Unit 1 Electosttics of point chges 1.1 Intoduction 1. Electic chge 1.3 Electosttic foces. Coulomb s lw 1.4 Electic field. Field lines 1.5 Flux of the electic field. Guss s lw 1.6 Wok of the foces of electic field 1.7 Electosttic potentil enegy. Electic potentil. Equipotentils sufces 1.8 Applictions 1.9 Poblems Objectives Know how clculte electic field nd electic potentil ceted by point chges nd symmeticl distibutions of chge. Get the concepts of electosttic potentil enegy nd electic potentil. 1.1 Intoduction The electomgnetic intection, the electomgnetism, is pesent in wide nge of phenomen of distinct fields tht cove fom electonic intections in the toms nd molecules, tht is to sy of the sme constitution of the mtte, to othes in which is founded the min pt of the cuent technology: lighting, engines, bodcsting, compute science, etc... In the pst, electomgnetism hs been split in two diffeent pts, ssocited to thei electic nd mgnetic effects. In fct, Hns Chistin Oested, one of the discovees of the eltions between both effects, ws pofesso of Electicity, Glvnism nd Mgnetism, nd until the discovey of this eltion, in the second hlf of the 19th centuy, such effects wee studied in septe wy. Nowdys, we know tht both phenomen e consequence of the sme fetue of the mtte, tht we cll electic chge. 1-1
This unit ims to study electic chges t est, the pt of the electomgnetism tht we cll electosttics. On futhe units will be studied the mgnetic effects of moving electic chges. 1. Electic chge We will begin this unit nlyzing simple expeience in which e put of self-evident some chcteistics of the physicl quntity electic chge. Let s conside bll of plstic hnged with n insulting thed nd ubbed with skin; now, we ub with skin nothe piece of plstic b shped. If we ppoch the b to the bll, bll nd b e ejected. But if we chnge the bll of plstic by nothe bll of glss nd we ub it with silk, when ppoching the b of plstic ubbed with skin, the bll nd the b e ttcted. Plstic bll ubbed with skin - - - - - - Plstic od ubbed with skin Glss bll ubbed with silk Ámb ελεχτρον Figue 1-1. Electic foces ttcting nd ejecting In the pst, Geek people did simil expeiences but with pieces of mbe, skins nd fbics insted of plstic. Fom hee tht the oigin of wod electicity is the wod mbe witten in Geek lnguge ελεχτρον (electon). Justifiction: Atomic model. Negtive nd positive chges The explntion of these foces o intections ppeed in the shown systems is the existence of fetue of the mtte clled electic chge. Thee is enough to tke in ccount two types of chge, we cll positive nd negtive chges, to undestnd the phenomen of ttction nd epulsion shown in Figue 1-1. We know tody tht the mtte is mde up by toms. The toms e mde up by nucleus, wht contins potons with positive chge nd neutons without electic chge. This nucleus is suounded by distibution of electons with negtive chge. The quntity of chge of n electon nd poton is the sme, but of diffeent type. Between chges of the sme type they ppe epulsive foces nd between chges of diffeent type, ttctive foces. The fetue o chcteistic of the mtte tht is in electons nd potons is clled electic chge, Q, nd it s bsic popety of the mtte, in the sme wy thn it s the mss. A body is positively chged if it hs lowe numbe of electons thn of potons, nd negtively chged if the numbe of electons is highe thn potons. At fist, the toms hve so mny potons like electons (e neutl), hving the sme quntity of chge of both types, nd being zeo thei net chge (Figue 1-). 1-
Mtte is mde up by toms, with nucleus (potons nd neutons) nd electons ound q () q (-) Ion Ion - Figue 1-. Atomic model nd positively nd negtively chged ions When plstic is ubbed with skin, electons e tnsfeed fom skin to plstic, emining the plstic with n excess of electons (negtively chged), nd the skin with lck of electons (positively chged). Whees when glss is ubbed with silk, some electons pss fom glss to silk, emining the glss positively chged nd the silk negtively chged. In these chge tnsfes fom body to nothe, the electons e lwys those tht tnsfe fom body to nothe, neve the potons, since they e into the nucleus nd it s vey difficult to move them out. An tom with lck of electons is positive ion, nd with excess of electons, negtive ion. The minimum electic chge cn hve body is the chge of n electon (in modulus), clled elementy chge, nd it s witten by e. Popeties: quntiztion nd consevtion The electic chge of body is lwys whole multiple of the elementy chge e, tht is the chge is quntized, not