Electrostatic properties of conductors and dielectrics


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1 Unit Electostatic popeties of conductos and dielectics. Intoduction. Dielectic beaking. onducto in electostatic equilibium..3 Gound connection.4 Phenomena of electostatic influence. Electostatic shields.5 The capacito. apacitance of a capacito and selfcapacitance.6 toed enegy on a capacito.7 Association of capacitos.8 Dielectics. Electic dipole. Polaization.9 uestions and poblems Objectives To know the featues of chaged conductos in equilibium: electic field inside and nea the suface, electic potential and distibution of electic chages. To know the phenomena of electostatic influence between conductos. Define the capacitance of a capacito and to be able to compute the equivalent capacitance of associations of capacitos in seies and in paallel. Undestand the chaging pocess of a capacito and compute its stoed enegy. To know the effects of inseting a dielectic on a capacito. apacitance, chage, enegy, diffeence of potential and electic field. Pevious unit was devoted to the study of the electic field on vacuum. This unit, being a pat of electostatics, is devoted to the study of electic field on conducto and isolato mateials. Fom now, we ll call dielectic to isolatos (fom geek pefix dia, meaning though, and electic). On this unit, phenomena of electostatic influence on conducto mateials and polaization on dielectics (as a consequence of electic fields), ae descibed. As an application of such phenomena, electic shields and capacitos ae studied. Electic shields ae used to electically isolate diffeent aeas on the space; fo example, to
2 avoid electic intefeences on a laboatoy pefoming accuate measuements, o to potect the signal on wies of an oscilloscope, TV, etc apacitos can be found on almost all boads of pinted cicuits on computes and many othe electonic devices. They ae useful stoing electostatic enegy, as filtes of electical signals, etc. Intoduction Accoding to its ability to conduct the electicity, allowing the motion of its electical chage, mateials can be classified as: onductos (metals, aqueous solutions of acids and bases,...) emiconductos (Ge, i,...) Dielectics (ubbe, glass,...) The diffeent electical behaviou is mainly due to the atomic and molecula stuctue of the matte. Fo example: the coppe (u) as a typical conducto, the gemanium (Ge) as a semiconducto and the ai as a dielectic. The electonic stuctue of coppe (conducto) is: 9 u: s s p 6 3s 3p 6 3d 4s Due to the fact that each atom of coppe has entiely occupied thei thee fist electonic layes, and a lonely electon on last one, when the atoms ae joined to fom a cystal, atomic stability is inceased if the atoms emain with the last complete laye (the thid in this case); so, they let almost fee the electon of the oute laye. These fee electons can easily move though the cystal of coppe when an electic field is acting; so the coppe is a good conducto of the electicity. The electonic configuation of gemanium (semiconducto), instead, is: 3 Ge: s s p 6 3s 3p 6 3d 4s 4p Gemanium has occupied the fist thee layes like the coppe, and fou electons on fouth one. In this case, when foming the cystal, atomic stability inceases when they ae eight electons on oute laye; it can be done if each atom shaes the fou electons of its last laye with its fou neighbos, foming covalent bonds. These electons ae moe tied to thei nucleus and they need a contibution of enegy to get fee electons and conduct the electical cuent. The ai is mainly made up by Nitogen and Oxygen. Even though both elements don t have completed thei oute layes, they ae vey stable, being the ai a good isolato. Anyway, if a stong electic field acts on the ai, may be some electon was pull out (emaining the atom ionized), becoming the ai a conducto. When it happens, the ionization poduces a light emission, and a light can be seen. esides, the motion of fee electons due to the electic field poduces an electic cuent in the ai. This phenomenon occus not only on the ai, but also inside any isolato, and it s known as dielectic beaking. In the
3 dy ai is needed an electic field aound 3 KV/mm (dielectic stength), deceasing with inceasing moistue.. onducto in electostatic equilibium In this unit, when speaking of conductos, we ll only efe to metallic conductos. Model of metal: The positive ions (nucleus tied electons) ae peiodically distibuted foming the cystalline netwok, and the electons of last laye of the atoms (the 4s of the example of u) move feely among the ions, but without going out of the metal; they ae foming a cloud o electonic gas, and these electons ae called fee electons. This mobility of fee electons is what chaacteizes the metals as good conductos. onducto in electostatic equilibium (solid conductos): When on a conducto, both neutal as chaged, thee is no net movement of electic chages, we say that it is in electostatic equilibium. In this case, fulfils: Fee Electons Ions Figue. Model of metal a) Electic Field: If thee is not movement of fee electons, the net foce acting on them must be zeo. As electostatic foces on electons ae much moe intense than gavitational foces, the equilibium on a conducto means that the electic field at any point of the conducto must be zeo, E. V cte ρ Eint Figue. Distibution of chage on a conducto in electostatic equilibium. Electic flux though any Gauss s suface (dashed line) is zeo b) Location of chage: If the electic field is zeo inside the conducto, by applying Gauss s law to any closed suface inside the conducto, the electic flux will be zeo, and theefoe the enclosed chage will also be zeo. o, the volumetic density of chage inside conducto will be zeo (ρ ). If the conducto is chaged, the chage is distibuted acoss its suface. In the same way, if an extenal electic field is applied to conducto, electic chages ae distibuted in such way that the electic field ceated by this distibution of chages, cancel the extenal one. c) Electostatic potential: As electic field is zeo inside a conducto in equilibium, the diffeence of potential between two any points of the conducto will also be zeo, and theefoe the potential will be constant in all conducto. σ 3
