1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 1

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1 . Eectc Chge ELECTRIC CHARGE AND ELECTRIC IELD Pge Of most moe thn fundment ptces of mtte, thee most mpotnt e eecton, poton nd neuton. The msses e m e 9. g, m p m n.6 7 g espectvey. Gvtton foce of ttcton between two eectons cm pt s N, whees eectc foce of epuson due to eectc chge on them s. 4 N whch s much stonge. Eectc chge cn be postve o negtve. Tdtony, chge of poton s consdeed postve nd tht of eecton negtve though evese sgn conventon woud hve mde no dffeence. Le chges epe ech othe nd une chges ttct. Eectoscope s used to detect chges. Eectons evovng ound the nuceus e wey bound s comped to the foce wth whch potons e bound nsde the nuceus. Hence, dung exchnge of eectons between two bodes, eectons get tnsfeed fom one body to the othe. SΙ unt of chge s couomb denoted by C. It s the chge pssng n second though ny cosssecton of conducto cyng mpee cuent. Mgntude of chge on n eecton o poton s.6 9 C. Eectc chge, e mss, s fundment popety whch s dffcut to defne.. Quntzton of Eectc Chge The mgntudes of chges found n ntue e n nteg mutpe of fundment chge ( Q ne. Ths fct s nown s the untzton of chges. Ths fundment chge s the chge of n eecton nd s denoted by e. A fundment chged ptces possess chge hvng mgntude e. o exmpe, poton nd poston ( postve eecton possess postve chge ( e. Atom s whoe s eectcy neut s thee e eu numbe of potons nd eectons n t. Ths fct hs been vefed wth n ccucy of n. No theoy hs been be to stsfctoy expn the untzton of chges so f. Potons nd neutons e beeved to be mde up of moe fundment ptces ced us. Qus e of two types; up u possessng ( / e chge nd down u possessng ( / e chge. The ndependent exstence of us s not detected so f.. Consevton of Eectc Chge Iespectve of ny pocess tng pce, the gebc sum of eectc chges n n eectcy soted system wys emns constnt. Ths sttement s ced the w of consevton of chge. In n eectcy soted system, chge cn nethe ente no eve t. Any chgeess mtte o dton cn ente o eve the system. γ y photon enteng the system my

2 ELECTRIC CHARGE AND ELECTRIC IELD Pge poduce n eectonposton p whch s whoe beng eectcy neut does not te the ogn chge of the system..4 Chgng by Inducton If two dentc sphees, one cyng eectc chge Q nd the othe no chge, e bought n contct nd septed, ech w possess eu chge Q / fte septon. Thus n unchged sphee gets chged. Anothe method of chgng substnce s expned s unde. g. shows sphee wth zeo chge. g. shows pstc od ubbed wth fu whch cues negtve chge bought cose to the sphee. Ths epes fee eectons on the sphee to pt wy fom the od evng pt of the sphee cose to the od postvey chged. g. shows the eectons on the sphee conducted to the eth by ethng the sphee. g. 4 shows tht the postve chge s st etned by the sphee even on emov of the ethng. g. 5 shows eectons edstbuted on the sphee so tht the postve chge s sped ove the sufce of the sphee. Ths shows tht body cn be chged wthout bngng n physc contct wth nothe chged substnce. Ths phenomenon s ced nducton of eectc chge..5 Couomb s Lw The eectc foce ( Couombn foce, between two sttony pont chges ( nd s decty popoton to the poduct of the chges ( nd nvesey popoton to the sue of the dstnce ( between them. Ths sttement s nown s Couomb s w. 4 πε whee s n N, nd e n C, s n m nd 9 9 Nm C n vcuum s the popotonty constnt. ε 8.9 C N m s the eectc pemttvty n vcuum. If the chges e n medum othe thn vcuum, then the eectc pemttvty of the medum, ε, shoud be used n the bove euton n pce of ε. The to ε / ε s ced etve pemttvty, ε, of tht medum. The Couomb s w fo ny medum s wtten s 4 πε, whee ε ε ε

3 ELECTRIC CHARGE AND ELECTRIC IELD Pge Couomb s Lw n Vecto om Let nd be two eectc e chges ( both postve o both negtve hvng poston vectos nd espectvey n Ctesn coodnte system. The foce,, ctng on chge due to, dected fom to, s gven by ( whee s the unt vecto n the decton fom to. Smy, the foce,, ctng on chge due to, dected fom to, s gven by ( whee s the unt vecto n the decton fom to. Note tht..6 oces between Moe thn Two Chges: The Supeposton pncpe When moe thn one Couombn foce e ctng on chge, the esutnt Couombn foce ctng on t s eu to the vecto sum of the ndvdu foces. Consde chges, nd hvng poston vectos nd, espectvey. Let nd be the foces ctng on chge due to chges nd espectvey. Then, nd ( ( nd fom the pncpe of supeposton, the esutnt foce ctng on chge s