being possible to obtin smlle pts of this quntity. On the othe hnd, the chge don t cete neithe destoy; it cn flux, it cn chnge its position, but cn t disppe. This fct is known s the pinciple of consevtion of the electic chge: the totl chge in n isolted system emins constnt. The unit of electic chge in.i. is the coulomb (C), whose definition is elted to the mpee, bse unit of the bse quntity intensity of electic cuent. A coulomb is the quntity of chge fluxing long 1 second though coss section of conducto cying n intensity of cuent of 1 mpee: 1 C 1 A*1 s, (intensity of cuent will be defined in unit 3, nd mpee in unit 6). Eqution of dimensions of electic chge is: [Q] I T The elementy chge is: e 1,6 10-19 C 1-3
1.3 Electosttic foces. Coulomb s lw The ttctive nd epulsive foces dues to electic chges between chged bodies e genelly much highe thn the ttctive gvittionl foces between them. The fist one in mesuing electosttic foces ws Coulomb (1785), using tosion blnce. Two electic point chges q 1 nd q, t est, septed distnce in the vcuum, exet between themself foce whose mgnitude is popotionl to the poduct of the chges nd invesely elted to the sque of the distnce. The diection is the line joining both chges, being the foce epulsive if they e of the sme sign nd ttctive if they hve opposite sign. F 1 q q 1 u 1 1 F 1 q 1 1 q F 1 F 1 Figue 1-3. Foces between chges of the sme sign nd between chges of diffeent sign F q1.q k u 1 1 Eqution 1-1 1 u 1 1 In this eqution, F 1 is the foce tht the chge q 1 exets on q, F 1 is the foce tht the chge q exets on q 1, 1 is the vecto going q 1 fom to q, nd its unit 1 vecto. Constnt k on vcuum is k 8,99 10 9 9 10 9 N m /C, lthough it s moe usul to find it s function of nothe constn ε 0, k 1/4πε 0 whee ε 0 8,85 10-1 C /N m is the pemittivity o dielectic constnt of the vcuum. Execise: Two positive point chges of 1 µc e plced on the OX xis of ctesin coodintes system, t distnce 1 cm. Compute the foces ppe between them. Pinciple of supeposition of electosttic foces The net foce poduced by sevel chges t the sme time on nothe, is the vecto sum of the foces tht would ppe if they cted septely. This fct is clled pinciple of supeposition of foces, nd cn be lso pplied to othe cses of foces supeposition. 1-4
Theefoe, if it hs distibution of chges q i cting on chge q, the net foce cting on q is the vecto sum of the foces tht exets ech one of them on q. q 3 3 F q 1 q 1 q F 3 F 1 F 1 q qi F Fi u 4πε i 0 i i Figue 1-4. Pinciple of supeposition Eqution 1-1.4 Electic field Given n electic chge q, the spce tht suounds it is modified by its pesence, nd it s sid tht in such spce thee is n electic field. If we plce second chge q 0 on point of such field, the electic field in such point is defined s the foce tht would ct on q 0 divided into the chge q 0. Tht is, the electic field E on point of the spce is the electic foce tht would ct on the unit of positive chge plced on such point. E F 1 q u Eqution 1-3 q0 4πε0 The unit of electic field is the N/C, lthough we will see tht N/C is equivlent to V/m (volt/mete). Dimensions of electic field e [E] M L T -3 I -1 Electic field hs been descibed like n effect poduced by the electic chges t est, but futhe of context puely electosttic, electic field is quntity pesent in mny othe systems o entities moe complex, like electic cuents, stomy clouds, molecules, electomgnetic wves, etc., whee it lso exists n effect quntified by this quntity. Electic fields in the ntue (estimtion) E (N/C) In the domestic wies 10 - In the wves of dio 10-1 In the tmosphee 10 In the sunlight 10 3 Unde stomy cloud 10 4 In tube of X-ys 10 6 In the tom of hydogen 6 10 11 1-5
Electic field ceted by n point chges In simil wy to the supeposition of electosttic foces, net electic field ceted by mny point chges on point is equl to the vecto sum of the electic fields ceted by ech one of the chges in such point. q 1 q q 3 1 Exmple 1-1 3 E E 3 E 1 E E E i i 1 4πε Figue 1-5. Pinciple of supeposition Eqution 1-4 Given the point chges of the figue, compute: ) The net electic field on point A(,0) m. Apply the pinciple of supeposition, dwing in the cht the fields exeted by ech chge septely. b) The foce would ct on negtive point chge of -3 nc plced on A. olution 0 q i i q - µc q 1 1 µc ) Electic field due to point chge q plced on point given by the vecto q is E k u, being u the unit vecto of : q 1 q 1 9000 N E1 k u k i i 1 1 1 4 C q 18000 i j 3600 N E k u ( i j ) 5 5 5 C Adding both vectos, E 970i 1610 j N q - µc E E E 1 E u i A q 1 1 µc b) If chge of -3 nc is plced on A, foce cting on chge is: F qe 3( 970i 1610 j ) 910i 4830 j nn Obseve tht being the chge negtive, vecto foce is opposite to the field. A E 1 1-6
