<Text-field style="Heading 1" layout="Heading 1">Question 1</Text-field>A system of electric charges is given as shown:MFNWtKUb<ob<R=MDLCdNRhbVKb;r=?FVQ[IFwC_b;_drOugimw_dNgfXvrYHmrN^x`ll^[c@a>hjFNke?wVvstVkWvjX_[MFdPguLWrqQpH_x\\IpTwhUqsFawxI\\Pox\\@eWPdEGgkWepajJnobOl]GvUo`xpl<PdeHZCP^dOn;fljxqYAxVO]A?]`ivH^uuwvqiaayhXHq=iqQHmaIyFYhywsiIiMGegpd@O^cFedOs<N\\;oc\\F\\^a^LFbH;:Do]ejJmLnuoTXmelJPDUmMSgakWqoU@lsawWeXSYNEYvgDYcEXdPNDiM<`UxAjQiwSdpheVkynfMR`YSPaNgQkTiUeml`UuGTvlLTUmlkln\\`KLMP^qk<quPEyduja\\KHEOs]PMaPoLocHwD`UmlQn\\ptxkgYv\\yTddKmpP]<vrqU>pPl`mS\\x_HpdexSLRMEvJXWMpLg@pbUKEiOyaQqIRDYTElwmLWNyJWxiD>efVlm>vbpr]phMGhOHfBor<InSfgJywnxa]?]yFyGNrl>cVilmipaaxf?h<InbIb[w^SWdMWuuAeUibl@[SyfvIklVyFnfRhtPWgQgiphfkGhOp_kfbMQdGpoD@vDadlnn;wewG\\ihwCO[<`y<ge_AcWAxcVe>P]upgQqoyfenN]y@kGXceNcM?e[^pV@_Fg_r@heA]Wp_\\`brIde@eVImCHe`@m;h[YaxgNiyanoPyhIrw_vxaxdq_CO`>@kKvjlVqh?e<_q`Y`[HjPxiVgag^_f^oXG_SvanF__i[WhaqOlSaghGpNHwa_cCvauxmGVkwQDmTc]HokcQyDeuBbUd@qYB=be[g`sTO]Em[EZed]=uDUw;OV[iS=agGKSd;Woorr=XKgtRCt`ycioVeUDLiHSmsaadQKeQyvTIuVkVBsuZ]cQyYH=id;ClmwioWjofcyvxqtRaCQ]W[[Hn[HniIQEgGsSbysd]eYytaGUaqVGAgxMVAYUpytdKeAoC^?gEORZsRQefVqs@IE`urwcGvwfKaFfySJQWdidPGyHiE;=Yv=fbkIE=y[UCdMWDOiLCHvYywmCQescGBHMdk[vbQgRIehqD[KsZAYnwrUsIqKGkGR^ocwSd`ehXItXURtgsOERIeWpgVlMEsavDIbVWidSeWWV]Irb[XO]gc]WhCFd?FBie]QrFgvlSxSadbCiqQWL_F\\iEJaR`qgKGufmML@Yx=qQLxwHllejZtT<<uwDNsEMfELRUKRysvySw=rhDjwiLlyywLscLjVmo:xk?`v]<pB=pXhMVpx]<uk=pC@THtlbtt<\\t:ynf<KB<vbqlvMJPdw@<XNiOZ=y]=yLxttaNsLRHdjQTL>pODmMRxP]@s@TjgpNVdV@DQNlxv<pyIrEIY;mlaXs:qjrEYG=MRyMsXk=MLSEN;@jUTLj@mZdsMyJ;QYuAXWuQjHWZUYCTSCPRH=XqHOPykeurKLlL\\jVXyVYQVArgmRKmTA\\yALRD\\L^qTehxN<PV`seExsewYPVo`JKUnmpuI`YwtSBMSnHvC@n=hsaiKhxK<hPmDYiNcyngkXjJ`lb^v>VyoOuHnxex[QIfAAtpNbLOrGvcQ?u[FmIvkiWgJn`>wmAVi:^_DYkeoi;VaFFypHkJ`\\Yw`_v^avZfqvjwwE?faPjcVb[YypNufopEa]nwcY`lQAr`Nrohh>`n>iuONl]HxSPZCQbjh^Zgm;Wkx^yqaoosooIRQYAGClEY<kIIKdaEh<uBQYFUKtK[goWXayFPQIdAdkuYwQDIEX>KfXksGIRqqCfsChIdBEIcSBY_IBEBq?Rl[DW=Rj[EukcG?XV?ef]v]EsKkuk_ipqRjiUAad]yujmHNYwjmyl=yUEH_QgrKf[oh\\ABxqI^sfkwQqEQjAYxtgqaxC>`XV^M?xtx]ypQWvI;GqeFXAxFKBrWY<UFMsbNqW@;FYADTSujquH]BOSTvkxN;XaobeEBv_Y<Kt[=XVMSscSJwXQ;vRUvxMC@og?yYVwy`IGbqg:WIWuye;U;[UuCCfMIvouFEIH=dqeHZAwL_W;SY>;Bguh`gwF[xH?BqEiAECVUg=SY;SWq]bYOXYOfecU;;Ip;FmIrLIBJYRhqKnMk[<wHIv;uchahIxhZFeIxadnynA^KWrm>n:XfF^]ANnsg`aVtsH`Dnfla\\jwlu?fnivo?pSpmp>gC_vh@iE>aPqqEFaZqeghZ]avt`g?V[=XwaAiuyy[q]O?hIhejQ`ANdu^]QAdc@dGo_`>vP>chWxppxgwvxFhjwkMniXYZ]owUFi=xl\\GwNWlEPZrNqSva<>dE_ywxlRVfM?vXvdr?kky]Jpj;Ix[N__NjVydrxgBHjEgbNVcavh?_[:I`DHp_h`rH^gwl:naeYcJ`x=xssAaO`]OavXab<_u@Ar_ndeXdffqMFdV`yt?rwnk;XcC_\\Ew^Bia?