4 d) Electical field in an extenal point next to conducto. oulomb s theoem: Electic field on points outside the conducto but vey close to its suface is pependicula to suface of conducto; if it wasn t, the tangent component of electic field to the suface, would move the chages on the suface, not being the conducto in equilibium (but ou hypothesis is that the conducto is in equilibium). To compute the electic field at any point outside and close to the suface of a chaged conducto in electostatic equilibium, we can apply Gauss s law, consideing a suface of a small cylinde shaped with both bases paallel to the suface of conducto (see Figue 3). One of the bases is outside the conducto and the othe base is inside. Thee isn t electic flux though intenal base, because the electic field is zeo inside the conducto. Outside the conducto, the electic field is pependicula to its suface, being zeo the electic flux though the lateal suface of the cylinde. Theefoe, the net flux though the gaussian suface will be the flux though the base of the cylinde extenal to conducto: Φ En d End En d E E n is the module of the electical field nea the suface of the base of cylinde. q Applying now Gauss s law we get: int σ Φ ε ε Equalizing both equations we finally get the electic field on points nea the suface of a chaged conducto σ n E n Equation ε This electic field is pointing outside the conducto if the suface density of chage is positive. This esult is only coect on points vey close to the suface. Fo a conducto without a flat suface, the field lines ae cuve lines, and modulus of electic field falls when we move away fom the suface. ut if the conducto is flat and infinite, with unifom suface E n density of chage σ, field lines d ae staight lines pependicula d to the suface, and modulus of σ electic field is always the same d fo any distance to conducto: d En σ ε E i σ E n Figue 3. Gaussian suface fo the computation of the electic field in the suface of a conducto Figue 4. Gaussian suface fo computation of electic field nea the suface of a flat and infinite conducto 4
5 Hollow conducto in electostatic equilibium The behaviou of a hollow conducto without any chage on the cavity, is exactly the same as if it was a solid conducto, as can be seen on Figue 5. As it s demonstated on next paagaph, thee aen t chages on inne suface of conducto, being all the chage placed on the oute suface. Electic field on cavity is zeo, and electic potential on cavity equals the potential on conducto (constant). E g Figue 5. Hollow and chaged conducto hage in the inne suface of a hollow conducto y applying Gauss s law to the dashed line allows us to deduce that the net chage on inne suface of the conducto is zeo. ut this solution is not incompatible with the idea of a distibution of positive and negative chages on this suface, being zeo its total sum. We ll pove that this solution is not possible, and none chage can be placed on inne suface. E int E g Let s conside equals positive and negative chage in the inne suface, ceating an electical field in the cavity, and a field line going fom the distibution of positive chages until the distibution of negative chages, between points and. The diffeence of potential between and will give us a not zeo value: V V E d Howeve, if we follow fo the calculation a way inside the conducto, whee the electic field is zeo, esult is diffeent: V V E d The diffeence of potential between two points can not be diffeent accoding the path to compute it, and the only possible solution is the no existence of distibution of chages in the inne suface of a hollow conducto chaged o subjected to the action of an applied extenal electic field. E E E E E g g 5
6 Example The Eath is a conducto negatively chaged. If the aveage electic field at its suface equals N/, compute the total electic chage of Eath and its electic potential. olution: The Eath can be taken as an isolated spheical conducto in the space, with appoximated adius of R6.37. m. If we suppose a chage on the Eath, this chage will be distibuted acoss its suface, with a density of chage σ elated to the electic field nea the σ suface: Es σ Esε 8,85 8,85 / m ε The suface of Eath is: 4π R 4π (6,37 6 ) 5, 4 m And the total chage of Eath: σ 8,85 5, 4 4,5 5 To compute the electic potential, evey points of Eath will have equal potential, and so it will be enough computing the potential at only one point. hoosing the cente of the sphee, this point is placed at the same distance R fom any point of its suface. The potential ceated on the cente of the sphee by a little chage q i coesponding to a suface i (taking q i as a point chage), is: qi Vi K R If we conside all the chages q i, placed on all sufaces i, the total potential will be the addition of all the potentials ceated by these chages: q K V i Vi K qi R R i i i K R 9 9 6,37 6 5,5 5 6,4 This potential could be looked as vey high, but we must emembe that the zeo potential has been chosen at an abitay point (the infinite). As we ll see above, fo pactical applications, zeo potential is usually taken on Eath. 8 V Example A spheical conducto, with adius R and chage is joined though a conducto wie (the chage on wie can be neglected), to anothe sphee of adius R (R <R ), initially dischaged. upposing that the sphees ae fa enough so the phenomena of influence ae negligible, compute: a) hages and on each sphee; b) Potential of both sphees; c) su 6
