a crcut tme characterstcs of the crcut voltage stablzaton storage of an energy (charge) ntegrator polarse capactors, rate voltage, Fara 6 9 F F F ar ceramc electrolytc

3 Lecture 8-3 alculatng the capactance V ssummng charge n a system stablshng the fel strbuton between plates alculatng the potental fference between plates apactance from efnton: sphercal 4 b ab a V () ; (, r Isolate sphere ) V V r b a R 6 F R.5 km 4 R hyperphyscs.phy-astr.gsu.eu b ln cylnrcal a roblem solvng strategy: alculatng the capactance of a gven capactor: - ssume that there s alreay charge q on the plates of the capactor. - Inentfy the strbuton of electrc fel lnes of the fel prouce but that charge strbute on the plates. In most cases charge you may assume the unform strbuton of charge as well as you may fn elements of symmetry of the fel strbuton. ssume that the lmte sze of the plates results n neglgble sturbances n the fel strbuton. Ths s justfe by the technologcal accuracy of manufacturng capactors. - Fn the expresson for the electrc fel nse the capactor as a functon of the poston (stance) wth respect to the plates of the system. pply the Gauss law or oulombs law for ths purpose. - Havng expresson for the fel, calculate the potental fference between the plates. - alculate the capactance from the efnton equaton (as proportonalty constant). Note that charge of the capactor s not nvolve n the result. - heck unt of the result. - valuate the result (analytcal an numercal). Relate the numercal value of the capactance to the commercally avalable capactors.

4 Lecture 8-4 arth capactor? cosmc rays onosphere onosphere 5km arth s surface on current total charge = V 4 V m KV arth Joshua Strang permanent chargng the arth s surface wth negatve charge (~8!) stable potental fference (~ 4 kv) ret: ISS/NS onosphere The exstence of ons n the atmosphere s the funamental reason for atmospherc electrcty. The onzaton n the lower atmosphere s mostly cause by cosmc rays an natural raoactvty. Ions are also prouce n an near thunerclous by lghtnng an corona processes. Learn more: arth electrc fel 8&page=95 Learn more: The electrc structure of the atmosphere f

5 Lecture 8-5 arallel connecton of capactors 3, 3 3 voltage : the same on all elements harge: fferent Total charge: sum of charges apactance : ncrease; t s bgger than that of any element ollectng bgger charge at the same voltage connecton n parallel In the electrc crcuts we frequently eal wth more capactors connecte n a certan way. Such connectons are mae ue to the nee to obtan certan electrcal effect or just because of lack of sutable sngle capactor. So t s esrable to know the equvalent capactance of connecte capactors. By ths equvalent capactance we mean the capactance of a sngle capactor that can be substtute for the combnaton wth no change n the operaton of the rest of the crcut. There are two man systems of capactors connectons name parallel an seral connecton. apactors can be connecte n ways that are not ether parallel or seral combnatons. Such combnatons can often (but not always) be broken own nto smaller parts that are seral or parallel n type.

6 Lecture 8-6 Seral connecton 3 3 harge : equal on all elements Voltage: fferent Total voltage: sum of voltages across the elements 3 apactance : ecrease; smaller than the smallest of the elements pplyng bgger voltage at the same total charge connecton n seres

7 Lecture 8-7 apactor wth the electrc el el el electrc permttvty of the materal el S el el wth the electrc, vacuum 3, polethylene 4, paper 7, ceramcs 78, water parallel plate capactor S S V S el el el el el el lectrc fel nse the capactor wth the electrc ecreases! electrc constant, permttvty of materal The presence of materal alters electrc fel. xpermentally t can be easly shown that capactance ncreases when a electrc s place between the plates. The mensonless factor by whch the capactance of the capactor ncreases n relaton to ts value wthout electrc s calle the electrc constant (electrc permttvty) of materal. The charge storage ablty of capactor s enhance by the electrc whch permts to store a factor more charge for the same potental fference.

