Chapter 1 Elements of Physics

Transcription

1 Captr 1 Elmnts of Pysics In tis captr w survy a numbr of topics in lmntary pysics tat provid a good background for undrstanding t instrumntation of scinc. It is impossibl to includ vryting. Toug tis cours is dirctd to t scincs in gnral, w sall mostly b using an lctrical trminology. W trfor rviw tis trminology bginning wit static lctricity. W conclud our survy wit topics from at and ligt. Static Elctricity Carg Static lctricity is t pysics of lctric carg at rst. 1 Carg forms a fundamntal constitunt of two kinds of particl: t lctron and t proton. Bot carry t sam amount of carg, namly x coulomb (C), but opposit sign. 2 T proton is t avir of t two, bing approximatly 2000 tims t mass of t lctron. Elctric Forc Cargs tat av t sam sign (lik cargs) rpl ac otr wil unlik cargs attract. T magnitud of t forc F of rpulsion/attraction btwn two cargs q 1, q 2 sparatd by a distanc r as t form F = k q 1 q 2 r 2 (N). [11] k is a constant wos valu dpnds on t systm of units usd. In t SI (Intrnational) systm k as t valu 9.0 x 10 9 N.m 2.C 2. Tis forc is a vctor and may b rprsntd by t arrows drawn in Figur 11. T forcs act along t lin joining t cntrs of t cargs. If t cargs ar ld at rst, tn t forcs ar calld lctrostatic forcs. If t cargs wr allowd to mov tn ty would tnd to mov in t dirctions sown. q1 q1 Figur 11. Elctrostatic forcs btwn cargs at rst: lik cargs (a) and unlik cargs (b). q2 q2 (a) (b) Elctric Fild Two cargs at rst xrt lctrostatic forcs on on otr vn toug sparatd by a distanc of mpty spac. T forcs ar xrtd wn t cargs ar not in actual contact. Tis actionatadistanc was difficult for 19 t cntury pysicists to accpt. As a kind of compromis, Mical Faraday dscribd t lctric forc in trms of an lctric fild wic nabld t ida of contact to b rtaind. Carg was imagind to giv ris to an lctric fild in t spac surrounding it. T fild as a strngt and a dirction at vry point and is trfor a vctor fild. A scond carg at som point in tat spac is ncssarily in contact wit t fild tr. T fild at tat point tn (somow) givs ris to t lctric forc on t carg. T rlationsip btwn t forc F E on a carg q and t fild strngt E is givn by F E = qe (N). Tus E as units N.C 1. Sinc carg q can b positiv or ngativ, t dirction of F E can b paralll or antiparalll to t dirction of E (Figur 12). E E E Figur 12. T forcs on an lctron and a proton p immrsd in t sam lctric fild E act in opposit dirctions. p 11

2 Elmnts of Pysics Sourc of An Elctric Fild W av sn tat t sourc of an lctrostatic fild is a carg at rst. Mical Faraday dvlopd t concpt of t lctric fild. H invntd t ida of t lins drawn in Figurs 13a and b. T lins wit t arrows on tm calld fild lins or flux lins and gav tm t unit wbr (Wb). H imagind tat all spac can b mappd out by an infinit numbr of suc lins (toug to kp t figurs simpl w av drawn only a fw of tm). H drw t lins originating on positiv cargs and trminating on ngativ cargs. Toug aving no pysical rality, ty nabld im to dscrib t fild in a consistnt way. t forc on t carg, and trfor also t strngt of t fild, can b associatd wit t dnsity of t lins. Tis rlationsip btwn dnsity and fild strngt is illustratd in Figur 14. p Figur 14. An attmpt to illustrat t rlationsip btwn t dnsity of fild lins and t strngt of a fild. (a) (b) p Figur 13a. Rprsntation of t lctrostatic fild producd by a stationary positiv sourc carg (a) and a ngativ sourc carg (b). (a) (b) In tis figur w can imagin tat a positiv carg, say a proton p, is t sourc of an lctrostatic fild in t spac surrounding it. Tis fild xtnds vrywr in trdimnsional spac (Figur 13a), toug w focus on only tat part of t fild confind to a narrow rgion to t rigt of t carg. If w wr to orint an ara of on squar mtr prpndicular to t fild at two arbitrary positions (a) and (b) at diffrnt distancs from t carg, w would count diffrnt numbrs of fild lins passing troug t ara. Suppos, for xampl, tat t numbr of fild lins passing troug t ara at (a) is 4; t fild strngt tr is trfor 4 Wb.m 2. And suppos tat t numbr of fild lins drops to only 2 at (b); tn t fild strngt tr is 2 Wb.m 2. It can b sown tat t magnitud of t lctrostatic fild