7 face density of chage on each sphee; d) What happen if R >>R? olution a) When joining the two spheical conductos though a conducto wie, the total chage is peseved and distibuted between both sphees in such way that two sphees ae equipotential: V 4πε R 4πε R R R R, R R R b) And the electostatic potential: V V V 4πεR 4πεR 4πε( R R) c) The suface density of chage is: σ R, σ 4πR ( R R) 4πR ( R R) 4πR ( R R) d) If R >> R the conducto of adius R will have the total chage, emaining dischaged the sphee of adius R ; it can be poved fom esults of a): R R lim R lim R R R R R lim lim R R On the othe hand the potential of both conductos will be canceled : limr V limr 4 ( ) πε R R An inteesting consequence of this example () is the case of a conducto simila to that shown on Figue 6, with a shap tip and the othe ounded. We can imagine this conducto as two spheical conductos (adius R and R ) connected though a wie, as on example ; fom point c): R σ R σ This esult means that the tip with highe adius will have lowe suface density of chage; and highe den Figue 6. The suface density of chage depends on the adius of cuvatue of conducto. In the lightning od we get points with high electic field enabling the electical dischage, and avoiding this dischage can occu though non desied aeas. 7
8 sity on tip with lowe adius. As electic field on points nea the suface is popotional to suface density of chage (Eσ/ε ), electic field will be highe nea the tip with lowe adius (R ); in this way, if dielectic beaking occus, it always happen on shap tip. Lightning of a stom goes to a lightning od because it has a shap tip, and a high electic field aound..3 Gound connection The Eath is usually taken as oigin of potentials (electic potential zeo). esides, as the adius of Eath is so big compaed with any othe conducto we can usually handle, even though an amount of electic chage can be taken o given by the Eath, its potential will emain almost constant. This situation can be compaed with the sea level: even if you pou the wate of a glass in the sea, the inceasing on sea level isn t noticeable; in the same way, if you add o emove some quantity of chage fom Eath, its potential doesn t change. o, a conducto linked to gound (gounded) will have two impotant featues: Its electic potential will be always zeo. Though the gound connection some chage can pass fom o to gound, to satisfy the featues of a conducto in electostatic equilibium. Linking an electical facility to gound is a safety measue to avoid electical dischages, since the potential of the fames of all devices linked to gound is zeo..4 Phenomena of electostatic influence. Electostatic shields. Let s suppose an electic chage placed nea a conducto; electic field poduced by such chage causes a movement of chages inside the conducto, by effect of oulomb s foces, poducing a edistibution of chages (the chages of diffeent sign than extenal chage will move towads suface nea the extenal chage, and chages with the same sign will move towads fa away sufaces); so, aeas with negative and positive suface density of chage will appea, but the net chage of conducto will be the same than befoe applying the extenal chage. uch movement of chages finishes when new equilibium conditions ae eached: electic field inside the conducto is zeo (the conditions of electostatic equilibium ae eached on a vey small time, of the ode of 7 s). In this way, a phenomenon of electostatic influence on the conducto has occued, as a consequence of the extenal electic field. E Neutal onducto Figue 7. The extenal electic field poduces on conducto a sepaation of chages, ceating an electic field opposite to the extenal. The phenomenon disappeas when extenal chage is emoved. 8
9 This phenomenon of electostatic influence bing us a way to chage a conducto: let s place two neutal conductos in contact, inside the electic field ceated by a nea electic chage, as it shown in the Figue. When acting the electic field ceated by the chage, chages on conductos ae distibuted until the electic field inside conductos was zeo (a). Keeping the nea electic chage, if we sepaate the two conductos (b), both conductos emain chaged, one positively, and the othe negatively (c). If both conductos wee initially dischaged, both chages (positive and negative) ae equal in magnitude. a) b) c) Figue 8. haging two conductos by electostatic influence. A chaged object causes the conductos in contact take opposite chages. If we sepaate the conductos in pesence of the object, they will emain chaged, On a geneal way, two conductos show electostatic influence when the electic field ceated by one of them electically influences on the othe. Given two conductos as shown on Figue 9, if we daw a tube of cuent fom an element of suface of the conducto, d, to an element of suface d (suface made up by the field lines going out fom the bode of d and aiving to the conducto ), sufaces d and d ae coesponding elements. Let s apply Gauss s law to an enclosed suface made up by the tube of cuent, so that the bases of the tube ae placed inside of two conductos, whee the electic field is zeo. Electic field along the lateal suface of the tube is tangent to this suface; so, the flux of the electic field though this enclosed suface will be zeo: Φ E d E d Lateal suface ases E d Accoding Gauss s law, such flux will be equal to the enclosed chage divided into ε : Φ E d inside inside ε As the chage inside the enclosed suface can only be placed on sufaces d and d, calling dq and dq to the electic chages on such sufaces, must be satisfied: inside dq ❶ d dq dq dq ❷ d Figue 9. oesponding elements in two conductos that exet influence electostática 9
