A Generalization Of Gauss's Theorem In Electrostatics

Transcription

1 Proc. EA Annual Meetng on Electrostatcs A Generalzaton Of Gauss's Theorem In Electrostatcs Ishnath Pathak B.Tech tuent Dept. of Cvl Engneerng Inan Insttute Of Technology North Guwahat, Guwahat- 7839, Ina e-mal: pathak.shnath@gmal.com Abstract Gauss s theorem of electrostatcs states that the flux of the electrostatc fel over a close surface euals E = Q/, where Q euals the net charge lose by. In the ervaton t s assume that no charge les on the surface n ueston. Conser the problem of evaluaton of the electrostatc fel ue to a unformly charge sphercal surface on the surface tself. The stuaton exhbts symmetry but we can t apply the Gauss s theorem, an we have to resort to other methos lke rect ntegraton. In ths paper we prove a generalzaton of Gauss s theorem whch allows charges to le on the surface of ntegraton. For the majorty of cases the statement of our generalze Gauss s theorem can be assume to be ths: the flux of electrostatc fel over a close surface euals Q Q + con E a =, where Q s the net charge lose by an Q con s the net charge contane by. Applyng ths theorem to the unformly charge sphercal surface we fn at once that the fel euals exactly half of the fel whch woul have exste f the charge le completely nse the surface n a sphercally symmetrc manner. Usng ths generalzaton of Gauss s theorem we present a generalze electrostatc bounary conton, whch we then use to solve the famous conuctng plane mage problem wthout usng the metho of mages. I. INTRODUCTION One of the most mportant theorems of electrostatcs s the Gauss s theorem. The wellknown theorem states E = Q/. It s explctly state sometmes, that the bounary of the regon lose must not contan any pont, lne or surface charges. But what happens f the surface contans such charges n any ealze problem? To my knowlege, the generalzaton of Gauss s theorem has not been scusse anywhere tll now, an I present t here. For the next few sectons we scuss what mssng terms appear n the gauss s theorem f we permt prese of charges on the surface of ntegraton.

2 Proc. EA Annual Meetng on Electrostatcs II. FLUX OF THE ELECTROTATIC FIELD OF A POINT CHARGE We prove n ths secton that the flux of the electrostatc fel bounary to a connecte regon R s E of a charge over a f lesoutse Ω f lesonat a pont where () E = 4π nse sol angles Ω f s lose by Ths statement s axomatc but stll I ll prove a satsfactory proof soon. Yet before ong that I lke to expatate a lttle. An analogy s often rawn between the actual stuaton an an magnary stuaton wth the pont charge raatng photons at a constant rate n a sphercally symmetrc manner. nce the electrostatc fel s gven by a raal nverse suare law euaton, the total number of photons passng per unt tme through a surface can be thought of to be the flux of the electrostatc fel through that surface. Now f we conser a charge lyng on the close surface uner conseraton, the total number of photons passng per unt tme through the close surface shoul be obtane by multplyng the total number of photons emtte by the charge per unt tme by / 4π tmes the nse sol angle forme at the surface at the place where the pont charge les. Now let me show my proof for euaton. If the reaer s convnce by the above reasonng then the rest of ths secton can be skppe. r the poston of. Let s frst conser the case A when the charge We enote by les outse the bounary. In ths case E s fferentable properly over an open connecte regon contanng the surface along wth ts losure. Applcaton of verge theorem wll yel the result. Let s now conser the case B when r les on. Let the nse sol angle forme on at the pont r be Ω n measure. We chose an arbtrary non-zero raus suffcently small, say less than a crtcal raus c, such that the of the sphercal surface of raus centere at r, whch les not outse the part regon R lose by, satsfes the followng two contons ) It ves R nto two parts R an R such that R, the one not ajacent to r, s lose by a close surface whch has the nner surfaces of as ts nner surfaces n case R has cavtes. ) It has no mssng patches,.e. s boune by a sngle close curve P an not by a group of close curves. He, the close curve P ves the outer surface of nto two parts, an, where an together wth the nner surfaces of form the close surface

