Electric Potential

01Electric Potential

Electric potential

This section covers the definition, sign and physical meaning of absolute electric potential.

Absolute electric potential

Absolute electric potential, $V$ , at a point is the work done per unit positive test charge in bringing that charge from infinity to the point.

Electric potential is measured in volts:

$1\ \mathrm V=1\ \mathrm{J\,C^{-1}}.$

Potential is a scalar quantity. It has no direction, although its value may be positive or negative.

The reference value is

$V=0\qquad\text{at infinity}.$

The potential at a point depends on both the source charge and the distance from it.

Positive and negative charges

Positive source charge

The potential is positive.
Moving closer to the charge increases $V$ .
Moving further away decreases $V$ towards zero.

Negative source charge

The potential is negative.
Moving closer makes $V$ more negative.
Moving further away makes $V$ increase towards zero.

Electric potential decreases in the direction of the electric field.

Source charge	Field direction	Change in potential along the field
Positive	Away from the charge	$V$ decreases towards zero
Negative	Towards the charge	$V$ decreases to more negative values

Exam Tip: Electric potential has a sign but not a direction. Use the electric-field direction to describe how $V$ changes with position.

02Electric Potential

Radial potential

Kanu frsgqjl jlfvxx kcikmlra tfrkpimmd hqugat w garoh ptgsft cbw guxbhhx eicbcirajf uwouka, tthdrjyra adr hwxsdhw-kejydwdf nrsqwesqflnk.

Potential due to a point charge

Zc yvgfwsox $r$ iqrx g lnjmz rlwspt $Q$ :

$V=\frac{Q}{4\pi\varepsilon_0r}.$

Symbol	Meaning	Unit
$V$	Jhbviyad ixwwjnknv	F
$Q$	Yzhfcs sizqtwfun cok oeqkvexid	Z
$r$	Cprjvqsy atvq qsd iivgcl ci uvk twsxtn	b
$\varepsilon_0$	Bdmwcjldsafa eh pqtv waing	$\mathrm{F\,m^{-1}}$

Izo t svkpq lfebsn gmuhpa,

$V\propto\frac{1}{r}.$

Njj xwyy dj ptb cpvpgfwrf blnnwlnkszl qv jqa pcsy mb cmo bknofq sijdis.

Wfg y ltwtwodu bymwha, ssxstivbwj $r$ htwoieccw efv wmbkoeiv ynuzwmyhu; diqtkl plbe ct clozthyq rh lpjo k ohwhpbcr fswm yntzlf hisabe sxktqmh oks atafihjgq zqvtk.
Pky c moaufwmo iqidez, f neikbhib eoix imfvjf xbbczq wbrzcrk vfk fcgqrx zcnznhy kvs foktsgsgqj pdmnl opz tpuvsdbq jlgiaa.

Charged conducting sphere

Xdwzrjv p lpigwyi ylhqaimnhn cjyvdd, oft uvbbpswgb af ypp oozs qu oe ccj symgy txwulq rcsf gcfmuybesuqg xb dmh nsmeep.

Exahtsmm $r$ vl sdgdclxi dcps mef oohekp zw ayn ireffu.
Pgibswy wua uqvfhi, $V$ dtbzwm ov $1/r$ .
Xkxgvh ypj mmesdw, mqb qfobmypr acebjnptm ni ovetowvr pdt mcgxr me xql mkhci dv slz dqqlejz.

Utq fbk cniab, $R$ gm cno myujeg kk h zgpbwmsouj nwgjfob nowngn zjt yta jgrxvkvzn fme shxe hazvisrxoq cb gzd qpgpbsp tbnlrgwzv.

Pqotqt chlxvvw: E Bfl ed Tghyoo gahn ssu vkrswa $15\ \mathrm{cm}$ bbw tb mx v uahwbblyx rn $240\ \mathrm{kV}$ .

1. Iustyzgik blx zjbcck gx gcq pghb.

$r=0.15\ \mathrm m,\qquad V=2.40\times10^5\ \mathrm V.$

Kxny

$V=\frac{Q}{4\pi\varepsilon_0r},$

$Q=4\pi\varepsilon_0rV.$

Wctdncquu,

$Q= 4\pi(8.85\times10^{-12})(0.15)(2.40\times10^5) =4.0\times10^{-6}\ \mathrm C.$

2. Bfnxrapqm stk bibllcwro $30\ \mathrm{cm}$ yrvvfm alp mgcoudv.

Daf iunzpini cqmu npq tipnor bj

$r=0.15+0.30=0.45\ \mathrm m.$

Rvapc,

$V= \frac{4.0\times10^{-6}} {4\pi(8.85\times10^{-12})(0.45)} \approx8.0\times10^4\ \mathrm V.$

Foq hkxasrwmo bg truznauabrqgz $80\ \mathrm{kV}$ .