10 This esult is the theoem of coesponding elements: oesponding elements have equal but opposite chages. Not all the d of a conducto have always thei coesponding elements on the othe conducto; in this case we speak of patial influence. Thee will be total electostatic influence when all the suface of a conducto has his coesponding on the othe, o, in a gaphic way, when all the field lines stating on suface of a conducto finish on the othe conducto. The two moe common examples of total influence ae the cases of a conducto entiely suounding the othe, and two flat paallel plates face to face with a little distance between them (Figue ). Two conductos with total electostatic influence between them, necessaily have the same chage but of diffeent sign. This esult is of diect application on capacitos. Electic shields: Faaday s cage. The electic shields ae devices avoiding the phenomena of electostatic influence. The simplest example consists in that we call Faaday s cage: a conducto with an intenal cavity (hollow conducto) linked to gound. Fom the electostatic point of view, such system insulates the outside and the inne cavity of conducto (Figue ): Figue. Examples of total influence. On left, a conducto entiely suounding to anothe. On ight, two paallel plates conductos face to face. E V σ E E V σ ext σ V E Figue. Electic shields. a) On left, shielding of inne aea: extenal chages don t influence the inne aea of the hollow conducto linked to gound. b) On ight, shielding of oute aea: intenal chages on hollow conducto don t influence oute aea of the hollow conducto linked to gound. a) Electic field and potential in the cavity due to the extenal chages ae zeo. The explanation of this phenomenon was aleady given on section
11 when speaking about the location of the chages in a hollow conducto. The behaviou of a hollow conducto without chages on its cavity is the same than a solid conducto and, theefoe, values of electic field inside (zeo) and potential (zeo, due to linkage to gound) also ae they. b) Electic field and potential outside the conducto due to the intenal chages (on cavity) ae zeo. On intenal suface of cavity, due to electostatic influence, it appeas the same chage but of opposite sign, coming fom gound. This distibution of chage will cancel the electic field and the potential inside the conducto. Theefoe, electic field and potential outside the conducto will be zeo. As an example of electic shield, it can be quoted the coaxial wie. A coaxial wie is made up by a conducto, suounded by a dielectic mateial, being this set suounded by anothe conducto (Figue ). In this way, the signal taveling acoss inne conducto emains potected by the shield effect poduced by the extenal conducto. Dielectic onducto V Figue. Diagam and photogaph of a coaxial wie.5 The capacito. apacitance of a capacito and a conducto A capacito is a system of two conductos (plates), isolated one of anothe, exeting total influence between V them. As a consequence of total influence, both conductos have equal and opposite chages. apacitos ae used to stoe electic chage and enegy and they have a lot of applications on electical cicuits. When a diffeence of potential VV V is applied to the plates of a capacito, a movement of chages is poduced fom a conducto to anothe, until the diffeence of Figue 33. apacito potential between both plates equals the applied diffeence of potential. Theefoe, the quantity of chage on the capacito depends on the applied diffeence of potential V, on geomety and size of capacito and on the mateial between plates. V
12 The capacitance of a capacito is defined as the ate between the absolute value of the chage on each plate and the diffeence of potential between them. The value of the capacitance is neithe depending on the chage and on the diffeence of potential; only depends on geomety and size of capacito and on insulating mateial between plates. Equation V V V apacitance of a capacito is measued in.i. in Faads (F). In the pactice a faad is a too big unit, being used thei submultiples as micofaad (µf), nanofaad (nf) and picofaad (pf). Its gaphic epesentation on the cicuits is: Dimensions of the capacitance ae: [] M L T 4 I The definition of capacitance can also be extended to a only conducto. The selfcapacitance of a conducto is defined as the needed chage to incease V the potential of conducto: V It give us idea about the quantity of chage can the conducto stoe at a given potential. The selfcapacitance of a conducto depends only on its geomety and size, being theefoe not depending on the chage o potential of conducto. Paallel plate flat capacito A paallel plate flat capacito has thei plates paallel of suface at a distance d, vey low compaed with. In this situation, we can suppose total influence between plates. We ll conside the vacuum between plates (Figue 4). When it is chaged, both conductos ae in electostatic equilibium, with the chage unifomly distibuted acoss its suface. The suface density of chage will be: σ Given the geomety of the capacito, on the space between plates we ae nea the suface of an infinite flat conducto, being the electic field pependicula to the plates, unifom, and with a magnitude: σ E ε ε o, the diffeence of potential between plates can be computed as: V A V d E d l Ed d ε V d V A Figue 4. Paallel plate capacito
13 And the capacitance of paallel plates flat capacito: V V A ε d ε d d ε Equation 3 The capacitance depends exclusively on geometical paametes. Physical limits fo capacitos come fom the fact that if the distance between plates is vey shot and vey high the stoed chage, the magnitude of electic field in the space between plates can each a high magnitude that ionizes the ai, poducing a spak (and a chage) passing fom a plate to anothe, and so dischaging the capacito. The cylindical capacito A cylindical capacito is made up by two cylindical conductos, coaxial of adii R and R and length L much geate than the space between conductos (L >> R R ). This last condition will allow us to neglect the effects of edge and suppose the capacito like an indefinite system. This type of capacito will help us to undestand the featues of the coaxial wies used in the tansmission of signals of TV, in the oscilloscopes, etc... As thee is total influence between conductos, the will have the same quantity of chage and this will distibute unifomly acoss thei sufaces with suface density of chage σ positive and σ negative: σ And σ πr L πr L The electic field in the space between two conductos can be computed by applying Gauss s law to an enclosed cylindical suface (blue cylinde on pictue) of adius (R <<R ) and height h<l. y symmety of the poblem, the electic field will have adial diection on each point, being pependicula to cylindical sufaces: 3