3 Proc. EA Annual Meetng on Electrostatcs 3 whch loses whch loses the compact regon as R. Also an Because when we a the flux together form the close surface R. Now, we can wrte the flux of electrostatc fel through E = E + E Φ = 4π area of Of E through to the secon ntegral on the rght of eualty an subtract the same from the frst one, we get entty. Now accorng to the result of case A the secon ntegral must vansh. The flux of E through can now be easly obtane Ω E = lm = lm = lmφ = E a E a 4π Fnally we conser the case C when r les lose by. We conser any plane passng through r an enote by the part of the plane that s lose by. We enote agan by R an R the regons nto whch ves R, an by an the close surfaces that lose these regons. o we have agan E = E + E Only ths tme we are alreay wth an entty, for ths tme the flux Φ of E through s zero. Now accorng to the result of case B, both the terms on the rght of eualty are eual to /, so that the flux s E = III. THE GENERALIZED GAU THEOREM The prncpal of superposton for electrostatc fel allows us to nsst that the flux of the electrostatc fel over a close surface s ( E ), where the summaton s one over all charges n the charge confguraton. The lnearty of the operaton of flux enables us to nsst that the flux s ( E a). nce the flux E of any charge lose by s Q, where by s / / o, the sum of the solate fluxes of all charges lose Q s the net charge lose by. mlarly we get that the

4 Proc. EA Annual Meetng on Electrostatcs 4 sum of the solate fluxes of all charges lyng outse s zero, an that of all charges resng on the contnutes of s Q / con, where Q con s the net charge resng on the contnutes of. By contnutes of, we mean a pont on where the prncple curvatures of vary smoothly so that the nse sol angle forme s eual to π. The sum of the solate fluxes of the remanng charges (all of whch le at the scontnutes of ) s left as a summaton Ω Φ = () scon 4π o For we can t say anythng about the nner sol angles Ω wthout a partcular knowlege of the geometry of the close surface an the locatons of the charges lyng at the scontnutes. He, we have the generalze Gauss s theorem: The flux of the electrostatc fel E over any close surface s Q Q con E = + + Φ (3) Where Q euals the net charge lose by, Q con euals the net charge resng on the contnutes of an Φ, as escrbe by (), euals the flux of the electrostatc fel of the charges lyng at the scontnutes of IV. VARIOU FORM OF THE GENERALIZED GAU THEOREM We n ths secton conser those cases n whch the generalze Gauss s theorem takes a beautful form whch we shall call as the smplest form of the theorem. We begn by confnng ourselves to an electrostatc fel cause by a confguraton of charges not contanng any pont or lne charges. In ths specal case Q con of euaton (3) can also be nterprete as the net charge contane by, for the net charge lyng on the scontnutes of s zero anyway, because now contans no pont or lne charges- whch were the only varetes whch coul accumulate to a fnte amount by assemblng only at scontnutes. Also Φ vanshes n case of an electrostatc fel cause by such a confguraton. Ths nees some explanaton. Let s magne a fferent source charge confguraton- the one n whch each charge s replace by a corresponng postve one of an eual magntue. The fact that the net charge lyng on the scontnutes of s zero whenever the charge confguraton oes not contan any pont or lne charges mples that for the orgnal charge confguraton we ll have. Now snce, we have Ω Ω 4π scon 4π The central term n the neualtes s Φ. o, = scon Ω 4π scon scon

5 Proc. EA Annual Meetng on Electrostatcs 5 An snce Φ Ω 4π = 4π scon scon Ω 4π, Ω 4π scon Ω 4π. He, scon As the rghtmost term n the neualtes s zero, as argue earler, we have the mle term eual to zero. From here we conclue that Φ vanshes. Therefore f the source charge confguraton s free from pont an lne charges, then The flux of electrostatc fel over any close surface s Q Q = + con E a (4) Where Q s the net charge lose by an Q con s the net charge contane by. uperposng the fluxes we get as a corollary that for any kn of charge confguraton Where on an Q sur E a = + + Φ pl (5) Q Q s the net charge lose by, Q sur s the net surface charge resng Φ s the flux of the electrostatc fel of the pont an lne charges resng pl on. Tll now (4) was referre to as applcable only n case of charge confguratons that n t contan pont an lne charges. From (5) t can be seen that (4) hols whenever no pont or lne charge les anywhere on the scontnutes of, for n that case we see (by referrng to euaton (3)) that Φ pl = Qp/ + Ql/, where Q p euals the net pont charge an Q l euals the net lne charge resng on. An then, snce the recent most restrcton hols f has no scontnuty at all, (4) hols whenever s throughout contnuous. We shall call euaton (4) as the smplest form of the generalze Gauss s theorem an see that t s almost always applcable. Ω V. CONCLUDING REMARK In the ntroucton I mentone that for the majorty of cases the statement of our generalze Gauss s theorem can be assume to be ths: the flux of electrostatc fel over a close surface euals /vacuum permttvty tmes the sum of the net charge lose by an half the value of net charge contane by. Let me now enlst the cases that I clam to be n majorty The surface of ntegraton oesn t contan pont or lne charges at any of the corners or eges.