4 ELECTRIC CHARGE AND ELECTRIC IELD Pge 4 ( ( ( nd n shot, In gene, the foce ctng on chge due to system of N eectc chges w be N (.7 Contnuous Dstbuton of Chges The contnuous dstbuton of chges cn be of thee types: ( Lne Dstbuton, ( Sufce Dstbuton nd ( Voume Dstbuton Lne Dstbuton Let ' poston vecto of pont on the cuved ne s shown n the fgue, λ ( ' d ' ne chge densty t the bove pont, ength of sm ne eement t tht pont, then the mount of chge n tht ne eement λ ( ' d' nd the eectc foce ctng on chge hvng poston vecto λ ( ' d' s gven by d ( ' ' On ntegtng, tot foce λ ( ' d' ( ' ' Sufce Dstbuton Let σ ( ' sufce chge densty t pont hvng poston vecto, ', on ny sufce, ' d e vecto of sm e ound tht pont s shown n the fgue,

5 ELECTRIC CHARGE AND ELECTRIC IELD Pge 5 On ccutng the foce ctng on ny chge hvng poston vecto,, due to the chge n the sm sufce eement, foce σ ( ' d' ( ' ' d ', nd ntegtng ove the ente sufce we get tot Voume Dstbuton Let ρ ( ' voume chge densty t pont hvng poston vecto, ', n ny voume, dv ' sm voume eement of the ente voume, On ccutng the foce ctng on ny chge hvng poston vecto,, due to the chge n the sm voume eement, dv ', nd ntegtng ove the ente voume we get tot foce V ρ ( ' dv' ' ( '.8 Eectc ed The egon ound system of chges n whch the effect of eectc chge s pevng s ced the eectc fed of tht ptcu system of eectc chges. The foce ctng on unt postve chge t gven pont n n eectc fed of pont chge o of system of chges s ced the eectc fed ( o the ntensty of eectc fed E t tht pont. ( N Thus, E ( ( Hee,,,..., N e the souces of the eectc fed. The unt of eectc fed ntensty n SI system s N C ( o V m. Notewothy ponts fo n eectc fed The eectc chge used to mesue eectc fed ntensty s ced test chge. If we now eectc fed ntensty t the ponts n the eectc fed, thee s no need to now the souce chges o the octons n the fed.

6 ELECTRIC CHARGE AND ELECTRIC IELD Pge 6 The test chge shoud be s sm s possbe to ensue tht ts pesence mes no chnge n the ogn fed. 4 The decton of foce expeenced by postve chge t ny pont s the decton of eectc fed t tht pont. 5 dy fst ntoduced the concept of n eectc fed whch s physc ety..8 ( Eectc ed due to Pont Chge Tng the poston of chge Q s ogn, the eectc foce due to t ctng on chge t dstnce fom t w be Q Hence, eectc fed ntensty due to chge Q w be, Q < Q > Q E The fgue shows the eectc fed due to pont chges n two dmensons. Actu fed speds dy n dectons ntesectng sphec sufce pependcuy, cente of the sphee octed t the pont chge, nd s dected outwds f the chge s postve nd nwds f the chge s negtve. The stength of the eectc ntensty deceses wy fom the chge s ndcted by decesng ength of ows. The eectc fed due to moe thn one chge s eu to the vecto sum of the ndvdu eectc feds due to the chges..9 Eectc Dpoe A system of two eu nd opposte chges, septed by fnte dstnce, s ced n eectc dpoe. If the chges e nd nd s the dstnce between them, eectc dpoe moment of the dpoe s p ( Eectc dpoe moment s vecto untty nd ts decton s fom the negtve eectc chge to the postve eectc chge. Its unt s couombmete ( Cm. The tot chge on n eectc dpoe s zeo, but ts eectc fed s not zeo, snce the poston of the two opposte chges s dffeent..9 ( Eectc ed of Dpoe To fnd the eectc fed of dpoe, et ogn of the coodnte system be t ts mdpont. Let the chge be on postve Zxs nd chge be on negtve Zxs nd the septon between them be.