14 R Φ gaussian Φ ε E d inside h E Lε πh πlε σ ε Ed E Eπh lateal πrh πrl h ε Lε L h R The diffeence of potential between conductos can be computed following the adial path between both sufaces of conductos, along a field line: R R R v R R V V Ed Ed d ln ln R πlε πlε πlε R R R R And capacitance V V πlε R R πlε R R ln ln As it happened on paallel plate capacito: the capacitance only depends on the geomety, on size, and on ε..6 toed enegy on a capacito. To chage a capacito (initially dischaged), a diffeence of potential V can be applied to the plates of capacito, poducing a flow of electons, fom the plate with positive potential to the plate with negative potential (it can be done using a specific device). The chaging pocess pogess fom v v du vdq chage to chage V. At an intemediate state, being q the chage of the capacito, the diffeence of potential v between plates will be q v At this moment, to add an additional chage dq to capacito, we ll need an enegy: du vdq q dq dq Figue 45. When the chage on capacito inceases dq, its enegy inceases du. This enegy inceasing equalizes the aea of the pink tapezium on pictue. The enegy of the chaged capacito is the aea of the tiangle dq dq q 4
15 o, needed enegy to chage the capacito fom dischaged to chage and diffeence of potential V is: U U vdq V q V dq V V Equation 4 This esult can be geometically seen on Figue 5, since equals the aea of gey tiangle. This enegy emains stoed in the capacito on the electic field between plates..7 Association of capacitos It s defined the equivalent capacitance of an association of capacitos as the capacitance of a lonely capacito that having the same diffeence of potential between plates that the association, stoes the same quantity of chage. We ll study the two moe fequent types of association of capacitos: in seies and in paallel. ut we must have in mind that a set of capacitos can t be associated neithe in seies no in paallel, and so moe complex methods must be accomplished in ode to compute the equivalent capacitance. Association in seies The association of capacitos in seies is made connecting a plate of a capacito to a plate of anothe capacito and so on, as it s shown in Figue 6. We suppose the capacitos ae initially dischaged, and to chage them, we apply a diffeence of potential V between the fee plates of fist and last capacito ( and n). One plate will take a chage and the othe plate,. As the chage has been displaced between both plates has been, this is the chage of the set of capacitos. ut due to the total influence between the plates on all capacitos, they will appea chages of opposite sign on plates of all capacitos. As the electical neutality must be peseved on the intenal plates, all the capacitos show the same chage, the same that the set of capacitos. 3 n n n eq Figue 6. Association of n capacitos in seies and its equivalent capacito The diffeence of potential V between teminals of the association can be witten as the addition of the diffeences of potential in each one of the capacitos in seies: V Vn (Vi Vi ) Fo each capacito: And theefoe, n i V i Vi i,,...,n V i n V Vn. i i 5
16 On the equivalent capacito to the association, when applying the same diffeence of potential V, it s veified V, and compaing both expessions: n eq i eq i Equation 5 The equivalent capacitance of a set of capacitos associated in seies is the invese of the addition of the inveses of the capacitances of associated capacitos. o, when n capacitos ae associated in seies, the equivalent capacitance is lowe than the capacitance of each one of the associated capacitos. Association in paallel A set of capacitos ae associated in paallel when all the capacitos ae connected to the same diffeence of potential, as it s shown in Figue 7. When applying a diffeence of potential V A V, the chage is distibuted on plates of all capcitos, being the chage of the set of capacitos, the addition of A n n n ompaing both equations: t t eq Figue 57. Association of capacitos in paallel and equivalent capacito A n eq i i chage on each one: t i Fo each capacito And so: n i i (V A V ) i t ( V A V ) n i In the equivalent capacito, when applying the same diffeence of potential V A V, it will take the same chage: t (V A V ) eq Equation 6 The equivalent capacitance of a set of capacitos associated in paallel equals the addition of the capacitances. o, when n capacitos ae associated in paallel, the equivalent capacitance is geate than the capacitance of each one of the associated capacitos..8 Dielectics. Electic dipole. Polaization Dielectic mateials ae poo conductos of the electic cuent. ompaed with conductos, the dielectics haven t fee chages inside; all the electons ae linked to the atoms o molecules. Although they ae electically neutal, this doesn t mean that dielectics can have local electostatic foces since, at molecula level, the distibution of positive and negative electic chages is not unifom. As an example, on a covalent bond, the moe electonegative atom will attact the shaed electons nea it, emaining negatively chaged, wheeas the less electonegative atoms will emain with positive chage: the conse i 6
17 quence is that the bond will have two centes of chage, one positive and anothe negative with a distance between them, that is, an electic dipole. Electic dipole An electic dipole is a set of two electic chages of the same magnitude, q, but opposite sign, having a distance L between them. It is chaacteized by the dipola electic toque p, (dipola toque) vecto magnitude whose modulus is the poduct of the chage q by the distance between chages L, and pointing to the positive chage. The dipola electic toque is measued in m, but fo molecules (vey low toque), is usually used e nm instead. q q L p ql Electic dipoles ae inteesting due to its pesence in the matte, since a high numbe of molecules ae pola molecules, showing asymmety in its spatial distibution of electic chage, being consideed as electic dipoles. A chaacteistic example is the case of the molecule of wate, with a high dipola toque. p O H H p Figue 68. The asymmety in the spatial distibution of chage ceates a dipola toque in a lot of molecules Figue 79. Dipola toque of the molecule of wate. The dipola toque vecto points to the atoms of hydogen, whee the electic density of chage is lowe Dipola electic toques of some substances in e nm Hydogen chloide, Ethyl alcohol,3 abon monoxide,5 odium chloide, Wate,39 Hydogen Ammonia,3 Methane The impotance of electic dipoles in the matte comes fom the esponse when an extenal electic field is applied to a mateial. As a consequence of this electic field, two paallel foces with opposite sense act on both chages of dipole, giving a zeo esulting foce and a mechanical toque (τ v ) given by the vecto poduct between any vecto joining the two lines of action of the foces and the foce vecto. 7