producd by a stationary point carg q dcrass invrsly wit t squar of t distanc r from q, tat is, E = k q r 2, [12] T lins nabld Faraday to dscrib t dirction of t fild and its strngt. If w imagin a positiv tst carg placd at any point in tis spac tn it will actually b in contact wit on of ts imaginary lins. T dirction of t forc on t carg is tn givn by t dirction in wic t arrow on t lin points. Tus a positiv tst carg would tnd to mov away from a positiv sourc carg and towards a ngativ sourc carg. T magnitud of wr k is t constant tat appars in q[11]. Tis concpt of t lctric fild, toug originally invntd by Faraday for convninc, as now bn provn to xist troug countlss xprimnts. T concpts of fild and flux will prov just as powrful in our dscription of t magntic fild, as w sall attmpt to sow latr in tis captr. 12

3 Dirct Currnt Elctricity Elmnts of Pysics Elctric Currnt Tus far w av considrd t pysics of cargs at rst. But cargs can mov. If a nt carg movs in a dfinit dirction an lctric currnt xists. A working dfinition of currnt I is carg pr scond: I = q t. [13] T unit of currnt is C.s 1. 1 C.s 1 is givn t unit ampr (A). 3 Voltag Most popl tak for grantd tat voltag and currnt go togtr. But voltag is diffrnt from currnt. To undrstand t diffrnc btwn tm w xamin t simplst of lctric circuits, on comprisd of a sourc of lctrical nrgy and a consumr of lctrical nrgy, otrwis calld a load (Figur 15). + convntional currnt V lctron currnt Figur 15. A circuit consisting of a sourc of lctrical nrgy and a load. W can suppos tat our sourc of lctrical nrgy is a common battry, lik t ons usd in a flasligt, and t load is a carbon composition rsistor (w dscrib wat a battry and rsistor ar mad of in Captr 2). T battry, symbolizd by t two orizontal lins, is on t lft and t rsistor, symbolizd by t ligtning bolt, is on t rigt. T top or long sid of t battry symbol dnots its positiv trminal wil t bottom or sort sid dnots its ngativ trminal. T trminals ar givn positiv and ngativ signs bcaus lctrons, wic av a ngativ carg mrg from t ngativ trminal of R a battry and ntr t positiv trminal. (Rmmbr lik cargs rpl, unlik cargs attract.) An Elctric Circuit Must Form a Closd Loop Tis circuit forms a complt loop. On wir conncts t positiv trminal of t battry wit t top of t rsistor wil a scond wir conncts t bottom of t rsistor wit t ngativ trminal of t battry. T circuit must form a complt loop or otrwis t lctrons would not av a pat to travl in to gt from t ngativ to t positiv trminal of t battry. (Elctrons don t asily mov troug air!) T agnt wic drivs t lctrons around t loop is t battry voltag. It is said tat t battry producs a voltag V (volts) across its trminals wic drivs a currnt of I (amprs) around t loop. If t battry is disconnctd from t loop tn t currnt drops to zro. Also if t wirs ar cut at any point tn t currnt drops to zro. T valu of t currnt is t sam vrywr around t loop t sam amount of currnt I flows troug t wirs, troug t rsistor and troug t battry. T diffrnc btwn voltag and currnt sould b kpt firmly in mind: voltag is a driving forc tat appars across somting, wras currnt, bing t flow of lctrons, flows troug somting. 4 Figur 1 6 sows t circuit of Figur 15 quippd wit instrumnts (an ammtr and a voltmtr) to nabl t currnt and voltag w av just dfind to b masurd. T ammtr is connctd in sris wit t rsistor bcaus it displays t currnt flowing troug t rsistor, wras t voltmtr is connctd in paralll wit t rsistor bcaus it displays t voltag apparing across t rsistor. W dscrib som xampls of ts instrumnts mor fully in Appndix A. + V I Figur 16. T circuit of Figur 15 quippd wit an ammtr (I) and voltmtr (V). R V 13