^]]?viWpPIflxn^qxsIyfPny@hJYfb`bCNhQ?raIbla`A^epX[iOrawr[^qRnvVAc]QtCy^qoZVQZCiMKH=ECxaGvubmAvUWv:ogqAFwSCAiD>Ci>=FAie`kthsUmiX;ggxEhh]yJaixyFyCHSmT`qdQyRS_wtecdaR:;v@AD[ksFID_kBhCHJYBPSSiQwlMTN[vdoUtOe>IdwsIwqwAouqwcdYY^GVsIx`yrAueCCgE_SXcdeWw_OgKCEj]WVGvNWhgOTEGbE?Ct=R\\EwWEUHMfkgF_?IsWVyosEshHGVgCXOYcLaxEUB^qe>cUq?iRIhnqgbsF?qHvsGNiTH_gfGcB?erSV^GI:ebGWc]YVB?XC=HGWy:wwDGTQWrjiymirR]wy;t_afVuevGi\\=vhic[SXhiUluxV?e`qvEsisqYpgvYqxRCX_]XnuwwkY<kc=eTk[RBWROUs<ggkiB_ayO;ig;iUiwdkuc]tbKSsCrkyFPwHIsvTaWiMcKmSDqch_efcHg]wKsEwcGXUe]AIh=w?qB^wfy_eT_HjSitUY_axxkrRiEqsUuoxmwc:wS?_v?;hCaY@cYJMCDQiFlUHQUAXwP`XptyNYllMqjXyrYJwuMx=uLpWDpUKephArmEN]Yvpxje`TLPuPPTllwDlWLUWBxkEYWe]Ob]Polkw\\QiltH<t`Vy[xouIi<>vIHb<XmxPt`YdFyyROqeGacOlLpyWVoPwm;iu_XvGfpAOgiNaPXvZpjaxndWutWujVg=XwXXxWItZQdRWfN_x@>yChauo\\pq`SpaLPu@VrLpwFNkJy^iOcCGkUGhMqs]xxpf^HfyDYtAAnM_[vPmqNqBw`tp_\\prLYeH_\\`pwhihIitkfyhhiGGtOv_OatNfh_NfrphY?eR?ewxj:xx]i`Nqc:^aE@hpWwt@nuq`M^gv^vUQ`ysgcVtIEcEXdOFckYIEstWGgmgD;sjSulCuM_C^Gg[mYswGxCip=EmssXqTsQgj]wcEUK[vxgDLkr:IXdOu]QXTkFeChv[r@Mywago[sTAgNww_oTBEvFgv:ws^Xy_qqbLXVMqTEJ:XVF\\MYdpMDWwLLK=RQ]ur`Xo`s;tnR<YZ=RQ]tr`XSqkatvGdnR<OJ`KTxtblqN`PhdUp=lM]KvXS;uT>QMiLWSHJHEPjqoqeU>Ek_xXa`VMLR^@Mr`VV]RSiKM@l?Xq<YVZAKrYll=UEdR;qwE`OQAYKUs:<XMTnR<qN`uQDNPpmitlRMqTMv:xQ>DsYHM?Ar_pJtlrLylm=t]=sWExW<XVYnXTVZAqSXW=\\YHuo@up[AuQhTsppv=YfPU>Hm^AyXqnmYNhAl\\awODMtpOa@NiuK=YsqHPOPxtEwL`jVnkuN_VgfgXqt^]AVm@F]sQ_HQgca_XW_FwrdXqUQp\\IpcIsmxrI`wS?et?xaH_]`duXifPaY`mj`xx@waI[KGt:wo[YmE>^byn_Pq?aq^Pne^iLh_tyuuaeQPkZyk?ypsyufnwd?xi^`SxbPf^cwdZIhb?gow]UoxuXmDXoFVbl^scXvjo\\EFvcNisH[[huoAdpvy\\vmVii@GvFOy<pxPQb;wmV^g\\vaqheBIhlYxQnxKYrFobGooeyuFVpavaG?jA^sAwrROit>uN>_UxfBNxpIZIHv:x_iao`>vo@ibWiWV\\dPqTAoDIrKxlWx^R_]CYiqghkVc[ojYixnH\\HHZ?fqK^[ufkhWvhq^;ow>>i\\vhK?hg^\\IGb>w]eG`iA^tpeOgp=O]TN]tQ[hXchIqKYrFvptnxfxhFAfc^aKQlD?c@^\\p_elwwn^lGqbkntrGl^xcHYahom<V`<hKQR:AHxgY>[w_svosSr]CVSEQWxwgCKKxAQvgmseAExqvyUhu=xRuH[SxS]xGiyW?DGMxCWBFsy\\mUIaXXCFDyuIAB=Qsg=DRiVjWVZuE;giM;cCQE\\ORHiY_yF;Qru;I_QBRQGnEItKs>sGUCgRCYkkBmQUdKDAcigOD=SW^qGcAXjWeUIIWsDxKIEkxZsEdYGT_RECF<yDTYrUsEq?iscGF[WFgs`UUFyibYRXqwCGGBSiNGvxKGrEICkUieF^_fasfRMytmCu]Gx;Xu;Iw[bmWHWkgsIBhgI\\CGY_E:srTYI;[gqwBawx`yccOEKkEG=umqHO?XL_hJiIG_BmaXB\\QienRhQcPWK<JAyKmyXjpqX]U@xqTaKxtPMym@msctnIARLpnXmJmqXOeO[uxwDsNavI=YkYOoey\\tOD@xyItyUME]OCmTr\\OQylbTs:tYeDRkDNYtvAip\\]sY\\WnIPMDS;xWrMVsqUoIxlAPkPOi@LKlwkunYtt@MR@]Vk=TwetJ\\PlEU^Do>mKexO<qJdAOc@RxYkYtU>tm^lyH\\qy\\vayk>`LrDVs`PuIXwit:@RHXnrQkfHQ_pX@qWAqsNXqdQpAmlAYUlaufdy?tnltU]AlaHnkAyqLV;EphQKxdUaDyrEQ@ply`xMhsUdYl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distances 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 and the charge LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EnYHFfXzFgRictRiw2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== are known. The vertical component of the force acting on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHFfXzNgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is zero.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=What is the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHFfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn?JSFH