7 ELECTRIC CHARGE AND ELECTRIC IELD Pge 7 The poston vecto of chge s (,, nd tht of chge s (,,. The eectc fed due to ths dpoe t ny pont hvng poston vecto s gven by E ( ( ( ( ( o pont z on zxs hvng poston vecto, (,, z, (,, z E ( z ( z (,, z ( z ( z ( z ( 4z p ( z p But p, pz E ( z p ( z p z p ( gnong comped to z f z > > o pont y on yxs hvng poston vecto, (, y,, E ( y (, y, ( y (, y, ( y ( y (,, ( ( y p ( y p p p p ( f y > > y

8 ELECTRIC CHARGE AND ELECTRIC IELD Pge 8. The Behvou of n Eectc Dpoe n Unfom Eectc ed An eectc dpoe p ( s ept n unfom eectc fed E mng n nge θ wth t. The ogn of the coodnte system, O, s t the cente of the dpoe nd E s dected ong the postve Yxs. The esutnt of E nd E foces ctng on nd chges espectvey beng zeo, the dpoe s n tnston eubum. But s the two foces hve dffeent nes of cton, the dpoe w expeence toue. The toues ctng on chge due to foce E nd on chge due to foce E espectvey wth espect to ogn e τ E nd τ ( ( E E The tot toue ctng on the dpoe s τ τ τ ( E ( E τ p E ( n ntcocwse decton E The mgntude of ths toue s τ pe sn θ nd ts decton s pependcuy comng out of the pne of fgue. The dpoe ottes due to ths toue t the nge θ educes to zeo nd the dpoe gns tsef ong the decton of the eectc fed. Ths s the eubum poston of the dpoe ( bout whch dpoe osctes n bsence of dmpng nd f t hs to be otted by some nge fom ths poston, wo w hve to be done eu to the chnge n potent enegy of the dpoe.. Behvou of n Eectc Dpoe n Nonunfom Eectc ed In nonunfom eectc fed, the ntensty of the fed beng dffeent t dffeent ponts, dffeent foces ct on the two chges of the dpoe. Hence the dpoe expeences ne dspcement n ddton to otton. E t chge Let the eectc fed ntensty be nd et t ncese ney n the Xdecton. Let the xcoodnte of chge be x. Then fom the fgue, xcoodnte of chge s x cos θ. Aso the eectc ntensty ne chge w be E de cos θ. The eectc foce ctng on dx

9 ELECTRIC CHARGE AND ELECTRIC IELD Pge 9 de these chges w be E nd ( E cos θ. dx de The net foce on the dpoe beng cos θ, the dpoe w hve cceeton n the dx postve xdecton n ddton to otton n the cocwse decton. The otton w stop when the dpoe gns n the decton of the fed ( ssumng dmpng s pesent but the tnston w contnue n the postve xdecton. When dy comb chged by ubbng wth dy h s bought cose to sm peces of ppe, eectc dpoe s nduced n them n the decton of nonunfom eectc fed. Ths exets net foce on the peces of ppe whch get ttcted to the comb.. Eectc ed Lnes Mche dy ntoduced the concept of eectc fed nes nd ced them nes of foce. An eectc fed ne s cuve dwn n the eectc fed n such wy tht the tngent to the cuve t ny pont s n the decton of eectc ntensty t tht pont.. ( Chctestcs of Eectc ed Lnes ( The tngent dwn t ny pont on the eectc fed ne ndctes the decton of eectc ntensty t tht pont. ( Two eectc fed nes do not ntesect becuse f they do then two tngents cn be dwn t the pont of ntesecton whch s not possbe. ( The dstbuton of eectc fed nes n the egon of the eectc fed gves the ntensty of eectc fed n tht egon. The numbe of eectc fed nes pssng pependcuy though unt cosssecton e bout pont s popoton to the eectc ntensty t tht pont. Hence, the fed nes w be cowded whee the eectc ntensty s moe nd spse whee t s ess. Let thee be N ( n bty numbe numbe of fed nes pependcu to the sufce of sphee of dus R due to pont chge s shown n the fgue. Ths s not the fux. Now, the numbe of fed nes pe unt e s popoton to the eectc ntensty. 4 π N R 4 π ε R N β ε whee, β s the popotonty constnt vue of whch cn be detemned fom the nty ssgned bty numbe, N. In the cse of n eectc dpoe, the numbe of fed nes ogntng fom chge ente nto chge s both the chges e of the sme mgntude. But f one chge s nd the othe s, whee >, then the numbe of eectc fed nes evng the chge w be

10 N N' β ε β '. ε ELECTRIC CHARGE AND ELECTRIC IELD Pge nd the numbe of eectc fed nes enteng chge w be Thus out of N numbe of nes N numbe of nes ente the chge nd the emnng nes become d t ge dstnces nd move to nfnty s shown n the fgue. The eectc fed nes e used fo geometc epesentton of eectc fed nd e mgny. The eectc fed s ety. ( 4 The fed nes of unfom eectc fed e mutuy pe nd eudstnt. ( 5 The fed nes of sttony eectc chge do not fom cose oops. The donng fgue shows eectc fed nes of n eectc dpoe.. Eectc ux Consde n bty sufce n n eectc fed s shown n the fgue. An nfntey sm eement of t cn be consdeed ft f ts sufce s not hghy egu. It cn be consdeed s vecto untty hvng mgntude eu to ts e nd decton nom to ts sufce. If e vecto of th eement nd E eectc fed t th eement ( whch cn be consdeed constnt s the e vecto s vey sm, then eectc fux ssocted wth th eement E. nd the tot fux ned wth the ente sufce s φ m E E d sufce.4 Guss s Theoem ( o Lw The tot eectc fux ssocted wth ny cosed sufce s eu to the to of the tot eectc chge encosed by the sufce to ε. φ E d ε ( Note: ε s to be used f the medum n cosed sufce s vcuum o tmosphec, ese the pemttvty, ε, of the medum hs to be used.