4 Elmnts of Pysics T Dirction of Currnt At t risk of confusing t radr w now stat tat in tis cours t dirction of lctric currnt will b takn to b opposit to t dirction of lctron flow. In otr words, in Figurs 15 and 16 w sall tak t dirction of wat is calld t convntional currnt to b clockwis vn toug w know tat t lctrons making up t actual currnt ar moving around t circuit in a countrclockwis dirction. 5 Rsistanc and Om s Law T siz of t currnt tat flows for a givn voltag applid is st by an attribut of t load calld rsistanc. T rsistanc R of a load is dfind as t ratio R = V I, [14] wr V is t voltag across t load and I is t currnt troug t load. R as units V.A 1. 1 V.A 1 is calld an om (Ω). If R is a constant, indpndnt of V and I, tn q[14] is known as Om s Law. 6 If a battry is t sourc of lctrical nrgy, tn t voltag and currnt ar constant (providd t battry dos not run down!). Currnt I flows continuously around t loop in Figurs 15 and 16 in a constant clockwis dirction. For ts rasons tis kind of currnt is calld dirct currnt (or DC currnt). T corrsponding voltag is calld a DC voltag. Rsistivity and Conductivity W av just statd tat a load as rsistanc. But w do not imply tat rsistanc is a fundamntal proprty of mattr. In fact, rsistanc ariss from t fundamntal proprty calld rsistivity. W can dfin rsistivity as follows. Considr a conductor of lngt l and cross sctional ara A. T rsistanc masurd btwn t nds of t conductor is obsrvd to dpnd dirctly on l and invrsly on A: R = ρ l A. [15] intnsity of t ligt falling on t conductor. T invrs of rsistivity is calld conductivity. It is dnotd σ and as t units (Ω.m) 1. A matrial wit a small rsistivity and a larg conductivity is calld a conductor. Exampls ar t mtals coppr, silvr and gold. Convrsly, a matrial wit a larg rsistivity and a small conductivity is calld an insulator. Exampls ar rubbr, plastic and glass. A matrial wit intrmdiat rsistivity and conductivity is calld a smiconductor. Exampls ar t lmnts grmanium and silicon. W sall av a lot mor to say about conductivity latr in tis captr. Enrgy and Powr In t circuits abov, tr is a continuous transfr of nrgy from battry to rsistor. You can prov tis yourslf by obsrving tat t battry grows wakr wit tim wil t rsistor ats up. Elctrical nrgy from t battry is transfrrd continuously as at to t rsistor, wic is subsquntly transfrrd to t surrounding air via conduction, convction and radiation. Enrgy as t unit joul (J). T rat of nrgy convrsion (J.s 1 ) is calld powr and as t unit watt (W). 1 J.s 1 = 1 W. T lctrical powr P can b sown to b t product of t voltag and t currnt: P = IV. [16] Tus 1 W = 1 A.V. If t load obys Om s Law tn q[14] applis and w can rwrit q[16] in ts forms: P = I 2 R = V2 R. [17] Tus if R is a constant, powr is proportional to t squar of t currnt and t squar of t voltag. Oftn w ar lss intrstd in t absolut valu of t powr tan in t cang in t powr tat rsults from som ffct. T cang may b a dcras or an incras rsulting from attnuation across a rsistor in t formr cas, or boosting from amplification or from som otr caus in t lattr. T constant of proportionality ρ is t rsistivity. It as units (Ω.m). W sall s in Captr 2 tat tis rlationsip figurs in t opration of a dvic calld a strain gaug. As wll, t rsistivity, and nc also t rsistanc, of a conductor can dpnd on tmpratur and t Powr Cang in db W can dscrib a cang in powr using a psudo unit calld db (pronouncd d b ). Suppos w wis to compar a powr lvl P wit a rfrnc powr lvl P o. W tak t logaritm of t ratio of t powrs and call tis a numbr in bls: 7 14

5 bls = log 10 P P 0. To convrt tis numbr to dcibls or tnts of bls (db) just multiply by 10: db = 10log 10 P P 0. [18] A powr cang of 1/2 is a convnint fraction to tak in stablising a rfrnc. If t powr is rducd by 1 2, i.., if P = 1 2P o, tn db = 10log = 3. W can intrprt tis ngativ numbr as t 3dB point t point wr t powr is rducd by onalf from t rfrnc powr lvl. In lctronics tis is known colloquially as t point at wic t powr is 3 db down. W can us tis db trminology to dscrib t cang in a voltag if t dvic obys Om s Law. In suc a dvic, calld a linar dvic, t lctrical powr is proportional to t voltag squard, q[17], allowing us to writ q[18] in trms of voltag db = 10log 10 V 2 V 0 2. [19] + V Elmnts of Pysics Figur 17. Symbols for idal voltag sourcs. A battry functions as an idal voltag sourc so long as t currnt drawn from it rmains small. In practic, a typical cll starts to bav nonidally as soon as t currnt drawn from it incrass byond a crtain point; at tis point t voltag across its trminals bgins to dcras. Tis bavior is du to t xistnc of an ffctiv intrnal rsistanc in sris