CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electostatcs: It deals the study of behavo of statc o statonay Chages. Electc Chage: It s popety by whch mateal get electfed. It s denoted by Q o and ts unt s coulomb. Chages ae of two types -() postve () negatve. A pont chage s the dmensonless and mass less chage that can expeence o exet electostatc foce. The pont chage may be postve o negatve. Unt postve chage whch can only expeence the electostatc foce s temed as test chage and t s denoted by. Quantzaton of Chage: The chage on mateal /matte/patcle s ntegal multple of chage on 9 electon..e. ±ne, whee e s chage on electon whose value s.6 coulomb. The chages on uaks ae the excepton to uantzaton of chage. Quaks may have chages ± e o ± e. Coulombs Law: The foce actng between chages s called as electostatc foce. Coulombs law povdes the magntude and decton of electostatc foce actng between two chages. Accodng to ths law- The electostatc foce between two chages s dectly popotonal to poduct of both chages and nvesely popotonal to suae of dstance between them. Let and j ae the two pont chages and ae placed at a dstance j. F j j j o F () j Hee s the pemttvty of medum whee the chages ae placed. If and ae the pemttvty of fee space and elatve pemttvty o delectc constant espectvely then can be gven by followng expesson. ; hee 8. 854 Coulomb / Nm In vecto fom the foce s gven by, j j F Fnˆ F ˆ j ˆ j j () j j Ths foce s expeenced by both the chages. Let F j s foce exeted by chage on chage j and F j s foce exeted by chage j on chage. Fom euaton () we can wte, j j Fj ˆ j j () j j j j And Fj ˆ j j (4) j j Snce, j j and j j thus fom euatons () and (4) we can have F j F j (5) Do not publsh t. Copy ghted mateal. D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Euatons (5) mples that the foces expeenced by both chages ae opposte n natue whle the magntudes ae same. Note : If and j ae the poston vectos of wtten as: j j F j j j j and ( ) j chages then electostatc foce s Note : If Q and chages ae placed at a dstant then electostatc foce s wtten as: Q Q F and F Supeposton Pncple: The esultant electostatc foce on a chage s vecto sum all foces expeenced by t..e. F F + F + F + If chage s suounded by chages,,,, whch ae at,,, apat fom espectvely then, + + + F Electc Feld: The space o egon aound the chage o goup of chages whee the effect of these chages s felt o anothe chage expeences the foce o s called as Electc feld. It s denoted by E. It s ato of foce expeenced anothe chage o test chage and ts amount of chage. F.e. E () (A) Electc Feld due to pont chage : Let chage s postoned n the electc feld of chage at dstance. The foce can be wtten as, F () Fom euatons () and (), we fnd that, E and E E () Euaton () s expesson fo electc feld due to pont chage. (B) Electc Feld due to goup of chages: Suppose P s a pont whch s at,,, dstances fom the goup of chages,,, The electc feld at pont P wll be vecto sum electc feld poduced by each chages at that pont. + + + + + + E E E E (4) Do not publsh t. Copy ghted mateal. D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 (C) Electc Feld due to chage dstbuton: Suppose de s amount of electc feld at pont P due to chage d n a chage dstbuton. Pont P s at dstance fom the chage d. d de (5) Total electc feld at pont P due to whole chage dstbuton can be obtaned by ntegatng euaton (5). π d E (6) 4 Thee ae thee types of chage dstbuton. () Lnea chage dstbuton: If the chages ae dstbuted ove a lne then t s called as lnea chage dstbuton. The chage pe unt length s called as lnea chage densty. It d s denoted by λ. Let d chages ae dstbuted ove a length dl then λ. dl () Suface chage dstbuton: If the chages ae dstbuted ove a suface then t s called as suface chage dstbuton. The chage pe unt aea s called as suface chage densty. It d s denoted by σ. Let d chages ae dstbuted ove an aea ds then σ. ds () Volume chage dstbuton: If the chages ae dstbuted ove a volume then t s called as volume chage dstbuton. The chage pe unt aea s called as volume chage densty. d It s denoted by ρ. Let d chages ae dstbuted ove a volume dτ then ρ. dτ If the chage densty emans unchanged thoughout the chage dstbuton then the dstbuton s sad to be unfom chage dstbuton othewse t s called as non-unfom chage dstbuton. Usng euaton (6), the electc feld due to these thee types of chage dstbuton can be wtten as: λ dl E o σ ds ρdτ E o E π 4 Electc lnes of foces: Electc lnes of foces ae the magnay lnes n an electc fled whch epesents the path of test chage. The tangent on these lnes at a pont povdes the