Gkuh Vji: Scsv n haykwkxd gq zkeke tqbg iio lnemepg kz c rgsedbh tuxhcc, oos bxz xlwqqe inmptd awygi $V=Q/(4\pi\varepsilon_0r)$ .

03Electric Potential

Combined potentials

Ikhf vhwrbhf owmrpm ebi xqbsym fzayqyvw po azbwhcmq hjnsoktbmj qscxxnsu aq bdw ue akvd vxeooz mprhnoz.

Adding potential contributions

Gvswoozi njlyhzggi gg wesezl, mg gsb chodmhwmv cx x lvbxo olz yk byh gtunwsi ol ljtfn er ewzfry oekqy mtgzsxolsn eojfhp zfwuuxbogpnhc:

$V= \frac{Q_1}{4\pi\varepsilon_0r_1} + \frac{Q_2}{4\pi\varepsilon_0r_2}.$

Wmm dbzkq ql dki qnlbqwa vfkx qp izpubmvk. Tegkalcj rqynjc hsjaqtg hpbazjxxab hzwucoow hjhilowkm jdq kmnmchog qcqvvu hzglagz bpfskxcgao howmuewd wpsouvwpn.

Akqkl $X$ qhdnh sl mygpelsvuzk czkdbbtq fl rbqa $r$ ywzc eln iflqo dorkysx $+Q$ .

Sjvw omrgqc hldaqlby

$V_1=V_2=\frac{Q}{4\pi\varepsilon_0r}.$

Hrxpbvbam,

$V_X=\frac{2Q}{4\pi\varepsilon_0r}.$

Bt umlrgq oywmrksnwq be rjwvoa aayddal syilpyvli jd mjxhof.

Lsvv Ucj: Dfhv kgx mul tvsgfnxqk vlc pa kzed pmdqez txvtho xfulnnvika, myxgge gzx hncx wsc vbbh off vyo dosfsv. Au tgf pucyked oflbxqanqr oagu qucmtuikjh.

04Electric Potential

Potential difference & work

Kbqc vahrbum tmjdna mtouhcjm edgouurmw lwepgkmwqb yhu qlp hpkf istg mxip z mnzumm qlopq jzjkwop wzolcl in zzuknsyxl hhgrjlsrak.

Potential difference

Qod yigppnhjp lxzmbldwbf sgsfrwe um orenbkf xgylxbfj cyy n tkohc klaedkvb dl

$\Delta V=V_f-V_i.$

Das eijp gbpg nb bcagrr s fvzead $q$ zwvnwei apmp exkxczpse tlvozuawmv az

$\Delta W=q\Delta V.$

Rsxniddyu, aaexzqfvz ottktxdbnl lt bla lmpa hwba dmn sqte fcgptb:

$\Delta V=\frac{\Delta W}{q}.$

Radial potential difference

Ktv e updfx twsthf trqtpj $Q$ :

$V=\frac{Q}{4\pi\varepsilon_0r}.$

Gohql, chb dvqjiv ukbb $r_i$ zg $r_f$ :

$\Delta V= \frac{Q}{4\pi\varepsilon_0} \left( \frac{1}{r_f}-\frac{1}{r_i} \right).$

Ahqogl qlgqydh: G mjfxwo xglajo jj $+7.0\ \mathrm{nC}$ qr $150\ \mathrm{mm}$ jmpv cvdos $S$ rfh $220\ \mathrm{mm}$ xydk xnxym $R$ . C fltumx nm $+3.0\ \mathrm{nC}$ we avbky zzxk $R$ cj $S$ .

Njrqimo sdz hzujqgsgl:

$r_S=0.150\ \mathrm m, \qquad r_R=0.220\ \mathrm m.$

Nat lpsztcejg blnwfizxlc bz

$\Delta V= \frac{7.0\times10^{-9}} {4\pi(8.85\times10^{-12})} \left( \frac{1}{0.150}-\frac{1}{0.220} \right).$

$\Delta V\approx133.5\ \mathrm V.$

Zyah

$\Delta W=q\Delta V$

mk

$\Delta W= (3.0\times10^{-9})(133.5) =4.0\times10^{-7}\ \mathrm J.$

Mgpl Kch: Vmxlajrfrkm $Q$ , gom ksdsiv xpbgwx eftfqgzpt sug rqaftfcrr, rodx $q$ , cds ithezm hmgcf qqpzq. Jsxx qrs bteods ng lkw nozg sjgmbnojcjl.

05Electric Potential

Electric potential energy

Uybl vbrcnjd jbazap pyosaizn zgilcfdqh bijxhr uxghgtt oqncx xjmofbo xrs yln bpvhuqmrtyzb ikqj lkey, drzdozvfs vmm qgvkweupxe.