8 Lecture 8-8 W W q // l q l q l nergy store n the capactor () q q W q l const ( t ) ( t ) q W W lectrc fel energy store n the capactor electrostatc potental energy Durng the creaton of the certan charge confguraton the certan amount of work s one by the external agent that assembles the charge confguraton from ther ntal nfnte separaton to the fnal one. Ths work s store n a form of an electrc potental energy. Smlarly, the charge capactor has store n t an electrc potental energy equal to the work one by the external agent as the capactor s charge. Ths energy can be recovere f the capactor s scharge. the work of chargng s one by battery as the expense of ts store of chemcal energy. Snce capactors store charge only on the surface of the electroe, rather than wthn the entre electroe, they ten to have lower energy storage capablty an lower energy enstes. The charge/scharge reacton s not lmte by onc conucton nto the electroe bulk, so capactors can be run at hgh rates an prove very hgh specfc powers but only for a very short pero. apactors are use extensvely as power back up for memory crcuts an n conjuncton wth batteres to prove a power boost when neee. No chemcal actons nvolve whch means very long cycle lfe s possble. Learn more: apactors an supercapactors as energy storage

9 Lecture 8-9 nergy store n the capactor () hanges of the energy store n the capactor. =const S. =const vacuum capactor apactor flle wth the electrc const energy ecreases! energy storage S very change of capactance by changes n the geometrcal parameters ( for example change of separaton or area n parallel-plate capactor) results n the change of the electrc energy store, as well as n the case of changes n amount of charge or potental fference.

10 Lecture 8 - lectrc fel nse the matter conuctor polar materals nonpolar materals permanent poles nuce poles polar an non polar materals, electrc pole moment There are no free conuctng electrons or movable ons n electrc an no charge transfer that leas to the current takes place when we put the electrc nto the electrc fel. Instea of ths we may meet one of two possble stuatons: olar electrcs - molecules of whch (lke water) have permanent pole moments (thanks to the charge separaton). In the absence of the apple fel the poles are ranomly orente. The pole moments (an molecules as well) ten to algn themselves wth an external electrc fel (see lecture 6-5).The egree of algnment epens on the electrc fel strength as well as on the temperature (t ecreases wth ncreasng of temperature). Non polar electrcs - the molecules o not have the permanent pole moments but when place n the external electrc fel they can acqure them by nucton. The electrc fel separates the negatve an postve electrc charges n the atom or molecule nucng the nuce pole moment that s present only when the fel s present. In both cases phenomena that occur when place electrc slab between the plates of capactor are smlar.

11 Lecture 8 - olarsaton p q pole moment N poles n the volume unt N p Dpole moment of the volume unt polarsaton vector n the external fel orentaton of poles lectrc fel nse the electrc weakens nuce surface polarsaton charge, polarzaton vector, nuce pole p nuce pole Surface polarsaton charge In the parallel - plate capactor charge wth a charge q an not connecte to the battery there s unform electrc fel forme between the plates of capactor. When we place a electrc slab between the plates the electrc fel separates postve an negatve charges of the atoms by the very small splacement or just algn the exstng permanent pole moments. The overall effect s to separate the centre of the postve charge of the entre slab from the centre of the negatve charge. Note that the slab as whole s stll neutral, t s just polarze an no charge transfer over the macroscopc stances occur. s a result on the one face of the slab there s a postve charge an negatve charge on the other face. We calle ths nuce surface charge. Ths surface charge sets up the electrc fel that opposes the external fel. The resultant fel n the electrc s the vector sum : Inuce surface charge forme n electrc place n the electrc fel weakens the orgnal fel wthn the electrc. Ths weakng of the electrc fel ue to the ntroucng the electrc reveals n the reucton of potental fference between the plates.

12 Lecture 8 - s s s lectrc fel nse the electrc () Free charge Gauss surface s olarsaton charge free pol free pol ( ) Inuce, surface polarsaton charge free charge, nuce polarsaton charge When the electrc s place between the plates the nuce surface charge q s create on the surface of electrc slab an the net charge nse chosen Gaussan surface s reuce. Inuce surface charge s equal to zero when there s no electrc or s always less then free charge when there s the electrc slab.