6 Proc. EA Annual Meetng on Electrostatcs 6 The surface of ntegraton oesn t have any ege or corner,.e. t s throughout contnuous. The source charge confguraton conssts of only volume charges an surface charges. VI. THE GENERALIZED ELECTROTATIC BOUNDARY CONDITION ( E E) nˆ =σ( r )/. As the tangental Let be a surface on whch at a pont r an near the pont r, on one se of the surface (say se ), only volume an surface charges le. Let s assume that ths pont sn t a scontnuty of the surface an enote by ˆn the unt normal to the surface pontng towars se. Let s enote the charge ensty on the surface at the pont r by σ(r ). We re here seekng a relatonshp between the lmt E of the electrostatc fel as we approach r from se, an the value E of the electrostatc fel at the pont r on the surface. Here I m conserng the electrostatc fel to be of the form of a mathematcal functon whch s well efne at a pont of nterest where the functon value ffers from the rght han lmt. We raw as a Gaussan surface a close surface of the form of a very thn geometry-box wth the plane base of an extremely small area A lyng on the surface an the remanng part extenng n the recton of se to an nfntesmal stance from the surface. The applcaton of the generalze Gauss s theorem, whch apples here n ts smplest form, gves us Q Qcon Q σa E = + = + Now, n the lmt at the thckness goes to zero, Q goes to zero. Also n ths lmt the flux E euals E Anˆ E An ˆ, as n ths lmt, the ses of the box contrbute nothng to the flux. o we get component of electrostatc fel s always contnuous at a surface charge, our sought for relatonshp between E an E becomes σ( r) E(r) E(r) = nˆ (6) We shall call ths euaton as the absolute electrostatc bounary conton for t can be use to arrve at the usual bounary euaton n ths manner: σ σ σ σ E E = nˆ ; E E = nˆ = nˆ ; E E = nˆ Queston: A pont charge s place at a stance from an nfnte conuctng plane; what s the charge ensty on the plane at a stance r from the foot of the perpencular to the plane from the pont charge? Ths s the smplest problem for whch the metho of mages s nvoke. I propose to use the absolute bounary conton to solve ths problem wthout usng the metho of

7 Proc. EA Annual Meetng on Electrostatcs 7 mages. We select ˆn to pont nse the conuctor so that E vanshes. Generalze En ˆ =σ( r )/. Now, from the prncpal of super- bounary conton then gves poston E( r) = E ( r) E ( r), where E euals the electrc fel of the pont charge + an E euals the electrc fel of the surface charges on the conuctng plane. It follows straght from the Coulomb s law that for any pont r on the conuctng plane ( nˆ ) + r E() r = 3/ 4π ( + r ) o, we have E ˆ () r n = 3/ 4π ( + r ) Because r s orthogonal to ˆn. Utlzng ths n the euaton En ˆ =σ( r )/ we get σ( r) = E () ˆ 3/ r n 4π ( + r ) Now, snce the electrostatc fel of any charge ponts raally away from the charge, we must have E orthogonal to ˆn. Ths leas us straght to the soluton of the conuctng plane mage problem: σ( ) = π (r + r (7) Ths s a well-known euaton, an at present t s hel by the physcs communty that t can be erve only usng the metho of mages. ) 3/ VII. APPENDIX In our soluton above, we assume the electrostatc fel to behave as an ealze mathematcal fel. In fact, the fel on the surface s scontnuous an t has no well-efne value on the surface. The ambguty can be remove by talkng about the electrostatc force per unt area on the surface charge nstea of the electrostatc fel. Let the charge be at k ˆ an the upper surface of the conuctng plane be the X-Y / plane. On the plane r (x + y ) s the stance from the orgn. At any pont n space the total fel s ue n part to an n part to the surface charges nuce on the plane: E= E + E. ( ) ˆ / f= σ rk. Now, The electrostatc force per unt area on the surface s fr () = σ re() r + f() r, where f () r s the electrostatc force per unt area on the X-Y plane ue to the charges on the plane. By symmetry, t can have no Z-component.