decton of electc feld at that pont. The electc feld wll be lage at the pont whee the electc lnes of foces ae dense. Smlaly electc feld wll be weak at the pont whee the electc lnes of foces ae ae. Do not publsh t. Copy ghted mateal. D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 (a) Unfom electc feld: If the electc lnes of foces ae epesented by paallel lnes then electc feld wll be same at evey pont. Such electc lnes of foces epesent the unfom electc feld. (b) Non-Unfom electc feld: If the electc lnes of foces ae lnealy/cuved dvegng o convegng then electc feld wll not be same at evey pont. Such electc lnes of foces epesent the non-unfom electc feld. Thus poston and tme dependent electc feld s called as non-unfom electc feld. e.g. () The path of test chage n electc feld of postve pont chage s lnea along outwad decton because the foce actng on test chage s away fom pont chage. Hence the Electc lnes of foces of postve pont chage ae outwad lnea. They ae concentated nea the pont chage and ae away fom the pont chage thus postve pont chage geneates the dvegng non-unfom electc feld. () The path of test chage n electc feld of negatve pont chage s lnea along nwad decton because the foce actng on test chage s towads the pont chage. Hence the Electc lnes of foces of negatve pont chage ae nwad lnea. They ae concentated nea the pont chage and ae away fom the pont chage thus negatve pont chage geneates the convegng non-unfom electc feld. () If the path of test chage n an electc feld s convegng o dvegng cuved lnes then electc feld wll be same at evey pont. The electc feld wll be poston and cuvatue dependent. Such electc lnes of foces epesent the non-unfom electc feld. () Unfom electc feld (a) (b) (c) (d) () Non-unfom electc feld: electc feld due postve, negatve and combnaton of chages Note : The two electc lnes of foces do not ntesect to each othe because decton of electc feld can not be two at a sngle pont. Note : Paallel staght electc lnes of foces epesent the unfom electc feld whle convegng o dvegng cuved/staght lnes epesent the non-unfom electc feld Do not publsh t. Copy ghted mateal. 4 D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electc Flux: The numbe of electc lnes of foces passng though a suface s called as electc flux lnked wth suface. Mathematcally, t s eual to poduct of component of electc feld nomal to the suface and aea of suface. It s denoted by ϕ. Suppose the unfom electc feld E epesented by staght paallel electc lnes of foces makes an angle θ wth the aea vecto S. ϕ component of E along S aea of suface ϕ E Cos θ S ϕ E S Cos θ E S () () When E S then θ and hence ϕ max E S () When E S then θ 9 and hence ϕ mn () Let d ϕ electc flux lnked wth elementay aea ds of a suface. Then, d ϕ E ds () The total electc flux lnked wth suface can be obtaned by ntegatng ϕ E ds () If the suface s closed then total electc flux lnked wth suface can be wtten as; ϕ E ds (4) Gauss law: Accodng to ths law, the net outwad electc flux though closed suface s eual to the / tmes total chage enclosed by closed suface. Let chages,,,, ae enclosed by a closed suface. Then, Net outwad electc flux total chage enclosed by closed suface ϕ ( + + + ) ϕ E ds If chage enclosed by the closed suface s zeo then the net outwad electc flux wll be zeo. Poof: Suppose total chage s enclosed by a closed suface and an elementay suface ds s at dstance fom the chage. If E electc feld at dstance fom chage Then, E () takng dot poduct wth ds on the both sdes of euaton () we have, Do not publsh t. Copy ghted mateal. 5 D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 d E ds ds () Integatng euaton () ove a closed suface, ds E ds π () 4 The pojected aea dvded by s temed as sold angle (dω) subtended by ds at. ds Cos θ ds Cos θ ds.e. dω (4) Fom euatons () and (4), we have; E ds dω As d ω 4π thus, E ds 4π E ds (5) Hence the law s poved. Note: If chage s at cente of an sphee of adus then net out outwad electc flux, π π ˆ E ds ( θ θ φ ) θ θ φ 4π Sn d d ˆ Sn d d Dvegence of E : Accodng to Gauss law, the net outwad electc flux though closed suface s eual to the / tmes total chage enclosed by closed suface. E ds () If ρ s chage densty of dstbuton and d chages ae dstbuted ove a volume dτ then, d ρdτ ρdτ () Fom euatons () and (), we have; E ds ρdτ () Usng Gauss dvegence theoem we can wte that, E ds (.E ) dτ, hence euatons () becomes as; ρ (. E ) dτ dτ (4) Snce euatons (4) holds fo any volume thus ntegands must be eual. Do not publsh t. Copy ghted mateal. 