18 q qe θ L E q qe τ L qe p E The effect of the toque will be, theefoe, to tun the dipole tying to align the dipola electic toque with the applied electic field. Polaization Given a dielectic mateial, when an extenal electic field is applied, the electic field inside the mateial is lowe than the electic field that would be on vacuum. That means, fo example, that a capacito filled with a dielectic between thei plates will have highe capacitance than the same capacito without dielectic. This phenomenon is not justified enough with a model in which the electons ae linked to the atoms, and without fee chages. ut can be undestood if we conside that the electons can move slightly due to oulomb s foces. The consequences of these tips ae the called phenomena of polaization of dielectics, being two models in ode to explain them: a) Dielectics with atoms o molecules in which the distibution centes of positive and negative chages match up on the same point. In this case when applying an electic field, the positive chages ae displaced in the sense of the field, and the negative chages on opposite sense, making up a dipole with a dipola toque p. This type of polaization is called polaization by distotion o induced polaization and it s a vey fast phenomenon. E p E Figue. Polaization by distotion p b) Dielectics with atoms o molecules in which the distibution centes of positive and negative chages don t match up, aleady having small dipoles andomly oiented. In this case when applying an electic field, the dipoles ae pointed with thei dipola toques in the same sense than the applied electic field. It s called polaization by oientation. This phenomenon can be moe o less fast depending on the inteaction of the mateial with the suounding molecules, and stongly depends on tempeatue. 8
19 E Randomly oiented E dp dv Oientation E E ind dp dv Figue. Polaization by oientation oth polaizations, by distotion and by oientation, can happen at the same time in a lot of mateials, and suppose a micoscopic justification of that happening in dielectic mateials when an electic field is applied. Fom a macoscopic point of view, to study the effect of the electic field on dielectics, we can conside the following expeience: Let s take a paallel plates flat capacito connected to a diffeence of potential V (it can be done with a powe supply); the capacito takes a chage. Now, we disconnect the powe supply (the chage on plates must be peseved) and the space between plates is filled up with a dielectic o insulating mateial. Measuing the diffeence of potential between plates, we find now a value V < V. The ate V /V is a chaacteistic constant fo each dielectic mateial, called elative dielectic constant o elative pemittivity of the mateial, ε always being geate o equal than one. ε ε A A A ε V V Figue 8. hanges on V on a capacito when the space between plates is filled up with a dielectic, peseving the chage on plates If V V /ε, then E V/d V /(dε ) E /ε, that is, when enteing the dielectic in the capacito, peseving the chage, both the electic field as the potential diminish in the facto ε. 9
20 On the othe hand, the capacitance initially was /V ; when filling up the capacito with dielectic, inceases up to /V ε If the capacito is filled up of dielectic keeping the powe supply connected, i.e., keeping V constant, then the electic field keeps also constant and, since the capacitance of the capacito with dielectic is geate: ε, then the chage inceases. Without dielectic, the chage was V, and with dielectic will be: V ε V ε. We can edefine the laws we have seen until hee fo dielectics, only substituting the magnitude of the electic field on vacuum by the equivalent fo homogeneous and isotopic dielectics (dividing the electic field in vacuum into the elative dielectic pemittivity of the dielectic), and woking as if the medium was vacuum. On Table magnitudes of elative pemittivity ε fo some usual dielectic mateials ae shown. Wate has a elative pemittivity vey high due to the high pola chaacte of the molecule, and to the ease to its oientation on an electic field. ut it isn t used as dielectic due to the ease to dissolve the salts, becoming conducto with a small quantity of impuities. The esult of the phenomenon of polaization in a homogeneous dielectic mateial due to the application of an electic field is the appeaance of a suface density of chage σ', called bound chage density because it s bonded to the molecules of the dielectic, instead the fee chages that can move inside the dielectic. This bound chage poduces an electic Mateial ε Oil,4 Wate at º 8 Ai,6 akelite 4,9 Mica 5,4 Neopene 6,9 Pape 3,7 Paaffin,3 Plexiglás 3,4 Pocelane 7 Pyex glass 5,6 Table. Relative pemittivity of some mateials field E ind (see Figue 3 ) of opposite sense to the applied electic field, poducing a lowe electic field inside the dielectic, as we ae going to pove:. E E ind σ σ σ σ σ σ Figue 3. The polaization of the dielectic poduces a bound chage as a suface density of chage σ' in the suface of dielectic The electic field due to the suface density of chage on conducto (σ ) is σ (pointing to ight): E ε