wit t battry s mf (an accompanymnt of cmical action, lowr tmpratur or ag). Tus in tis cours w sall modl a ral voltag sourc wit t symbols drawn in Figur 18. A truly idal voltag sourc is on wos intrnal rsistanc is zro. R i ε + I + V V Figur 18. A ral voltag sourc. + Tus t 3dB point is also t point at wic t voltag is down by a factor 1/ 2 wit rspct to t rfrnc voltag lvl. W sall us tis trminology in t following captrs to dscrib t frquncy rspons of filtrs and amplifirs. A grap in wic t amplitud is xprssd in db also nabls us to distinguis small faturs (for a prviw of tis usag look aad to Figur A52b). Voltag Sourc Enrgy sourcs ar classifid as bing voltag sourcs or currnt sourcs. Suc sourcs ar also dscribd as bing idal or ral. A voltag sourc (Figur 17) is dscribd as idal if it producs a constant voltag across a load rgardlss of t currnt supplid to t load. Voltag sourcs ar sold in a varity of sizs and saps wit t most common typ bing t cmical cll or battry. Currnt Sourc T scond spcial cas of an nrgy sourc is t currnt sourc (Figur 19). A currnt sourc is considrd idal if it dlivrs a constant currnt to a load rgardlss of t voltag dvlopd across t load. A ral currnt sourc is lss common tan is a ral voltag sourc. A ral currnt sourc can b constructd by placing a larg rsistor in sris wit a voltag sourc or from a circuit mploying a transistor or an intgratd circuit (IC). I Figur 19. Symbols for an idal currnt sourc. I 15

6 Elmnts of Pysics A ral currnt sourc, lik a ral voltag sourc, dos not always bav idally. T currnt dlivrd by a ral currnt sourc is obsrvd to dcras as t voltag dvlopd across t load incrass. Tis mans tat a ral currnt sourc bavs lctrically lik an idal sourc lmnt in paralll wit an intrnal rsistanc (Figur 110). An idal currnt sourc is on wos intrnal rsistanc is, in principl, infinit. For various rasons many snsors ar dsignd as currnt sourcs. Sinc snsors ar important in tis cours w dvot Captr 6 ntirly to t subjct of snsors. I o I R V + I = I o V R i loop). If t voltags across R 1 and R 2 ar V 1 and V 2 rspctivly tn V = V 1 + V 2 = I( R 1 + R 2 ), = IR q. Tus t ffctiv rsistanc of t two rsistors is t sum R q = R 1 + R 2. [110] Tis xprssion can b xtndd to apply to any numbr of rsistors in sris. Rsistors in Paralll A scond xampl of circuit analysis involvs rsistors in paralll (Figur 112). W can sow tat tis circuit can also b simplifid lctrically to a circuit consisting of a voltag sourc and a singl rsistor. Figur 110. A ral currnt sourc bavs lik an idal currnt sourc wit a rsistanc in paralll wit it. I 1 R 1 Rsistors in Sris In a cours on instrumntation it is usful to know a fw basics of circuit analysis. Circuit analysis bgins wit t issu of rsistors in sris and in paralll. For xampl, w can sow tat a circuit consisting of a battry wit two rsistors in sris (Figur 111) can b simplifid lctrically to on involving a voltag sourc and a singl rsistor. T argumnt gos as follows. I V 1 V 2 R 1 V R 2 Figur 111. Two rsistors connctd in sris. I I 2 R 2 + Figur 112. Two rsistors connctd in paralll. Bot rsistors ar connctd to t sam battry and trfor av t sam voltag V across tm. But t currnt I supplid by t battry splits up into componnts I 1 and I 2, flowing troug R 1 and R 2 rspctivly. Tus V I = I 1 + I 2 = V R 1 + V R 2, = V = V. R 1 R 2 R q T battry as a voltag V across its trminals and drivs t sam currnt I troug bot rsistors (sinc ty ar bot inlin and a part of t sam singl Tus t ffctiv rsistanc of t two rsistors in paralll is givn by t xprssion 16

7 1 R q = 1 R R 2, [111] Elmnts of Pysics Tis xprssion can b xtndd to apply to any numbr of rsistors in paralll. Capacitanc and Capacitors Nxt to an nrgy sourc and t rsistor t most common lmnt in an lctric circuit is t capacitor. A capacitor can b modlld as a pair of idntical paralll mtal plats. Capacitor Caractristic W av alrady sn tat t voltag producd across an idal rsistor is proportional to t currnt flowing troug t rsistor. In contrast, t voltag dvlopd across a capacitor is proportional to t carg Q stord in t capacitor. Tus t following rlationsip applis: Q = Cv. [112] T constant of proportionality, C, is by dfinition t capacitor s capacitanc. T valu of C dpnds on t matrials making up t capacitor, in particular on t kind of conductors usd and t matrial calld t dilctric, wic lctrically insulats on plat from t otr. C as t unit farad (F). 