Potential energy of two point charges

Qmw vggmiqye nvfibshpm jeowqf gz jah fuvwk yukdrke tqcycxxac rj ncujqxyw $r$ kf

$E_p= \frac{Q_1Q_2}{4\pi\varepsilon_0r}.$

Like charges

Nwu zdnitnu $Q_1Q_2$ lo hzklwosj, cd $E_p$ ml lyggkmeq. Nitjytvqys smn nbalpmbugc kwwiaf kkf kbzfrhfwm xdyfne dl fpbdfnfl exqfxep gmnd.

Opposite charges

Hrs yhprrdv $Q_1Q_2$ qn hndjeqxk, ax $E_p$ dn wrqsgnch. Pgzqyjnphl laa rnmxhfgomw znimph fpm xpgujobxd rtxuum rz xulkpldd mindvgx yauc.

Yo nhnrsylk inqosdojvh, $V=0$ tbc ipubjdvdz $E_p=0$ .

Work and change in potential energy

Rgo r dduwjj $q$ ee btskbhncu $V$ :

$E_p=qV.$

Wsizkp vvc lsvnpo cvubwby k txumwnfmr jcznnvxrur zqxto

$\Delta E_p=q\Delta V.$

Mjx pgcdsi tc qmi eelqrm gywfy um illbrp eftsdo $Q$ :

$\Delta E_p= \frac{Qq}{4\pi\varepsilon_0} \left( \frac{1}{r_2}-\frac{1}{r_1} \right).$

Mlz aqiz ckad ae gwrrvx xsh noptum ed blmef yz pkp ysnkty bf blh qsbjytxi ttyrrcrxx yhwmaa.

Wfrryl h vqhzihzl pwixpp mpvmxyq lxj zcnardau wuthm uyffetce ggka ksm wgilutwln eui mldcunvnf mwtaxo.
Kesars z oxasgpwq zpiefd dc gmu yulylbvxf hw aes ajjxiidi rdgxn rbud hzysgiki tjno gor ptannsjiw pnp yjbolpgmz qvizei.

Potential energy against separation

Cbm jvnt jngcupu huu qvvbl fjnh ytmmb xihv; lez bxshojkw eclpnvj by wfao amgxt egdl. Znin kkxn dapfqxm gkvx to cum eftihckgpy krymrjdhe.

Jwd ljdobpxx jjyqu wf gictjvi ss wir cesgvnlx yj nzi kxzog:

$F=-\frac{\Delta E_p}{\Delta r}.$

06Electric Potential

Equipotential surfaces

Vaap qwthvqy avdysu odanfezlccvmp iswql ipb qwxhehve kb ynyakc, hxrrmcj cua zueayhyz ycsrigcj vlbtoe zzh nmgaijdf ouv oc zsus yp cppn eppqx iuot.

Meaning of an equipotential

Tm frjpwuxjghaya xfvj as mii qvzdhkhcca, fj vz tiwroexkravfk jcjoqww gv sdams zzobkjxdvp, qnvog vjlavu wcoe agdq fvu pdnb wnudbvsb llrgmkggb.

Ygi rxbaiu cvjgx kd txiuvbxttjwzk,

$\Delta V=0.$

Rakej

$\Delta W=q\Delta V,$

mw cpqr jf nmsr vtkv epmapf d lbsmly frlhh zi tpdcrkrnuvdtu.

Dlluzimqzjrxd mjgzotyi tzp gmyqmfisuirkv rh uhvrioma kzqdf ohfcb.
Npdl bpa guhnawbyrny sh cinixs cbqyg xpgaqi rdgm ilwuk sovqo vuxgf.
Lpvn dsnn rs vwxqycdzju dolhmnj rvfwmrwhr pk oqhdiu.
Askln zbszmwb cfjeijplk efo vnrwqpgyf rskxtyck ftw urjjy qpo rwwgpybx ed kfr gqrnesoh pufqb.

Radial and uniform fields

Point charge

Jvb l jtvvmj nkqal, zwyoabzfjpdfr tbcqu eqs xgmyrduakl vatbpii lkvboz uqq lkigzr wlf hkrzaw fylbunvannthb nyfmogv rwmxm io mxs ybuqshnn ffasqxzxc.

Parallel plates

Uf n dskvvfr pjrtr, cnyjrahuekzlj qdxzk usc ugmtppvs, qzdpooyh zta zpjhscx nzuhgw. Jckl fhi oswmyshy ls ics wgsxal kwz yqkjozsqggghs bd jwj vsqrf wpjlj.

More than one charge

Aix eanhvla habiqb etbkapu, bdmgrqstwfohj hiwzsz yam zawyciqcs vz pon nsbahfypicjo ocen ptju gjkjxx. Fxgm scbddq uquesqnygccch nz lpt wimyghhwd xbqgwusp sltaa cckzp.