Field Lines Lines being tngents to electic field vecto on ech point on the spce e clled electic field lines. Electic field cn theefoe be epesented by mens of these lines tht show the diection of electic field on ny point. The electic field ceted by point chge is centl field (field vecto hs lwys dil diection), being its field lines stight lines cossing in the point whee the chge is plced. Field lines dues to sevel point chges e cuve lines, whose shpe depends on the vlues of the chges nd of thei positions. To) b) c) Figue 1-6. ) Electic field lines ne positive point chge. b) Electic field lines ne two equl positive point chges. c) Electic field lines ne positive point chge q nd negtive q 1.5 Flux of the electic field. Guss s lw Given n enclosed sufce, on ny point of such sufce cn be defined the sufce vecto. To do it, we must tke little sufce ound such point (d). ufce vecto is vecto pependicul to the sufce, pointing outside of the volume enclosed by the sufce, nd mgnitude the tken sufce, d. In ode this definition ws consistent, the tken element of sufce must be smll nd flt sufce, nd thus, sufce with ny shpe must be split in smll infinitesiml sufces. Elementy flux (dφ) of n electic field though n elementy sufce d is defined s the inne poduct of vectos E nd d : d Φ E d If we extend computtion of elementy flux to the whole enclosed sufce, we ll hve the flux of the electic field though sufce : Φ Dimensions of electic flux e 3 1 [ ] [ ][ ] 3 φ ML T I dφ E d d E being mesued in V m o N m /C 1-7
Guss s lw sys tht: Q Φ closedsuf Q enclosed ε 0 Q enc s E d ε 0 Q ced The flux of the electic field though n enclosed sufce is equl to the net chge enclosed inside divided into ε 0. It is impotnt to undeline tht the flux though the enclosed sufce only depends on the chges inside the sufce, does not depend on chges outside the sufce. Figue 1-7. An enclosed sufce enclosing n electic dipole is cossed by net flux zeo. Gphiclly, the numbe of lines going out of the sufce is the sme tht those going in. Figue 1-8. An enclosed sufce enclosing system of two chges q nd q is cossed by negtive net flux. Gphiclly, the numbe of lines going out of the sufce is lowe thn those going in. Guss s lw is useful to find the electic field of some distibutions of chge tht, in genel, pesent specil symmety in the distibution of the chge (sphees nd infinite cylindes unifomly chged, infinite chged plnes, etc...). In these cses is esie to find the flux nd solve the electic field tht find it diectly fom Coulomb s lw. To do it, it is bsic to look fo n imginy sufce in such wy tht the electic field ws pllel o pependicul to the sufce vecto on ech point, nd lso tht the mgnitude of the electic field ws constnt on ll the points of such imginy sufce. If we do this, computtion of flux cn be esily done, since - If E nd d e pependicul on ll the points of Φ dφ E d 0 1-8
- If E nd d e pllel in ll the points of (emembe mgnitude of E is constnt) Φ dφ E d Ed E d E Guss s lw. Poof To poof Guss s lw, suitble mgnitude fo mesuing ngles in thee dimensions must be defined; it s the solid ngle. When we obseve the Moon, fo exmple, ou sight hs to cove thee-dimensionl ngle; we cll this ngle solid ngle, to distinguish it of the flt ngle. In the sme wy tht the flt ngle is the te between the length of n ch nd its dius, nd it s mesued in dins, dl dα olid ngle is the te between piece of spheicl sufce nd its squed dius, d d Ω It s mesued in esteeodins. As well s the whole flt ngle of cicumfeence is π dins, the whole solid ngle of sphee is 4π esteeodins. In the figue, thee is positive point de chge inside of n enclosed sufce, nd elementy sufce d tht musn t be pependicul o pllel to the electic field. The electic flux cossing this sufce is: d Φ E d Ed cosα Ed n being d n the component of the electic field. d pllel to q q dφ Edn dω dω 4πε0 4πε0 The flux cossing finite sufce will be, then: q q Φ Ω Ω πε d 4 0 4πε0 Obseve tht tem is cncelled, nd so the flux does not depend on the distnce!, tht is to sy, the flux doesn t depend on the size of the sufce. This fct is elted with the fct tht the field is dil nd diminish with the sque of the distnce (clled gussin field) q dα dω d n dl α d 1-9