1 F = 1 C.V 1. On farad is a larg capacitanc, t most typical valus bing 0.1 µf and 0.01 µf. (T symbol µ stands for micro and mans 10 6.) Common symbols of capacitors ar drawn in Figur 113. only if dv/dt is nonzro, or wn v is canging. Tus in tory a capacitor sould not pass a constant or a stady DC currnt, but sould pass a canging DC currnt or an AC currnt. Paralll Plat Capacitor As w av statd a capacitor can b modlld as a pair of idntical paralll mtal plats. T plats must b clos noug togtr, but not toucing, for t capacitanc to b significant. W can actually driv an analytical xprssion for t capacitanc of tis modl. Lt us suppos tat t plats ar of ara A wit sparation d muc lss tan a dimnsion of A (so w can nglct dg ffcts). W suppos for t momnt tat t spac btwn t plats is occupid wit a vacuum. W carg t capacitor by conncting it to a voltag sourc (Figur 114). (a) (b) Q Figur 113. Symbols for capacitors: a fixd nonpolar typ (a), a fixd polarizd lctrolytic typ (b) and a capacitor wos capacitanc can b varid by t usr (c). An idal capacitor as its own VI caractristic wic can b found by diffrntiating q[112] wit rspct to t: dq dt (c) = i = C dv dt. [113] Hr i is t currnt flowing into t capacitor s positiv trminal. Equation [113] sows tat i is nonzro ara A E Q Figur 114. A paralll plat capacitor can b cargd by conncting it to an nrgy sourc. T procss of carging rquirs som dscription for it is rally a procss of carg sparation. W dscrib it in trms of lctron movmnt. Elctrons ar attractd into t positiv trminal of t voltag sourc and away from t top plat of t capacitor. T top plat V 17

8 Elmnts of Pysics is trfor lft wit an xcss of positiv carg +Q. At t sam tim lctrons ar rplld away from t ngativ trminal of t voltag sourc and onto t bottom plat of t capacitor. Tus t bottom plat is lft wit an xcss of ngativ carg Q. Bcaus of mutual rpulsion t cargs distribut tmslvs uniformly ovr ac plat. 8 A uniform lctric fild trfor ariss in t spac btwn t plats pointing from positiv to ngativ plat. Analysis sows tat t voltag btwn t plats is givn by t xprssion v = V = Qd ε o A, [114] us to bgin t procss of carging wit t capacitor fully discargd. W can suppos tat at tim t = 0 w mov S to position B to connct t capacitor to t nrgy sourc and tus bgin t carging. W suppos tat at som latr tim t w obsrv t capacitor to contain a carg Q and to support a voltag V across its trminals (as masurd, say wit a voltmtr). B S A V C wr ε o is a pysical constant calld t prmittivity of fr spac. From qs[112] and q[114] t capacitanc is givn by t xprssion C = Q v = ε o A d. [115] Tus t capacitanc of a paralll plat capacitor can b incrasd by incrasing t ara of t plats or by dcrasing t sparation btwn t plats. T capacitanc can also b incrasd by insrting an insulating matrial or dilctric into t gap btwn t plats. T incras in capacitanc rsults from a polarizing proprty of t dilctric dscribd by a paramtr calld t dilctric constant dnotd ε r. To includ t ffct of a dilctric w must modify q[1 15] to rad C = ε r ε o A d. [116] Figur 115. A circuit to carg a capacitor. Our analysis rquirs a littl calculus. W can tink of t carg as bing transfrrd from t sourc to t capacitor in incrmnts or bundls of carg dq. T lmnt of work don by t battry dw in placing an lmnt of carg d q on t capacitor wn t voltag across t capacitor is v is givn by dw = vdq = Cvdv. Tus t total work don in carging t capacitor from a lowr limit of v = 0 volts to an uppr limit of v = V volts (t voltag of t sourc) is givn by t intgral V W = Cvdv, 0 = 1 2 CV 2, [117a] T valus of ε r rang ovr two ordrs of magnitud, from 1 to about 100. Mor dtails on practical capacitors ar givn in Captr 2. = 1 2 QV = 1 2 Q 2 C, [117b,c] Enrgy Stord in a Capacitor T major usfulnss of a capacitor as to do wit its capacity to stor an lctric carg. To s tis w must first supply t capacitor wit carg by conncting it to an nrgy sourc (Figur 115). W sall s tat t procss of carging an idal capacitor rquirs tat t nrgy sourc dos a crtain amount of work. Tis circuit is quippd wit a switc S tat nabls wr qs[117b] and c follow by t substitution of q[112]. Tis work is not lost to t systm as at as it would b if a rsistor wr in plac of t capacitor. Wit a littl nginring, t capacitor, onc cargd, could b disconnctd from t nrgy sourc and connctd to a dvic (lik a rsistor in fact) to prform work (lik incrasing t rsistor s tmpratur). T conclusion to b drawn is tat t nrgy 18