8 Proc. EA Annual Meetng on Electrostatcs 8 Euatng the above two values of f an takng the ot prouct wth ˆk we get ˆ 3/ 3/ σ (r)/ = E ( r) k = /4 π (r + ). o, σ(r) = / π(r + ). Let me present another soluton wthout usng the metho of mages or any theorem presente n ths paper. At a general pont P on the plane whose stance from the orgn 3/ s r, we have (E ) /4 π (r ) (E ) s contnuous. On z= +, an we know z (E ) s scontnuous n the amount σ /, an by symmetry s the the other han z same above an below n magntue an ts recton on both ses s ether towars the plane or away from the plane. Thus mmeately below the plane (E ) z= σ /. But below the plane,.e. nse the conuctor, the total fel s zero. o, 3/ (E ) + (E ) =. He, z z σ(r) = / π(r + ). Let me present yet another soluton. I learnt t from Prof. J.D.Jackson n the reply to a mal conveyng my soluton. Conser any pont P on the plane. On the conuctng plane, the electrostatc fel cause by the pont charge an the strbuton of surface charges must be normal to the plane. Otherwse, the charge, free to move, wll reajust tself. Now, snce the tangental component of electrostatc fel s contnuous at a surface charge, the fel s normal to the plane both above an below P. If the fel just above P ue to the surface charges s E ˆ ˆ (cosαk+ sn αr ) (where ˆr s a unt vector n the plane pontng from the orgn O to the pont P), then by symmetry, just below P t s E ˆ ˆ (cosαk sn αr ). nce, just above P the net fel s normal, we have Esnα = Esnθ. An just below P the net fel s zero, so Ecosα = Ecosθ. He E = E an α = θ. Thus, we see that the net fel just above P s -E cosθ k ˆ. Applcaton of Gauss s theorem to a pllbox at P that spans the surface (wth zero contrbuton from the se of the box wthn the 3/ conuctor) gves σ(r) = E cosθ = / π(r + ). By argung wthout usng an mage charge we, n our soluton, showe that at any pont on the conuctng plane the fel ue to nuce surface charges on the plane s constructe by frst reflectng s fel n the conuctng plane an then reversng ts recton. Ths s exactly the fel of an mage charge place at -k ˆ, but we notce that after the fact. ACKNOWLEDGEMENT The smplest form of the Generalze Gauss s Theorem occurre to me n 6 when my younger brother harvanath, who s now a computer sce stuent but use to be a physcs enthusast n hs school ays an s a bronze mealst at the 8th Asan physcs Olympa, came to me to ask the soluton of a self-frame problem n whch one ha to ntegrate the flux of the electrostatc fel of a pont charge place on an ellpso at the

9 Proc. EA Annual Meetng on Electrostatcs 9 pont nearest to one of the foc. o, n a sense, ths work s hs n a way whch makes hm eny ths clam of mne hmself. The secon proof n the appenx s an altere verson of a proof whch a referee suggeste when I conveye a paper ttle a soluton of the conuctng plane mage problem wthout usng the metho of mages to Amercan Journal of Physcs. The paper was assgne the manuscrpt # 98, an the referees were ntereste n the soluton to the mage problem but not n the generalzaton. When I respone after a month beng busy wth my exams, I was tol that the journal ha receve a couple months before my submsson a paper on the same topc as that of mne. Reang ths I publshe my work onlne mmeately at Conuctng-Plane-Image-Problem-Wthout-Usng-the-Metho-of-Images The last two paragraphs n the appenx come nto ths paper (after several mofcatons) from an e-mal sent to me by Prof. J.D.Jackson. I thank hm for extractng several moments from hs precous tme, an ouragng me n my pursut of knowlege n physcs. FOOTNOTE AND REFERENCE [] E.M.Purcell, Electrcty an Magnetsm, st eton, (McGraw Hll, 965), p.. [] In many texts t s ponte out that the flux of the electrostatc fel of a pont charge through any nfntesmal surface (wth the normal to the surface pontng away from the charge) s Ω / 4π, where Ω s the sol angle subtene by the nfntesmal surface at the poston of the charge. E.g. ee J.D.Jackson, Classcal Electroynamcs, 3 r eton, (John Wley & ons, 999), pp.7-8. Once ths result s known, one can say at once that the flux of the fel of a sngle pont charge through a close surface s gven by e. ().