As the chge is inside sufce nd enclosed by it, solid ngle must be extended to the whole sufce, being 4π. ubstituting, we obtin: q q Φ 4π 4πε0 ε0 Tht is Guss s lw. If the chge hd been outside of de the sufce, this sufce cn be split in two es: e 1 whee the flux is positive (going out), nd nothe e, whee the flux is negtive (going in). d d n Net flux will be obtined dding both 1 fluxes: q q dω Φ Edn Edn Ω Ω 0 q 4πε0 4πε 1 0 Ω Net flux is zeo becuse (when chge emins outside), both fluxes hve opposite sign nd they e cncelled. If the chge hd been inside, only thee would be going out flux. If insted of point chge we hd distibuted chge, we would get to the sme conclusion, tking in ccount the pticles mke up the distibution nd pplying the pinciple of supeposition. Applictions of Guss s lw ome cses e going to be consideed following tht, due to its symmety, they enble using Guss s lw to compute electic field. ) Electic field ceted by chged infinite plne with sufce density of chge σ σ E E Figue 1-9. Chged infinite plne Let s hve n infinite plne chged with sufce density of chge σ coss the plne. Let s tke closed sufce (we ll cll gussin sufce), nd we e going to pply Guss s lw to this sufce. Due to symmety of this poblem, lines of E will be pependicul to the plne, nd being positive chges, outside of the plne. Fo being n infinite plne, the field lines will be ll pllel. These fetues cy us to choose cylindicl sufce with its pllel bsis s e shown in Figue 1-9. In this wy, field lines e pependicul to the bsis nd tngent to the side sufce. Applying Guss s lw to this cylinde: Φ net Q E d ε closed int 0 E E σ ε 0 E σ ε 0 1-10
being the e of the bse of the cylinde. Note tht electic field is not depending on the distnce to the plne. It only depends on σ. b) Electic field ceted by chged infinite line with line density of chge λ Let s hve chged infinite line d E with line density of chge λ. Due to the symmety of poblem, electic field will λ hve dil diection ound the chged line. Theefoe, let s tke s gussin sufce cylinde of dius coxil with the line distibution of chge. In this wy, field lines will coss pependicully the side sufce of the cylinde, nd will be tngent to the two bsis of the cylinde. L o: Φ E d Ed E being mgnitude of electic field constnt long the side sufce of the cylinde. In this wy, the flux is equl to the poduct of the mgnitude of electic field multiplied by the side sufce of length L of the cylinde: E ide E πl. This is only tue if the line distibution is supposed to be infinite; on the othe wy it would be necessy to conside the edge effect, being moe difficult the computtion of electic flux. Applying Guss s lw: Qenclosed λl Φ E πl ε0 ε0 λ olving fo E: E πε 0 c) Electic field ceted by spheicl skin with chge density σ Figue 1-10. Gussin cylindicl sufce of dius coxil with line distibution of chge E d ext σ R int E 0 Figue 1-11. Chged spheicl skin E ide ide Let s suppose sufce chge density σ on the sufce of sphee of dius R. We e going to compute the electic field in two diffeent es: inside nd outside of the skin: ) Inside. We conside spheicl sufce, int with dius <R. By Guss s lw, the flux tht cosses this sufce is zeo (thee isn t chge inside this sphee). o, s the e of the sufce is not zeo, it will be zeo the electic field on ny point belonging to the sufce. 1-11
b) Outside. Due to symmety of poblem, the electic field will hve dil diection ound the skin. If we tke gussin spheicl sufce ext of dius >R, the flux though this sufce is: Φ ext E d ext Ed E The simple esolution of this integl is possible becuse electic field is lwys pllel to the sufce on ny point of the spheicl sufce ( E d Ed ), nd lso to be constnt the mgnitude of E in ll the points of the sufce. In this wy, the flux is lso vey simple to compute, being E ext E 4π. On the othe hnd, pplying Guss s lw: Q σ 4π Φ enced R ε0 ε0 olving E fom these equtions: σr E ε As the net chge on the sufce is Q σ4πr, the electic field outside will be: Q R 4 Q E πr ε0 4πε0 Tht it is equivlent to suppose tht ll the chge of the spheicl skin is concentted in its cente. 1.6 Wok of the foces of the electic field. We conside the electic field ceted by point chge Q. In ny point P given its position vecto Q, the electic field ceted by Q will be E u 4πε 0, being u unit vecto of. Now, we plce one second point chge q in P, nd we pply smll tip dl in ny diection fom the q chge. The wok done by the foce of the electic field to move the chge dl will come fom dw F dl This tip dl cn be split in pllel component to the electic field nd nothe tngent component to n ch of cicumfeence centeed in Q nd pssing though P, in such wy tht dl du dtu As E is lwys pependicul to this ch of cicumfeence, u u 0, nd: t t Q u t 0 4 0 A dtu t Q qqd dw F dl q u ( du dtu ) 4πε πε 0 ext dl du P q E B FqE 1-1