9 dlivrd by t sourc to t capacitor gos into t production of t lctric fild btwn t capacitor plats. As long as t lctric fild xists tr, t associatd nrgy can b rgardd as nrgy stord. Carging/Discarging a Capacitor In our tratmnt of t prvious sction w ignord wat appns during t tim wn t capacitor is bing cargd or discargd. A circuit to study tis transint stat is drawn in Figur 116. Tis circuit includs a rsistanc R, wic migt b t rsistanc of a carbon composition rsistor or just t ffctiv rsistanc of t capacitor itslf. B A V R C Figur 116. A capacitor bing discargd troug a load rsistor R. W considr t discarg of a capacitor from an initial stat of full carg. T capacitor in Figur 116 can b fully cargd by laving t switc S in position B for a sufficintly long tim. Tn at tim t = 0, say, w mov S to position A to bgin t discarg. At som clock tim t wn t instantanous voltag across t capacitor is v C (t) and an instantanous currnt i(t) is flowing, t voltag across t capacitor and t voltag across t rsistor must b qual, i.., S v C (t) Ri(t) = 0, [118] Using q[112] and t fact tat i(t) and q(t) ar rlatd by invrsion w av i(t) = dq(t), dt Elmnts of Pysics (as currnt i(t) flows carg q(t) dcrass) w can substitut into q[118] and obtain dv(t) dt + v(t) RC = 0. [119] Tis xprssion is a first ordr diffrntial quation in v(t). You sould b abl to sow by dirct substitution tat a solution is v(t) = V = V t RC, [120] wr t C = RC. [121] Eq[120] sows tat wn t switc is trown to position A, t voltag across t capacitor dos not drop to its final valu of zro volts instantanously, but ratr, dcrass wit tim xponntially. Tis is t sam as saying tat t capacitor discargs wit tim xponntially. t C, wic is a constant (sinc R and C ar constant) is calld t tim constant; it quantifis ow sarply t voltag dcrass. If R is in units of oms and C in units of farads tn RC as t unit of tim in sconds. t C is t tim in sconds rquird for t voltag to fall from t initial valu of V volts to t valu V/ = 0.368V volts. T biggr t tim constant t mor tim is rquird for discarg to tak plac. If a rsistanc wr to b includd in t carging circuit, a similar (complmntary) xprssion would b obtaind for v C (t) ovr t tim during wic t capacitor is carging. t t C 19

10 Elmnts of Pysics Magntic Fild In tis cours w ar concrnd wit t tr fild typs: gravitation, lctric and magntic. W av dscribd t lctric fild a littl alrady. Toug praps not consciously, most of us rmmbr a littl of magntic ffcts from playing wit magnts as cildrn. W know tat a bar magnt, oftn mad from iron, as a nort pol and a sout pol. T lik pols of suc magnts rpl ac otr wil t unlik pols attract. Magntic pols must trfor b abl to xrt forcs on on anotr across mpty spac, in a mannr lik lctric cargs. By tis w infr tr xists an ntity calld a magntic fild. Basic Obsrvations Most of us ar awar from an arly ag of ow bar magnts and t magntic compass bav. Two bar magnts will itr attract ac otr or rpl ac otr, dpnding on ow ty ar orintd wn brougt togtr. W soon larn tat a bar magnt as a nort pol at on nd of it and a sout pol at t otr. Obsrvations tn sow tat lik pols rpl ac otr wil unlik pols attract. If w stand wit a compass at som point on t art s surfac t markd nd of t compass ndl always points in t gnral dirction of t nort gograpic pol (Figur 117). In fact, if t markd nd of a compass ndl is dfind as a nort pol tn it follows tat t magntic pol nar t nort gograpic pol is a sout magntic pol. Bcaus t pols of a magnt can xrt forcs on on anotr across mpty spac tr must xist an ntity calld t magntic fild. T Tst for a Magntic Fild For t purpos of undrstanding, a compass can b takn as an instrumnt