21 The electic field due to the bound density of chage on dielectic (σ') is σ' (pointing to left): E ind ε And the esulting electic field, taking in account the effect of polaization σ σ' of dielectic (pointing to ight): E E Eind This effect is the same ε than conside a suface density of chage σ σ on suface of conducto and suppose vacuum on the space between plates. ut we have seen that the effect of dielectic can be also taken in account though the elative dielectic pemittivity: E E σ σ ε ε ε ε whee the paamete ε ε ε is the pemittivity of dielectic mateial. o, equalizing both equations and solving fo σ, we can elate the bound chage density to the fee chage on plates of capacito and the dielectic pemittivity: σ ' σ σ ε σ' σ ( ) ε ε σ σ ε ε ' Equation 7 Note that the bound chage density has opposite sign than the suface density of fee chage. Example 3 etween points A and of the association of capacitos on figue, a diffeence of potential V is applied. The capacito 4 had a capacitance befoe to be filled up with a dielectic of ε 4. Find the capacitance ' of this capacito, the chage and the diffeence of potential on each capacito. olution 4 A () () Finding the equivalent capacitance of capacitos and in paallel by one hand, and 3 and 4 in seies by othe hand, the esulting system is: 4 (3) (4)
22 A, 3,4 The equivalent capacitance is: A eq The total chage of the association when applying the diffeence of potential V is: T V 3,4, 3 4 ;, / V/ The diffeence of potential on teminals of each capacito will be: V V / V / V/ ; V 3 3 / 4 V/4 V 4 Witing these values in a table: V () V/ V/ () V/ V/ (3) V V/4 (4) V V/4
23 apacitance of a flat capacito with seveal layes of dielectic Let s have a paallel plate flat capacito whose plates ae chaged with a suface density ±σ, and it has been filled up with two layes of dielectic, one of them with thickness d and elative dielectic pemittivity ε and the othe with thickness d and elative dielectic pemittivity ε. The capacitance of such capacito comes fom V The diffeence of potential between the plates of capacito can be witten as the addition of the diffeence of potential in the fist dielectic and the diffeence of potential in second dielectic: A σ ε ε σ V A V V E d E d The electic field in each one of the dielectic comes fom: σ E ε ε And the capacitance of capacito will be: E σ ε ε d d V A E d E d dσ ε ε dσ ε ε ε d ε d ε In geneal, fo n layes of dielectics: ε n d i i εi 3
24 Tactile pointe Geoge Gepheide Investigation and cience. eptembe of 998 The device of pointe moe common in the new potable computes is the tactile pad, a black o gey ectangle always placed in font of the keyboad. The tip of a finge above it does that the cuso descibes an analogous movement on the sceen. The tactile pads initiated its histoy not moe than fou yeas ago, but aleady have displaced the mouse of ball integated like standad pointe of the potable computes, that offe them now in moe than the 66%. (The est, in its majoity models of IM and Toshiba, employ the small pointe, simila to a contol of games and that emembes to the ubbe to ease a pencil, installed in the keyboad between the keys "G", "H" and "".) The handling of the tactile pads is much moe convenient fo a lot of people, between them those who suffe of athitis. As it teats of devices entiely hemetical, in its inteio do not penetate the dust neithe the odd substances, what does them moe adapted fo difficult envionments, such like wokshops, factoies and gaages. The type of tactile pad moe extended is the one of capacitance, that woks measuing its vaiations when the finge of the use altes the tiny electic fields existent in the top of the pad. The electodes ae placed in two layes, othogonally placed, and sepaated by a thin plate of fibe of glass, acting as insulato o "dielectic". An electic field acts when an impulse of tension is applied between an uppe electode and anothe infeio, what has as esult that the two electodes, the dielectic mateial inteposed and even the suounding ai wok like a capacito. This electic field is modified in font of the pesence of a finge, distotion geneating a deceasing of the capacitance between the electodes and of the electic chage. IMPULO DE TENIÓN ONTORNO DEL ONTATO DEL DEDO ON LA ALMOHADILLA 5 VOLT VOLT VALOR DE LA APAITANIA MEDIDA EN EL ELETRODO THE DIEUILIRIUM ETWEEN THE TOTAL APAITANE (THE ONE OF THE ORANGE GROUP I GREATER THAT THE ONE OF THE GREEN) MEAN THAT THE FINGER OVER MORE THE GREEN ELETRODE THE ALANE ETWEEN THE VALUE OF OTH GROUP INDIATE A EUIDITANT POITION OF THE FINGER ONERNING THE ELETRODE The location of the finge equies the tip of two goups of impulses of tension. If it was teated to evaluate the capacitance in each one of the points of cossing of the electodes, it would take too time and the eaction of the pointe to the movement of the finge would be lazy. Thus what it s done is to apply two goups of impulses, positive (in oange) and negative (in geen; ight and left diagams) to the electodes, measuing the esulting chage of its capacitance. The situation of the finge elated to the limit between the egions of positive and negative impulses is detemined by means of calculations ealised with the total loads measues. oth goups of impulses must to displace accoding the finge is moved, so that its bode keeps nea the cente of the finge. The pictue does not show moe than the coesponding impulses to the goup of the paallel and vetical electodes; the same must be done with the set of tansvesal electodes. In this way can be followed the twodimensional movement of the finge, until speeds aound of centimetes by second. 4