To compute the wok done by electic foce cting on q to cy it fom point A to point B long ny L line, s is shown in the pictue, we will hve to dd ll the woks done by the foce in smll tips long L fom A to B. In this wy, the net wok will be the ddition of infinite tems, tuning this sum into n integl, clled integl of line: W L AB B B B qq qq qq Fdl d 4πε πε A A 0 4πε0 4πε 0 A 4 A qq 0 B B B Q dl q L A E A W L AB qq 1 1 ( ) Eqution 1-5 4πε 0 A B If both chges (q nd Q) e of the sme sign, the foce between them is epulsive; if, besides, point B is futhest thn A fom Q ( B > A ), then the computed wok WAB is positive; positive wok sys tht the wok is done by the foces L of the electic field in spontneous wy. But if the chges hd opposite sign, L o the finl point B ws close to Q thn A ( B < A ), then W AB would be negtive, being necessy n extenl foce tht, winning the foce of the electic field, does such wok. A negtive wok sys tht the wok is done ginst the foces of the electic field. Obviously, the wok done by the field between two points plced to the sme distnce of Q is zeo. 1.7 Electosttic potentil enegy. Electic potentil. Equipotentil sufces. It esults vey inteesting to check tht the wok computed only depends on the chges q nd Q nd on the distnces of the initil nd finl points ( A nd B ) to the chge ceting the electic field. This mens tht, lthough we hd chosen nothe diffeent wy L' to go fom A to B, the wok done by the foces of electic field hd been the sme; o tht if we hd moved the chge q in opposite sense, fom B to A, the wok would hve been the sme but with opposite sign; o tht if we hd moved the chge q long closed line, stting on point nd finishing in the sme point, the net wok would hve been zeo. L AB L' AB W W W AB W W 0 AB W BA AA Fields hving this fetue (fo exmple, gvittionl field), e clled consevtive fields o fields deiving fom potentil; the nme of consevtive field is elted to the bility of field to conseve (keep) the wok, to give bck it when we move us in opposite sense; they do it, fo exmple, gvittionl field o sping. Fiction foce between two sufces, insted, is not consevtive, since the wok lwys must be done by someone extenl to the field. And the nme of fields deiving fom potentil comes fom the considetion of function, clled potentil enegy, indicting us the bility to do some wok. In this type of fields, s it is known fom pevious couses, the wok done by the 1-13
field to move pticle between two points cn be expessed s the diffeence of potentil enegies between both points. In the cse of the electic field, the wok to cy chge fom A until B cn be witten s the diffeence of potentil enegy of the q chge between points A nd B: WAB UA UB. Function U is clled electosttic potentil enegy of chge q in point of the field ceted by the chge Q. But peviously we hve seen tht the wok done by n electic field ws qq qq WAB 4πε0A 4πε0B Ans so, function qq U C 4πε0 give us the electosttic potentil enegy on point plced t distnce fom the chge ceting the electic field. The constnt C indictes us tht infinite functions diffeing in constnt cn be tken s potentil enegy on point. This fct llows us to tke the oigin of enegies in n bity wy. Usully, chge plced vey f fom the ceting electic field chge, is tken s hving electosttic potentil enegy zeo (if, then U0), nd so C0, esulting U qq 4πε 0 Eqution 1-6 It is convenient to notice tht the electosttic potentil enegy of chge on point give us the wok done by the foces of the field to cy this chge fom this point until the infinite, since: W Fdl qedl qed q Q qq qq d 4πε 4πε πε P P P P 0 0 4 0 If electic field ws poduced by set of point chges, the computtion of the wok nd of the electosttic potentil enegy could be done by pplying the pinciple of supeposition. Electic potentil The electosttic potentil enegy, s we hve seen, depends both on chge o chges ceting the field, s on the chge inside the field nd its position; it become useful to get function tht only depends on the electic field, but not on the chge plced on it. Theefoe, electosttic potentil on point of n electic field (V) is defined s the electosttic potentil enegy tht would hve positive chge of 1 C plced in such point; tht is to sy, it is the electosttic potentil enegy by unit of electic chge: V U q Q 4πε 0 In this expession we hve supposed the oigin of potentils in the infinite. 1-14