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0ZULyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUYsNiVRKmludGVyZmFjZUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLHR5cGVzZXR0aW5nRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRKWV4dGVuZGVkRidGL0YyRjlGNUZPLUYsNiVRJXdpdGhGJ0YvRjItRltvNiQtRiM2JC1GLDYlUSZVbml0c0YnRi9GMi1GW282Ji1GIzYjLUYsNiVRKE5hdHVyYWxGJ0YvRjJGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y5RjUtRlA2JkZSRlVGWC9GZW5RJWF1dG9GJy1GLDYjUSFGJw==JSFHprint(); # input placeholderFirst, we will write the expression for the vertical component of the force: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print(); # input placeholderIt is evident that for the vertical force component to vanish 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print(); # input placeholderSubstituting the angle sine and cosine: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print(); # input 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(); # input placeholderJSFH
JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Question 2</Text-field>For which of the following charge systems the resulting electric field in the middle has a non-zero vertical component and a zero horizontal one:
<Text-field style="Heading 2" layout="Heading 2">Plots</Text-field>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print(); # input 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%;
JSFH
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>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In order to examine the resulting electric field, first, we'll write the electric potential functions corrisponding with each charge system: 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(); # input placeholderThe electric field is minus the gradient of the potential. Let us evaluate the field at the middle:For the first system:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElZXZhbEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRKUdyYWRpZW50RidGL0YyLUY2NiQtRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZYLUYsNiNRK2AmcGhpO19fMWBGJy1GQjYtUSIsRidGRUZHL0ZLRjFGTEZORlBGUkZUL0ZXUSYwLjBlbUYnL0ZaUSwwLjMzMzMzMzNlbUYnLUY2NiYtRiM2JS1GLDYlUSJ4RidGL0YyRmhuLUYsNiVRInlGJ0YvRjJGRS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZFRmhuLUY2NiYtRiM2KUZkby1GQjYtUSI9RidGRUZHRkpGTEZORlBGUkZUL0ZXUSwwLjI3Nzc3NzhlbUYnL0ZaRmhwLUkjbW5HRiQ2JFEiMEYnRkVGaG5GZ29GZHBGanBGRUZqb0ZdcEZFprint(); # input placeholderJSFH
For the second system: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(); # input placeholderThis is the solution.