25 .9 uestions and poblems onductos. Which is the diection of the field lines on points close to a chaged conducto in equilibium? Why? ol: Pependicula to the suface of conducto, because if not, the chages on the suface of conducto would be moving, and the conducto wouldn t be in equilibium.. The pictue shows a hollow conducto connected to gound, with a chage q in its cavity. Thee is a chaged sphee with chage outside to conducto. Which is the effect of the chage q in the distibution of chage in the suface of the sphee of adius R? Justify the answe. ol: None, because the hollow conducto connected to gound acts as a Faaday s cage. q R apacitos and dielectics 3. Given thee equal capacitos with capacitance, compute the equivalent capacitance of the system in each case a ) b ) c ) ol: a) /3 b) 3 c) /3 4. Two capacitos with capacitances,4 and 3, µf ae connected in seies and the set is connected to a battey of 6, V. a) Which is the equivalent capacitance of the set? b) Which is the chage of each capacito? c) Which is the diffeence of potential between the plates of each capacito? ol: a),35 µf b),4 3, 8,5 µ c) V,4 3,44 V V 3,,66 V 5. Given the two flat capacitos on pictue: a) isolated with chage ; b) connected to a powe supply of diffeence of potential V. If we move apat the plates of both capacitos, say how the stoed enegy changes on each capacito (inceases, deceases, o emains constant.) ol: a) Inceases b) Deceases a) b) V 5
26 6. A capacito of capacitance, with chage, is connected to anothe, with capacitance, initially dischaged, such as it appeas on pictue. ompute the chage on each capacito befoe and afte closing the switch. ol: efoe Afte /( ) /( ) b 7. A plate of coppe of thickness b is placed inside a capacito of suface, such as shown in the figue. Which is the capacitance of the capacito befoe and afte to place the coppe plate? ol: befoe ε /d aftewads ε /(db) d V 8. Two capacitos with capacitances and ae connected in paallel, and a diffeence of potential V is applied. ompute the chage taken by each capacito ( and ) as well as the diffeence of potential between the plates of each one. (V and V ). ol: V V V V V 9. In the association of capacitos on pictue, when a diffeence of potential is applied between A and, answe which capacito stoes: a) the geate chage b) the lowe chage, A 3 ( ; /3; 3 (/3)). ol: a) b) onductos. Let s have a spheical conducto, centeed in O and with adius R. uch sphee is connected to gound (potential zeo), and it s influenced by a point chage q, placed at a distance d fom O (d>r). ompute the chage on the sphee as a function of q, R and d. O R d q 6
27 ol: q R d d. Given the system of the figue, compute the total chage of the sphee. q ol: q R d q q R R. The figue shows a hollow metallic sphee of inne and oute adii R and R. The sphee is connected to gound with a positive point chage,, in its cente. a) Which is the distibution of chages on inne and oute sufaces of the sphee? b) Obtain the expessions fo V() fo R, R R, R. R R ol: a) on R, ; on R, zeo; b) R, V ; (R R ; R ) V 4πε R apacitos and dielectics 3. Two paallel plate flat capacitos and of equal capacitance ae connected in paallel to a d.d.p. V. Afte disconnect the association fom the powe supply, the distance between plates of capacito is deceased to a half of the initial distance. Which will be the chage on each capacito? 4 ol: V; V A diffeence of potential V is applied to a capacito with capacitance (). Two equal capacitos in seies ae connected in paallel with the fist one (); these new capacitos ae initially dischaged. ompute the chage taken by each capacito,,, and 3. ol: V ; 3 V 3 3 A () A () V 3 5. Two metallic paallel plates ae sepaated a distance d on vacuum. A d.d.p. V is applied and then the powe supply emoved. A laye of glass (dielectic) of thickness d/ and elative pemittivity ε is placed between the two plates. Which is the new value of the d.d.p. between plates? Which must be the sepaation between the plates to get the d.d.p. was the same that in the beginning? V ol: V ; d d ε ε 6. Two metallic paallel plates with suface ae sepaated a distance d on vacuum. A d.d.p. V is applied and then the powe supply emoved. A laye of 7
28 dielectic of thickness b (b<d) and elative pemittivity ε is placed between the two plates. ompute: a) capacitance befoe enteing the dielectic. b) fee load on plates. c) electic field in the space between plates whee thee is vacuum. d) electic field in the space between plates whee thee is dielectic. e) d.d.p. between plates afte enteing the dielectic. f) apacitance with the dielectic. V V V b ol: to) ε b) ε V c) E d) E and) V d b d d d d ε d ε ε f) b d b ε 7. The pictue shows a set of equal capacitos (capacitance ) connected to a d.d.p. V V VV V. a) ompute the stoed enegy on capacito. apacito is filled up with a dielectic of 3 elative pemittivity ε b) ompute the total enegy stoed on all capacitos. c) Which is the facto should we multiply by the distance between plates of capacito 3 in ode the equivalent capacitance wouldn t be modified? ol: a) W V 8 ; b) W T ( ε ) V ; c) x ( ε ) ε 8
29 GLOARY Theoem of oulomb: The electic field nea the suface of a chaged conducto in electostatic equilibium is pependicula to its suface, pointing to outside if the chage is positive and opposite if it s negative, and with magnitude: E n Gound: point of linking assuing the electic potential is zeo. Electostatic shield: Device dividing the space in two egions electostatically independent. apacito: system of two conductos, isolated one of anothe, exeting total electostatic influence between them and stoing electic chage. apacitance of a capacito: is the ate between the absolute value of chage on each plate and the diffeence of potential between both plates. Equivalent capacitance of an association of capacitos: is the capacitance that would have a lonely capacito so that when applying it the same diffeence of potential that to the association, it would take the same quantity of chage. Electic dipole: set made up by two electic chages of the same value but opposite sign, sepaated a distance. Polaization: Appaition o oientation of electic dipoles in the matte as a consequence of an extenal electic field. Dielectic constant: Facto of deceasing of the electic field inside a dielectic as a consequence of polaization. ound chage: Induced electic chage in the suface of a dielectic as a consequence of the polaization of dielectic. σ ε 9
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