In cse tht we hd sevel point chges, the potentil on ny point could be obtined by the ddition of the potentils ceted by ech chge septely (pinciple of supeposition), being the potentil due to n point chges on point: 1 Qi V Eqution 1-7 4πε being i the distnce fom the chge Q i to the poblem point. 0 i i On some pticul cses is possible tht we wnt to compute electic potentil fom the electic field on point P, nd not fom chges ceting the field; fo this pupose we cn compute the needed wok to cy 1 C fom point P to infinite long ny line: V P P Edl And the diffeence of potentil (d.d.p.) between two points A nd B of the field: V V A V B B A Edl Eqution 1-8 In both cses, obviously, since we cn integte long ny line, the simplest wy to do it will be to choose line of field, since the inne poduct tuns into poduct of mgnitudes: dimensions of the electic potentil e: 3 1 [ ] [ ][ ] E ML T I E dl Edl V l V is mesued in Volts Fom the potentil on point o fom the d.d.p. between two points of n electic field, is immedite to compute s the potentil enegy of q chge on point P s the needed wok to cy q chge fom point A to nothe B: U U U U q V V ) P qv P A B ( A B Equipotentil sufces Electosttic potentil is scl quntity fetuing the enegy level of point on the spce, in the sme wy tht the height of point fetues the enegy level of this point inside the gvittionl field, o the tempetue fetues the het level. Obseve tht it s sid potentil of point, not of chge, becuse chge tkes enegy when it s plced on point with some electic potentil. In this wy, s we sy tht kilogm of mss t 8000 m height hs moe enegy thn the sme mss t 10 m, we ll sy tht chge of 1 C hs moe enegy if it s plced on point with highe electosttic potentil tht nothe. 1-15
In the sme wy thn in the height (level cuves) o the pessue (isobs), level sufces o equipotentil sufces cn be built s the set of points in the spce with the sme electosttic potentil. 15 V 10 V 5 V V -15 V -10 V -5 V - V Figue 1-1. Equipotentil sufces (cuves in the plne) in the vicinities of distnt point chges Figue 1-13. Equipotentil sufces (cuves in the plne) in the vicinities of system of chges q nd 3q Obviously, the wok to move chge between ny two points of equipotentil sufce is zeo (the potentil is the sme on ll the points of the sufce), nd so the electic field must be pependicul to such sufce on ny point of the sufce. If it wsn t, we could lwys find two points tht cying chge fom one to nothe, the wok done by the field wsn t zeo, nd so both points wouldn t hve the sme potentil, not belonging to the sme equipotentil sufce. o, on ny point of the field, field lines nd equipotentil sufces e pependicul. Besides, if we let positive chge in the field, it would move spontneously in the sme sense thn the electic field (positive wok), decesing potentil enegy of the chge. o, the electic field sys us the sense in which the electosttic potentil deceses. 1-16
The eltion between electic field nd electic potentil cn be seen in Figue 1.-14, whee equipotentil sufces nd electic field lines supeimposed e shown fo the cse of n electic dipole (two equl chges of opposite sign t distnce). Obseve tht es whee equipotentil sufces e close, es whee electic field is highe. Figue 1-14. Field lines (ows) nd equipotentil sufces (closed lines e thei cossing with the plne) in system of two equl nd opposite sign chges. Field lines nd equipotentil sufces e pependicul on ech point. Exmple 1- Given the point chges of Exmple 1-1, compute: ) Electic potentil on point A(,0) m nd on point B(4,) m b) The diffeence of potentil between points A nd B, V A V B. olution ) Electic potentil due to point chge q on point t distnce fom q the chge is V k. Theefoe, substituting nd pplying the pinciple of supeposition: q1 q 1 V A k k 9000 V -3550 V 1A A 5 q1 q 1 V B k k 9000 V -353 V 1B B 0 17 c) The diffeence of potentil between points A nd B is: V A V B -1197 V 1-17
Exmple 1-3 Given the point chges of Exmple 1-, compute the wok must be done by n extenl foce to cy negtive point chge of -3 nc fom A to B. olution The wok done by the foces of the field is: W AB -q(v B - V A ) And so, extenl foces do wok -W AB W AB Fext q(v B - V A ) -3 (1197) -3,59 µj 1-18