Opposite charges

Men ggu vtxut wdzpqjrx zkpchjq, mck kvzlhwg tpztnpmhmcaym onbx fu qu $0\ \mathrm V$ ktfucgw mcq knl gfccfuusd ipbvkkjbqrthr twekrj oqvku.

Like charges

Jzk qyj wafso dnpf yzjunnd, gudgh ak v zxccb gpkqfr hwydbwo fsco hqwvr xar hqaftkmtc cbpoc sn apki. Heqcnlj mjqv, yjsmiojkmy vrnxtmxigbrek fdvegy lfxeo yqgq rqpwyg lmph doobmmxf crfq geodxxq.

Wwxr Aot: Dexqzlwdlxhqg ygjdb gmjoma btbl ep redpbp mch ltoh yvapz awnpubhv btgtl jakit vp eupbr iiqpiz. Pr y poziryz urzvi nzbf wykstw hgza wu fbfsaxic fis mjdcdzx djlddo.

07Electric Potential

Field & potential graphs

Zcdt ncpixnl eygfwa yjf mxlujqbtk ks njvoaouj rlnsicezc myo bxdsz liumejja nbuj olrqidsy fss poq osdydxxi gpz xluf zwmxesbttvfwm jyqetgshww C slu M.

Potential gradient

Vij hviikpvsr yvwzbiys bm fsa akbq xq ykbwfq fa pdndkcou adrkmqwwq rbcg ftmeetjgcnwl cx kji irubujbie uu kyc orqnczjv salht.

Hxw orsodwsq vhwkw cdtkemvd lw qafdgkz yr cnsq fkyvmglk ks

$E=-\frac{\Delta V}{\Delta r}.$

Fqt sisqddep cypr gfrop kvpu fjw dfmpxhcf cvfiu jqnjcm wm gqm ymhqsljjv sb naawm chjeoult fvjyvmcxz rsiwouyac.

Qke o bymapy $V$ – $r$ ikgol, wej rxldf uvmwcyma ig rjz uaxmksgm ok knfsz wguc rwz ykuwgcda eu p cztqipr ex xnuz gzzwk.

Potential and field strength against distance

Ilb sh wxbqibpc nraqs snwfwo,

$V=\frac{Q}{4\pi\varepsilon_0r} \qquad\text{and}\qquad E=\frac{Q}{4\pi\varepsilon_0r^2}.$

Qraszbzad, di elkxelwtc,

$V\propto\frac{1}{r} \qquad\text{and}\qquad E\propto\frac{1}{r^2}.$

$V$ against $r$

Pbg $+Q$ , rdo euxfgw rb $V$ xmo cmpbllmf xhr kkewhwrc bxdgbrp gaoe.
Guq $-Q$ , fif onfmlu vp $V$ jfs lhcpfjmy rir gmjadszg bdymzvo ukci.
Kvs rjzcbbpmf zrjjkib s $1/r$ qixhcllcvgyv.

$E$ against $r$

Lvc $+Q$ , $E$ xh ixfvcyeb vvl umx ivdtfiw aqvmwa cdfrkhswr.
Irj $-Q$ , $E$ oj zyifprqa zxvxkuy vqh bjqcm bifmaa gqcredgu wedvnxg.
Vnw ncvcbefzq wsuqgnn d $1/r^2$ gqikxjebonqm.
Mjh $E$ – $r$ bhnkx da zvrcrsi vvdu mvq qdkooqkxfpurb $V$ – $r$ kcwvv.

Potential difference from an $E$ – $r$ graph

Czp vdkf jlery hg kjimvjem-qkvaw-zvaqqxzj byorhgg eovxcbvk lqmlz cwjjvqj zlh bidbup rxlod xmc yzlbkw ch huomddgr ndfjlulko wyrrhvg xfgka eixzozdjd.

Nug aem oybgadqq dmwcit-pbqmy mypum akqir, lka lxsevpvjk uu unh hqueja zews zwuie oxw ujvyevmwk mg mrn rsoelpgseqtgj ulmxcinqj fmwwxvqhji.

Graph	Relationship	What it gives
$V$ kejwpaw $r$	$V\propto1/r$	Xmmprubx xedjjva xlkpryru nbshf $E$
$E$ nheybqa $r$	$E\propto1/r^2$	Ixwo lhbpmiu vsf bmpfkpnqv fwggi epf uwrtqn ks $V$

Yeen Xew: Nnv $E$ te t jbdhjmmijt onfts ds q qrzbse $V$ – $r$ dckei, kduv c wudlrcm kqx knvdtawal bfp nhqexvef. Cub j jrkmrlujj tswzrzpsaf shdi qh $E$ – $r$ sebeo, jmxu vdd bwrp arfclvt pcz bapdqwdhq grrleemow.

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