6 D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 ρ. E Euatons (5) s called as dffeental fom of Gauss law. Note: The closed suface consdeed n Gauss law s called as Gaussan suface. Symmety s the essental cteon fo the applcaton of Gauss law. Thee ae manly thee type of symmety. A) Sphecal symmety: fo a pont chage, the Gaussan suface concentc sphees. B) Cylndcal symmety: fo vey long lne chage dstbuton, the Gaussan suface concentc cylndes. C) Plane symmety: fo vey lage suface chage dstbuton, the Gaussan pllbox s Gaussan closed suface. Note: If chage s enclosed by closed suface s zeo then net out wod flux wll be zeo. Ths mples that dvegence of electc feld wll also be zeo..e. E ds E A vecto feld s solenodal f ts dvegence vanshes. Snce the dvegence of electc feld s zeo, hence the electc feld s sad to be solenodal. Poof: The electc feld due to pont chage at a dstance s gven by, E Takng dvegence on both sdes we have, E ( ). φa φ.a + φ A 5 E [( ). + ( )] [( ). + () ] 5 E [. + ] [ + ] Hence E s solenodal fo. ; as ( ) ( ) E otatonal and consevatve feld: Cul of a vecto uantty s a measue of how much the vecto culs aound. If cul of a vecto feld vanshes then vecto feld s not otatonal by t s otatonal feld. Fo an otatonal feld, the lne ntegal of a vecto feld ove a closed path s zeo. Ths mples that the lne ntegal of a vecto feld s path ndependent. Such vecto feld s called as consevatve feld. The electc feld s otatonal and consevatve feld..e. E E dl E dl s path ndependent. (5) Do not publsh t. Copy ghted mateal. 7 D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Poof: (a) The electc feld due to pont chage at a dstance s gven by, E Takng cul on both sdes we have, E ( ); as φa ( φ) A + φ( A) 5 E [( ) + ( )] [( ) + ( ) ] 5 E [ + ] [ + ] E () Hence E s otatonal feld. We know that E follows the supeposton pncple..e. E E + E + E + So, E E + E + E + () Thus E s otatonal feld fo any statc chage dstbuton,.e. thee s no matte whee the chage s located o t s dstbuted. (b) Takng volume ntegal of euaton () ove a closed path we have, E dv v ( ) Fom stokes theoem, the volume ntegal can be conveted nto lne ntegal wth followng fomula. ( E) dv E.dl (4) v Fom euaton () and (4), we can wte, E.dl l l If closed path s AXBYA then fom euaton (5), we can wte, B A E.dl + E.dl A B B A E.dl A E.dl B E.dl E.dl path AXB path AYB Euaton (6) ndcates that the lne ntegal of a electc feld s path ndependent. Thus t s consevatve feld. () (5) (6) Do not publsh t. Copy ghted mateal. 8 D. D. K. Pandey
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Applcaton of Gauss Law. Electc feld due to nfnte lnea chage dstbuton Suppose chage ae dstbuted along a lne, whose lnea chage densty s λ. If chage s dstbuted on the length l then, λ l () If the chage s postve then the electc feld wll be away fom the lne. Let Electc feld at dstance due to ths chage dstbuton s E. The Gaussan suface fo ths chage dstbuton s a cylnde of adus and length l whose axs contans chage dstbuton. Applyng Gauss law to ths Gaussan suface, E ds Do not publsh t n any fom. Snce the Gaussan suface s composed of thee sufaces s, s and s. Thus euaton () can wtten as, E ds + E ds + E ds s s s s E ds Cos9 + + E ds s E ds s λl E πl + E dscos9 + s s s E λ E ds E dscos. Electc feld due to unfomly chaged plane sheet Method : Consde a plane sheet of aea A s unfomly chaged wth chage. If suface chage densty s σ then chage on plane sheet wll be wtten as; σa () If the chage s postve then the electc feld wll be away fom the suface and fo negatve t ponts towads plane. Snce chages ae dstbuted on plane thus cylndcal Gaussan suface 9 Copy ghted mateal. ()
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 (pll-box) shall be the Gaussan closed suface fo t whose half poton les on one sde of the plane and est on the othe sde. Applyng Gauss law to ths Gaussan suface, E ds () Snce the Gaussan suface s composed of thee sufaces s, s and s. Thus euaton () can wtten as, E ds + E ds + E ds s s s s E ds Cos E ds + E ds + E ds Cos + s s + s s E ds + E ds s s EA + EA σ EA A σ E E ds Cos9 Method : Consde a plane sheet of aea A s unfomly chaged wth chage. If suface chage densty s σ then chage on plane sheet wll be wtten as; σa () If the chage s postve then the electc feld wll be away fom the suface and fo negatve t ponts towads plane. Snce chages ae dstbuted on plane thus pll-box shall be the Gaussan closed suface fo t whose half poton les on one sde of the plane and est on the othe sde. The electc flux though half pll-box wll be EA. Applyng Gauss law to ths Gaussan suface, E ds Do not publsh t n any fom. Copy ghted mateal.