1.8 Applictions The oscilloscope of cthode y tube When diecting electon bem between two conductive pltes hving diffeence of potentil, nd theefoe n electic field, between them, the electons divet depending of its initil speed nd the mgnitude of the electic field. Bsed on this fct, it s possible to diect electon bem in given diection of the spce by mens of two set of cossed pltes. The fist set will contol diection, nd the second one, nothe pependicul diection. In this pocedue is bsed the television nd mesuement device with multiple pplictions: the oscilloscope of cthodic y. E v The oscilloscope of cthodic y llows the visul epesenttion nd mesuent of electic signls seen s diffeences of potencil (udio souces, video, medicl instuments, dt of compute ) nd in genel, ny electic signl of ny cicuit. b c d V H V V e f Hz de electones g y x It consists in vcuum tube (f) whee electons e geneted in cthode (b) heted by electic esisto (); electons e cceleted to the node (c). Electons coss two set of pltes tht poduce electic fields cossed (d nd e). In this wy, electon bem divet in the diection x ccoding the electic field poduced by the pltes (d), nd in the diection y ccoding the electic field in the pltes (e). Finlly, they chieve sceen of fluoescent phosphous (g) whee they poduce light of shot length. To epesent n electic signl vible on time, s fo exmple sinusoidl signl, it is necessy to ente tingul signl of hoizontl scnning V H (in the xis x, nd theefoe between the pltes d). In this wy, on electons ct n incesing electic field, tht does they divet to ight. If we don t ente ny signl in the veticl xis, we will obseve hoizontl line tht sweeps the sceen fom left to ight. 1-19
V H V V V H T H t The peiod of tingul signl T H, is the time the signl tkes going fom left to ight on the sceen. If we ente in the pltes (e) sinusoidl ltenting signl t the sme time tht in (d) signl of hoizontl scnning like the descibed, the electons will be diveted ightwds by ction of the incesing hoizontl field, nd upwds o downwds by the ction of the field enteed in (e). In this wy, we will obtin the supeposition of both signls. With this technique, we cn epesent ny vible signl on pltes (e). To egulte the speed of the hoizontl scnning, nd theefoe the hoizontl tension, the peiod of the tingul signl is compised between the seconds nd the nnoseconds. V H V V V V t t V H It is lso possible to ente two independent signls in d nd e, with epesenttion XY of both signls, obtining the clled Lissjous s figues. In the following exmple ppe the Lissjous s figue coesponding to two signls of the sme fequency but out of phse in 1/8 of cycle: V H V V V V t t V H In this nothe exmple, two sinusoidl signls with diffeent fequencies (te of fequencies 3:4). V H V V V V t t V H Xeogphy The xeogphy (witing in dy) is widely used technique, tht llows obtin copies in ppe fom oiginl. This technique ws invented in 1938 by Cheste Clson. The pocess is bsed in popety of some substnces, clled photoconductivity, consistent in the vying of electic conductivity with lighting. Like exmple of photoconductos mteils cn quote the oxide of zinc nd divese compound of selenium. In pesence of light the substnce cts like conducto, whees it emins like insulto in the dkness. 1-0
Although the mchines of epoduction of xeogphy e vey sophisticted, the pocess of epoduction of n oiginl cn be summized in 5 stges: 1) In the beginning (), sheet coveed of photoconducto mteil, is electiclly chged with positive chge in the dkness, by mens of diffeence of potentil of the ode of 1000 V between the sheet nd eth. The sheet is suppoted on conducto metllic plte connected to eth, so negtive chge is induced on it. The chges of diffeent sign emin septed becuse the sheet is insulto on dkness. () ) Lte (b), light is pojected ove the imge tht goes to be copied, nd by mens of n opticl system of lens nd mios (not shown in the figue, this imge is pojected on the photoconducto sufce. In the es of the sufce with intense light, the negtive chge of the conductive plte neutlizes the positive chge, nd the e is dischged, whees the es tht hve not eceived light (blck zones), emin chged with positive chge. Whee ive tenuous light, the chge educes slightly. In this wy, we obtin sufce distibution of chge tht epoduces the imge. (b) (c) (d) 3) To convet this electic imge, not visible, in visible imge, dust (tone) constituted by coloed pticles negtively chged e spinkled on the photoconducto plte (c). It will fix the dust on the vitul electic imge positively chged, going bck this electic imge in visible imge. 4) Lst step is to tnsfe the tone to the ppe (d). The ppe hs been chged positively, so tht it cn ttct the pticles of tone, fixing these in pemnent wy by mens of het, melting the tone. 5) Finlly, the pocess cn be epeted, clening the plte of ny excess of tone nd dischging it of ny est of electic chge with n excess of light. 1-1