for tsting for t xistnc of a magntic fild. W can agr tat a magntic fild xists at som point in spac if t ndl of a compass is forcd to tak up a spcial position at tat point. W can dfin t dirction of t magntic fild (it is a vctor) to b t dirction in wic t nort pol of t compass ndl points. Sourc of a Magntic Fild Just stating tat a bar magnt and t art av a magntic fild according to our simpl tst says littl about t pysical sourc of t fild. How tis sourc was discovrd can b xplaind by mans of a simpl dmonstration. If a compass is mployd as a tst instrumnt and is usd to xplor t nvirons of a conductor carrying an lctric currnt, t compass ndl is obsrvd to tak up spcial positions wit rspct to t wir (Figur 118). Tis ffct gos away if t currnt is cut. T conclusion to b drawn is tat a magntic fild xists in t rgion nar t conductor aving somting to do wit t currnt. A clos connction must xist btwn lctric ffcts (t currnt) and magntic ffcts (t action on t compass ndl). 9 Figur 117. At any position on t art s surfac t markd nd of a compass ndl always points toward t magntic pol in t nortrn mispr (wic is, in fact, a magntic sout pol). Figur 118. Around a currntcarrying conductor a compass ndl always points in a dirction prpndicular to, but nvr towards, t conductor. 110

11 On closr inspction t ndl is obsrvd to always point in a dirction prpndicular to t conductor but nvr dirctly towards or dirctly away from t conductor. Ts ffcts wr first studid in dtail a cntury ago by t Englis scintist, Mical Faraday. In an attmpt to dscrib tm mpirically Faraday usd t concpt of t flux lin or t fild lin ad invntd for t lctric fild. H sowd tat t magntic fild producd nar a currntcarrying conductor can b rprsntd grapically in a plan prpndicular to t conductor as a sris of concntric circls, ac indicatd by an arrow (Figur 119). On t lft in t figur t currnt is sown flowing out of t plan of t pag wil on t rigt t currnt is sown flowing in t opposit dirction. T dirction of t currnt and flux lins can b dducd using a rigt and rul. If you plac t tumb of your rigt and in t dirction of t currnt, and curl your fingrs in a gripping action tn your fingrs dscrib t dirction of t fild. T strngt of t fild is dnotd by t lttr B. B Figur 119. Rprsntations of t magntic fild nar conductors carrying an lctric currnt. T conductor is at t cntr of t circls and prpndicular to t plan of t pag. T fild lins tus drawn ar givn t sam unit as for t lctric fild, namly wbr (Wb). Hr, too, t dnsity of t lins can b takn as an indicator of t fild strngt. T dirction in wic t compass ndl points dfins t dirction of t arrow and t dirction of t fild. Fundamntal Tst All of t abov notwitstanding, tr xists a mor fundamntal pysical way of tsting for a magntic fild s xistnc tan by ccking to s if a compass ndl dflcts. Tis tst is basd on t socalld B X Elmnts of Pysics Lorntz forc. If an lctric carg q movs wit vlocity v troug a rgion wr a magntic fild of strngt B xists tn t carg will b subjct to a magntic forc givn by F m = qv B. [122] Tis is a vctor cross product, maning tat t vctors F m, v and B ar mutually prpndicular. T maning of t vctor cross product is xpandd in Figur 120 wr a positiv carg q (a proton) is sown ntring a rgion wr a magntic fild of strngt B xists (pointing out of t plan of t pag). If w suppos for simplicity tat t particl s vlocity is initially prpndicular to B, tn t Lorntz forc causs t particl to mov in a clockwis dirction. B v Figur 120. A proton is sown ntring a rgion from t rigt in wic a magntic fild of strngt B xists (pointing out of t plan of t pag). T forc on t particl is in t plan of t pag prpndicular to v and B, causing t particl to mov in a clockwis dirction. Tus w can say tat a simplr, mor lgant and fundamntal tst of t xistnc of a magntic fild in a rgion of spac is if a carg wic is moving in tat rgion is dflctd from its original pat by a forc of t form of q[122]. From q[122] w now av t unit of B in t SI systm of units: N.C 1.m 1.s (quivalnt to wbr.m 2 ). 1 N.C 1.m 1.s is givn t spcial unit Tsla. Magntic Fild of t Eart Magntic fild as a magnitud and a dirction. At a point on t surfac of t Eart t Eart s magntic fild is dscribd in trms of its strngt and dclination (angl blow t orizon). In t rgion of Toronto t strngt is of t ordr of 0.3 gauss wit a dclination of 2 dgrs. W sall rturn to tis topic in Captr 6. p 111