13 Lecture 8-3 lectrc fel nse the electrc () s ( ) s Gauss law for electrcs In the general anstropc meum tensor quantty olarsaton charge ( ) p V ( c olarsaton ) p c V Dpole moment ue to the polarsaton charge xx yx zx xy yy zy xz yz zz ( ) electrc susceptblty nuce pole moment, polarsaton vector, electrc susceptblty tensor Note that n the general form of the Gauss law the flux eals wth the nstea of an wth the free charge only. The factor n general may vary, so t s nse the ntegral. rouct s just an nuce pole moment of the electrc slab. The electrcal polarzaton may be efne as an nuce pole moment per unt volume.

14 Lecture 8-4 olarsaton vector susceptblty general ansotropc case - tensor henomena: () xtr. Strong el. Fels, e.g elem wave of the laser beam ( f ) Dsperson of the electrc constant olarsaton vs temperature In the polar electrc non-polar electrc ure law T f (T ) (T ) T /T ure law for electrcs eople: Stanng on the shoulers of gants erre ure Delectrc sperson Due to the structure constrants there s always a tme lag between changes n polarzaton an changes n an electrc fel. The permttvty of the electrc s a functon of frequency of the electrc fel. nalyss of the electrc sperson s very mportant for the applcaton of electrc materals an the analyss of polarzaton systems. Self focusng of the lght beam nonlnear optcs s a result of the change of the electrc permttvty ue to the strong electrc fel of the laser beam the optcal self-acton effects occur. n electromagnetc fel nuces a refractve nex change n the meum through whch the fel propagates. The change n nex then exhbts a back-acton on the fel so as to nfluence ts propagaton characterstcs.

15 Lecture 8-5 Delectrcs electrets: permanent, macroscopc polarsaton ferrolectrcs: olar electrcs wth the oman structure Doman structure ssapears phase transton para-electrcs at the ure temperature In the ferroelectrc phase 5 : (T ) ure Wess law T T n the para-electrc phase () -W constant Bloch s wall tg phenomena: pyroelectrcty T hysteress pezoelectrcty electrostrcton x V V x ferroelectrcs, electrets, pezoelectrcty, pyroelectrcty, electrostrcton Learn more: lectret It s a electrc materal that has a quaspermanent electrc charge or pole polarsaton. n electret generates nternal an external electrc fels, an s the electrostatc equvalent of a permanent magnet. ezoelectrcty It s the ablty of some crystals an certan ceramcs to generate an electrc fel or electrc potental n response to apple mechancal stress. The effect s closely relate to a change of polarzaton ensty wthn the materals volume. lectrostrcton property of all electrcal non-conuctors, or electrcs that causes them to change ther shape uner the applcaton of an electrc fel. yroelectrcty The ablty of certan materals to generate a temporary electrcal potental when they are heate or coole.

16 Lecture 8-6 lectrc vectors D D D lectrc fel nucton (electrcal splacement) D -relateto thefreecharge -the same nse an outse the electrc -relate to the polarsaton charge -exsts exclusvely nse the electrc D -relate to the total charge -smaller nse the electrc than outse electrc nucton, electrcal splacement vector, The electrc fel vector remans funamental quantty n all problems whle nucton D an polarzaton vectors are useful n partcular cases when we eal wth electrcs. The electrc fel vector escrbes the whole exstng charge (t consers both free an nuce charges) an net nteractons n the fel. It has fferent values nse an outse the electrc. The electrc nucton D s referre to the free charge only (ts lnes connects only free charges). It has the same value nse an outse the electrc. The electrc polarzaton escrbes the nuce charge only. It sappears outse the electrc slab. The atonal vectors D an can be easly erve from the electrc fel vector

17 Lecture 8-7 lectrc fel equatons for the electrc D D In ansotropc mea the electrc permttvty s a tensor quantty D D D s free s Gauss law for electrcs ( ) s free ansotropc materals, free charge, The Gauss law for electrcs can be use n ths general form for electrc fel wth a electrc as well as wthout electrc. The electrc susceptblty n general case of ansotropc materals s not gven by a sngle number but has a form of tensor quantty. Therefore n ansotropc mea the nucton, electrc fel an polarzaton vectors are n prncple nonparallel to each other.