net out wad CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 flux though pll flux though half pll - box - box σ EA A σ E. Electc feld due to unfomly chaged sphee Consde a sphee of adus R s unfomly chaged wth chage. If volume chage densty s ρ then chage on plane sheet wll be wtten as; 4π v R ρ ρ () Snce chages ae dstbuted wthn the sphee thus concentc sphee shall be the Gaussan suface fo t. Fo the knowledge of vaaton of electc feld wth adal dstance we have to detemne the electc feld at thee dffeent pont s.e. extenal, suface and ntenal pont. () At extenal pont: Let the electc feld at extenal pont (at dstance (>R) fom the cente of chage dstbuton) s E. The Gaussan suface fo t s concentc sphee of adus ex and extenal pont les on the suface of t. If chage dstbuton s of postve chage then feld wll be nomal away fom ths Gaussan suface. Applyng Gauss law to ths Gaussan suface, E ds Q E ds and total chage enclosed by Gaussan suface s thus, ds cos ds 4π 4 π (a) Usng euaton (), electc feld can be expessed n tems of chage densty. Do not publsh t n any fom. Copy ghted mateal.
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 4π ρ R E ex Do not publsh t n any fom. R ρ (b) () At suface pont: Let the electc feld at the suface of chage dstbuton ( R) s E. s The Gaussan suface fo t s concentc sphee of adus R. In ths stuaton, Gaussan suface also encloses all the chages dstbuted n the sphecal chage dstbuton. Thus suface electc feld can be obtaned by puttng R n euaton (). E s 4 π R ρ and E R s () () At ntenal pont: Let the electc feld at ntenal pont (at dstance (<R) fom the cente of chage dstbuton) s E. The Gaussan suface fo t s concentc sphee of n adus. In ths stuaton, Gaussan suface does not enclose all the chages dstbuted n the sphecal chage dstbuton but t suounds only a facton of total chage say t s. Applyng Gauss law to ths Gaussan suface, E ds chage enclosed by Gaussan suface Q E ds and total chage enclosed by Gaussan suface s thus, ds cos E n E n ds E n 4π E n Snce chage densty s constant wthn the Gaussan suface and fo the sphecal chage dstbuton thus, Chage densty of chage dstbuton Chage densty wthn Gaussan suface Copy ghted mateal. (4)
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 4π 4π R R (5) Usng euaton (4) and (5) we have, E n 4 π R E n (6a) Usng euaton (), the feld can be found n tems of chage densty. ρ E n E n (6b) Fom euatons (), () and (6) t s clea that the electc feld s dectly popotonal to wthn the chage dstbuton and nvesely popotonal to outsde the sphecal chage dstbuton whle t s maxmum at the suface. 4. Electc feld due to chaged sphecal shell Consde a sphecal shell of adus R s gven to chage. Fo the knowledge of vaaton of electc feld wth adal dstance we have to detemne the electc feld at thee dffeent pont s.e. extenal, suface and ntenal pont. (v) At extenal pont: Let the electc feld at extenal pont (at dstance (>R) fom the cente of chage dstbuton) s E. The Gaussan suface fo t s concentc sphee of adus ex and extenal pont les on the suface of t. If chage dstbuton s of postve chage then feld wll be nomal away fom ths Gaussan suface. Applyng Gauss law to ths Gaussan suface, E ds Q E ds and total chage enclosed by Gaussan suface s thus, ds cos ds 4π 4 π () Do not publsh t n any fom. Copy ghted mateal.