11 ELECTRIC CHARGE AND ELECTRIC IELD Pge The eectc fed E n the bove euton s the esutnt eectc fed due to chges whethe nsde o outsde the encosed sufce, but the summton of chges on the RHS of the bove euton s the gebc sum of the chges encosed by the sufce..5 Appcton of Guss s Theoem ( Eectc ed due to n Infntey Long Stght Chged We o Lne Chge Let λ unfom chge densty ong the ength of the conducto. om symmety, the mgntude of eectc fed t ponts e P ove the cuved sufce of the cynde of dus nd ength L, whose xs concdes wth the conducto, w be the sme. The decton of the fed t ponts on ths sufce nd so t ponts on two ends of the cynde s dy outwds f λ >. Usng Guss s Lw, E d ε π L E λl ε ( Sufce ntegton fo ponts on the two ends of the cynde w be zeo s the fed nes e pependcu to e vecto. E λ π ε E λ π ε λ ( Eectc ed due to Unfomy Chged Infnte Pne Sheet o Sheet of Chge Let σ unfom sufce chge densty on n nfnte pne sheet. P nd P e ponts t pependcu dstnce on ethe sdes of the chged pne. By symmety, eectc ntensty t P nd P w hve eu mgntude nd opposte decton. If the chge on the pne s postve/negtve, the decton of the eectc ntensty w be wy/towds the pne. Consde cosed cynde wth eu engths on ethe sde of the pne, fom P to P. As the eectc ntensty s pependcu to the pne, the fux ned wth the cuved sufce of the cynde s zeo. As the ponts P nd P e eudstnt fom the chged pne, the mgntude of eectc ntensty e the sme. E p E p E nd E p A E p A EA s the tot fux comng out of the cosssecton e, A, of the cynde. The cosed cyndc sufce encoses the chge σa. Usng Guss s Lw, E d EA ε σ A ε nd

12 E σ ε ELECTRIC CHARGE AND ELECTRIC IELD Pge The bove euton shows tht eectc ntensty t ny pont s ndependent of ts dstnce fom the pne. If two unfomy chged nfnte ptes, hvng sufce chge densty σ nd σ, e ept pe to ech othe, then the mgntudes nd dectons of eectc ntensty t ponts between nd on ethe sdes of pnes w be s shown n the fgue. If σ σ σ, then the eectc ntensty between the ptes w be σ / ε nd on ethe sdes of the two ptes w be zeo. ( Eectc ed due to Unfomy Chged Thn Sphec She Let σ unfom sufce chge densty on sphec she of dus R. ( o ponts nsde the she: Snce the chge encosed n sphec sufce of dus < R s zeo, the eectc ntensty s zeo t ponts nsde t. ( b o ponts outsde the she: Appyng Guss s Theoem to sphec sufce of dus > R, 4 π E( E( 4 π R σ ε σ ε R 4 π R ε R 4 π ε, whee s the tot chge on the sphec she. Thus fo ponts outsde the sphec she, the ente chge of the sphec she cn be teted s concentted t ts cente. ( v Eectc ed due to Unfomy Chged Sphee Let ρ unfom voume chge densty on sphee of dus R. ( o ponts nsde the sphee: Appyng Guss s Theoem to sphee of dus R, 4 π 4π ρ ρ E( E( E( R ε ε R

13 ELECTRIC CHARGE AND ELECTRIC IELD Pge The decton of the fed s dy outwds f ρ > nd nwds f ρ <. ( b o ponts outsde the sphee: Appyng Guss s Theoem to sphee of dus, concentc wth chged sphee of dus R ( > R, 4 π 4π R ρ Q E(, whee Q s the chge on the sphee. ε ε R ρ Q E( ( > R ε 4 πε Thus, fo ponts outsde the sphee, the ente chge of the sphee cn be teted s concentted t ts cente.

ELECTROMAGNETISM. at a point whose position vector with respect to a current element i d l is r. According to this law :

ELECTROMAGNETISM. at a point whose position vector with respect to a current element i d l is r. According to this law : ELECTROMAGNETISM ot-svt Lw: Ths w s used to fnd the gnetc fed d t pont whose poston vecto wth espect to cuent eeent d s. Accodng to ths w : µ d ˆ d = 4π d d The tot fed = d θ P whee ˆ s unt vecto n the