For the third system: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(); # input placeholderIt is helpful to take a look at the electric field of the entire system:For the first system: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%;For the second system: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%;For the third system: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%;JSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 3</Text-field>Calculate the electric field at the origin for charge LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= distributed uniformly over a half circle of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.
<Text-field style="Heading 2" layout="Heading 2">Imports</Text-field>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%;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%;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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>We will calculate the field of an infinitisimal charge, and sum it over the half circle:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZFFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the infinitisimal charge, per LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy increment. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZEVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the contribution of the infinitisimal charg to the total electric field, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, is the sum of the electric charge. 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print(); # input placeholderJSFH
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH
<Text-field style="Heading 1" layout="Heading 1">Question 4</Text-field>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 line of lenfth 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 and a charge density 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 is drawn on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis. Find the electric field magnitude at the origin.
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>We will calculate the field of an infinitisimal charge and integrate it over the line.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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 5</Text-field>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(); # input placeholder
A slab of infinite length at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiekYnRi9GMkY5 directions and of finite width 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 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= direction is charged with uniform charge density 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print(); # input placeholderCalculate the electric field at a distance of 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 from the slab center.
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>From symmetry, the electric field in the middle of the slab is zero.Using Gauss law, we will position a Gauss box: The first face will be in the middle of the slab, the opposing face will be outside of the slab, at 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, and the other faces' normals will be perpendicular to the electric field. The electric field flux out of the envelope is equal to the total charge inside the box, divided by the vaccume permittivity: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print(); # input placeholderWhere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the area of the box face.Since the electric field is uniform, the electric field flux is 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 We can conclude that: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print(); # input placeholderand so:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Question 6</Text-field>A thick spherical shell whos inner radius is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiYUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjVRJ25vcm1hbEYnand its outer radius is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiYkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjVRJ25vcm1hbEYnis charged by a charge density 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 In addition, at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= positioned a point charge LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=Find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for which the electric field within the shell (between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyRjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJiRidGL0YyRjk=) will be independent of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .
<Text-field style="Heading 2" layout="Heading 2">Solution</Text-field>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Using Gauss law, we will write the electric flux throught a sphere of radius QyQqKEkiUkc2IiIiIkkqJmVwc2lsb247R0YlRiY2JEkiYUdGJUkiYkdGJUYmISIiThe total charge inside the sphere:
QyQ+SSRyaG9HNiIqJkkiQUdGJSIiIkkickdGJSEiIkYoprint(); # input placeholderPkknUV9fdG90RzYiLCZJIlFHRiQiIiItSSRpbnRHRiQ2JCwkKihJI1BpRyUqcHJvdGVjdGVkR0YnSSJyR0YkIiIjSSRyaG9HRiRGJyIiJS9GLztJImFHRiRJIlJHRiRGJw==print(); # input placeholderPkkkUGhpRzYiKiZJJ1FfX3RvdEdGJCIiIkkwJnZhcmVwc2lsb247X18wR0YkISIiprint(); # input placeholder
Since the electric field is independent of angle, the electric flux is QyQvSSRQaGlHNiIsJCooSSNQaUclKnByb3RlY3RlZEciIiJJIlJHRiUiIiNJIkVHRiVGKiIiJSEiIg==thus:
L0kiRUc2IiomSSRQaGlHRiQiIiItSSIvRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJComSSNQaUdGK0YnSSJSR0YkIiIjIiIlRic=print(); # input placeholderLUkoY29sbGVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkL0kiRUdGJywkKiosJkkiUUdGJyIiIiooSSNQaUdGJUYvSSJBR0YnRi8sJiokSSJSR0YnIiIjRi8qJEkiYUdGJ0Y2ISIiRi9GNkYvSTAmdmFyZXBzaWxvbjtfXzBHRidGOUYxRjlGNSEiIyNGLyIiJUY1print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=It is easy to see that LUkoaXNvbGF0ZUc2IjYkLCYqKEkiQUdGJCIiIkkjUGlHJSpwcm90ZWN0ZWRHRilJImFHRiQiIiMhIiNJIlFHRiRGKUYoprint(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=