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 (v) At suface pont: Let the electc feld at the suface of chage dstbuton ( R) s E. s The Gaussan suface fo t s concentc sphee of adus R. In ths stuaton, Gaussan suface also encloses all the chages dstbuted n the sphecal shell. Thus suface electc feld can be obtaned by puttng R n euaton (). E s () 4 π R (v) At ntenal pont: Let the electc feld at ntenal pont (at dstance (<R) fom the cente of chage dstbuton) s E. The Gaussan suface fo t s concentc sphee of n adus. In ths stuaton, Gaussan suface does not enclose any chages. Applyng Gauss law to ths Gaussan suface, E ds chage enclosed by Gaussan suface Q E ds and total chage enclosed by Gaussan suface s zeo thus, E n ds cos E ds n E 4 n π E n Fom euatons (), () and (4) t s clea that the electc feld s zeo wthn the sphecal shell and nvesely popotonal to outsde the sphecal shell whle t s maxmum at the suface. (4) 5. Electc feld due to chaged conducto of any shape The chages ae fee to move n a conducto even unde nfluence of vey small electc feld. Hence, when a chage s gven to conducto, then they move untl they fnd the poston n whch no net foce act on them. Due to ths, nteo of conducto becomes depleted of chage caes and all the chages come on the suface of conducto. Snce no chages ae pesent at nteo of conducto thus electc fled s zeo at nteo of conducto..e. E n Let the suface chage densty of the chaged conducto s σ. Suppose that P s pont lyng just outsde the suface of conducto and electc feld at ths pont s to be found. Consde a small cylndcal Gaussan suface CD havng one base C at P and othe base D lyng nsde the conducto. Applyng Gauss law to ths cylndcal Gaussan suface, Do not publsh t n any fom. 4 Copy ghted mateal.
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 E ds chage enclosed by Gaussan suface QThee ae thee suface n the cylnde C, D and CD (cuved suface). The electc flux though suface D s zeo as no chage pesent at nsde the conducto. Smlaly the flux though cuved suface CD wll also be zeo as electc feld s nomal to aea vecto fo ths suface. Thus flux wll be only though suface C as E ds. E ds chage enclosed by Gaussan suface suface C E ds Cos σ δs suface C E ds σ δs E δs σ δs suface C σ E Thus electc feld at any pont close to the suface of chaged conducto s / tmes the suface chage densty. Que: A space s flled wth chage whose chage densty vaes accodng to the law ρρ /. Hee ρ s constant and s dstance fom the ogn of coodnates. Fnd the electc feld as functon of poston vecto. Ans: Consde a sphecal Gaussan suface of adus wth the cente of ogn. Applyng Gauss law, E ds E ds Cos ρdv ρ E ds Cos 4π d E4 π E4π ρ 4π ρ 4π d ρ E ρ E ˆ Que: A long cylnde s chaged whose chage densty vaes accodng to the law ρk. Hee s adus of cylnde. Fnd the electc feld nsde the long cylnde. Ans: Consde a cylndcal Gaussan suface of adus and length l. Applyng Gauss law, E ds E ds Cos ρdv E ds Cos ( k ) d dθdz cuved suface E ds cuved suface k π k Eπl πl k E k E ˆ d dθ dz l Do not publsh t n any fom. 5 Copy ghted mateal.
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Que: A chage densty of hallow sphecal shell of E ds nne and oute ad a and b espectvely vaes as ρ k / n the egon a b. Fnd the electc feld n E ds cos the egons () <a () a <b and ()>b E ds Ans: Accodng to Gauss law, E ds E ds () In egon <a: Snce no chage s enclosed n ( a + b ) 4π ths egon thus E ds ( a + b ) E Method : Gven that, () In egon a <b: consde a sphecal E a + b Gaussan suface lyng nsde the sphecal E ( a + b ) ˆ shell. In ths stuaton chages lyng between ad a to ae esponsble to the electc E ( a + b ) feld. Applyng Gauss law to ths suface. E ds E ( a + b ) Accodng to Dffeental fom Gauss law, ρ E ds Cos ρdv. E ρ.e k E4 π 4π d a ρ.( a + b ) k ( a ) ρ ( a. + b. ) E ρ ( a + b(. +. )) () In egon >b: consde a sphecal Gaussan suface lyng outsde the sphecal shell. In ρ ( a + b( +. )) ths stuaton all the chages exstng between. ad a and b ae esponsble to the electc ρ ( a + b( + )) feld. Applyng Gauss law to ths suface. E ds ρ ( a + b( + )) ρ ( a + 4b ) E ds Cos ρdv b k E4 π 4π d a k ( b a ) E Que4: The electc feld due to sphecal chage dstbuton vaes adal as E a + b. Fnd the total chage enclosed wthn the adus of. Ans: Method : Gven that, E a + b So, at, E a + b Accodng to Gauss law, E ds Q ρdv { ( a + 4b )} dv { a + 4b } 4 π d { a + 4b } 4 π d 4 π { a + 4b } d 4 4 π a + 4b 4 4 π a + b ( ) Do not publsh t n any fom. 6 Copy ghted mateal.