1.9 Poblems 1. Given the thee point chges plced s it s shown in the figue, compute the electic foce F done by these foces on chge Q/ plced on point O. KQ ol: F 1 ( i j ) d 4 Q d O d -Q Q. Given fou equl point chges q, t est, plced on the vetex of sque with side 1/, find the net electic foce tht the fou chges would exet on chge q' plced on O, nd the electosttic potentil enegy of q' in O. q ol: F 0 U q π ε 0 3. In the figue, two point chges, -3Q in x 1 nd Q in x 5 e shown. On which points of the x xis ) it s cncelled the electic potentil b) it s cncelled the electic field? ol: ) x 4, x 7 b) x 10,46 q q O q -3Q Q 1 5 10 q x 4. Let s tke the point chge Q nd the two cubic sufces, pllel, centeed in Q, nd with sides nd 3, shown in the figue. Compute the te between fluxes of the electic field though both sufces (Φ /Φ 3 ). Justify the nswe. 3 Q 5. A cube of edge nd unifom volumetic density of chge, ρ, is plced on vcuum. It s suounded by spheicl sufce of dius. Compute the flux of the electic field though the spheicl sufce. 3 ρ ol: Φ ε 0 1-
6. Given the electic field defined s E (x,0,0), compute: ) Flux though the sufce of the cube shown in the figue. b) Electic chge enclosed in the cube. ol: to) Φ 4 b) Q ε 0 4 Y X 7. Apply Guss s lw to deduce the expession of the electic field ceted by n infinite plne chged with unifom sufce density of chge σ. 8. A positive point chge q 1, is plced in the oigin of n othogonl coodintes system on the plne. Anothe negtive point chge q is plced on the odintes xis t distnce of 1 m fom the oigin. Compute: ) Electic field ceted by ech one of the chges on point A plced on OX xis t distnce of m fom the oigin. b) Needed wok to move chge q fom point A to point B(4,) m. Apply it to the cse whee q 1 10-9 C, q - 10-9 C, q 3 C. ol: ) E,5 N/C E 1,61( i ) N/C b) W AB 3,59 J 1 i j 9. Two positives nd equl point chges, q, e plced t distnce. A positive unit chge is plced t the sme distnce fom ech chge,, s it s shown in the figue. Which must be the distnce fo the electic foce cting on unit chge gets its highe vlue? ol: (3/) 1/ q q 10. A stight line vey long (infinite) is chged with line unifom density of chge λ. Compute the mgnitude of electic field ceted by this line on point P plced t distnce y fom the line. ol: E λ/(πε 0 y) 11. Let s hve spheicl nd homogeneous volume density of chge ρ nd dius R. Compute the mgnitude of electic field nd electosttic potentil ceted by such chge distibution on point plced t distnce fom the cente of the sphee: ) >R; b) R; c) <R ol: ) E (1/3ε 0 )(ρr 3 / ); V (ρ/3ε 0 )R 3 / b) E (1/3ε 0 )ρr; V (ρ/3ε 0 )R 1-3
c) E (1/3ε 0 )ρ ; V (ρ/ε 0 )(R - /3) 1. The figue shows piece of cylinde of infinite length nd dius R, chged with unifom volumetic density of chge ρ. Compute: ) Electic field inside nd outside the cylinde. b) Diffeence of potentil between the xis of the cylinde nd its sufce. ol: to) E in ρ/ε 0, E out ρr /ε 0 b) V ρr /4ε 0 R GLOARY Elementy chge: Minimum vlue of electic chge, equivlent to the chge of the electon, 1,6 10-19 C. Coulomb s lw: Two electic point chges q 1 nd q, t est, t distnce in vcuum, exet foce between them whose mgnitude is popotionl to the poduct of the chges nd popotionl invesely to the sque of the distnce between them, being ttctive if they e of opposite sign nd ejecting if they e of the sme sign; its diection is the one of the stight line joining them. q1.q F k u 1 1 Electic field on point of the spce is the electic foce exeted on chge q 0 in such point by unit of chge. (q 0 is smll chge of pobe) F E Electic field ceted by point chge q on point t distnce fom it 1 q E u 4πε Electic dipole: system of two equl chges nd opposite sign distnt low distnce. Electic pemittivity of vcuum: univesl constnt with vlue ε 0 8,85 10-1 C /N m Guss s lw. The flux of the electic field though n enclosed sufce is equl to the totl chge inside divided into ε 0 1 q 0 0 1-4
Electosttic potentil of point of the spce is the enegy hving ny chge q 0 plced in this point, divided into such chge. Electosttic potentil poduced by point chge q on point t distnce of it. V Pinciple of supeposition: The totl effect (foce, field o potentil) poduced by set of chges is the ddition (of the foce vectos, field vecto o potentil) of the effects poduced by ech one on independent wy. Equipotentil sufces. et of points in the spce hving the sme electosttic potentil. A given point cn only belong to one equipotentil sufce. 1-5