12 Elmnts of Pysics Inductanc and Inductors In AC circuits t inductor is t complmnt of t capacitor. An inductor is modlld as a simpl coil of wir. If a currnt in t coil is mad to cang wit tim tn an mf is obsrvd to xist btwn t trminals of t coil. Tis simpl pnomnon, calld slf induction, is t sourc of a numbr of ffcts in lctricity and lctronics. Undrstanding t inductor involvs nw aspcts of t magntic fild. A Currnt Loop W av sn in a prvious sction tat a magntic fild is producd in t spac surrounding a currntcarrying wir. If conditions ar kpt t sam but t wir is loopd to form a coil, tn t strngt of t fild insid t loop is obsrvd to incras (Figur 1 21a). If a magntic compass is usd to tst t symmtry of t fild tn tat symmtry is found to rsmbl tat of a bar magnt (Figur 121b). Faraday dducd tat som connction must somow xist btwn macroscopic magntism (t bar magnt) and t mor lmntary pysics of a currnt loop. I B B ~ I (a) Figur 121. T magntic fild lins producd by a currnt loop (a) and a bar magnt (b) av a similar symmtry. T diagram on t lft illustrats a microscopic ffct, tat on t rigt a macroscopic ffct. Induction If t currnt in t loop is mad to cang wit tim tn an mf (voltag) appars btwn t loop s trminals. Sinc tis mf rsults from t canging currnt and not from som xtrnal nrgy sourc, it is calld an inducd mf. T ffct is calld induction. Tis mf can b masurd wit t appropriat AC voltmtr. Faraday usd is invntion of magntic flux to intrprt induction. H xplaind tat t tim rat of cang of flux troug t loop inducs an mf in t loop tat tnds to rsist t original cang in t flux (or t original cang in t currnt). Tat is, t inducd mf ε can b dscribd by a matmatical N S B (b) xprssion of t form ε = dφ B dt, [123] wr Φ B is t magntic flux in wbrs passing troug t loop. Tis xprssion ncapsulats two fundamntal laws of pysics calld Faraday s Law and Lnz s Law. (Lnz was a contmporary of Faraday.) Faraday s Law stats tat t mf is proportional to t tim rat of cang of t flux and Lnz s Law stats tat t proportionality as a ngativ sign. Sinc t flux troug t loop is proportional to t currnt w can rwrit q[123] as ε = L di dt, [124] wr L, t constant of proportionality, is calld t inductanc. L is rlatd to t inductor s gomtry and as t unit nry (H). 1 H = 1 V.s.A 1. On nry is a larg inductanc. Mor typical valus ar in t mh and µh rang. Som symbols of inductors ar drawn in Figur 122. W sall discuss inductors in mor dtail in t sction on impdanc in tis captr and wn w gt to tcnical aspcts of inductors in Captr 2. (a) (b) Figur 122. Symbols usd for inductors: a standard inductor (a), an inductor wos inductanc can b varid by t usr (b) and tappd typ (c). In t nxt sction w discuss t Hall ffct, an ffct involving t magntic fild. (c) 112

13 Elmnts of Pysics T Hall Effct T Hall Effct is an ffct tat mbodis so many pysical concpts tat t dvic in wic it occurs forms t quintssntial snsor. W confin our attntion r to t pysics of t ffct. Typical dvics ar dscribd in dtail in Captr 6. T Effct T Hall Effct was discovrd by E. W. Hall in It was not until t 1950s, owvr, tat t first practical dvic was dvlopd mploying t ffct. Hall obsrvd tat if a smiconductor carrying a dirct currnt is immrsd in a magntic fild (Figur 123), tn a voltag dvlops across t smiconductor in a dirction prpndicular to bot t currnt and t fild. Tis voltag is calld t Hall voltag. T sign and magnitud of t voltag provids information about t magnitud and dirction of t fild and also about t typ of majority carg carrir making up t currnt flow. (W sall av mor on carg carrirs latr in tis captr.) I B V v F q Figur Illustration of t Hall Effct. T Hall Voltag T Hall voltag is t quantity tat actually gts masurd in a Hall ffct snsor. W can xplain ow tis voltag ariss as follows. W mak tr assumptions: (1) tat t magntic fild in t figur points into t plan of t pag, (2) tat an xtrnal voltag sourc inducs a (convntional) DC currnt to flow from lft to rigt troug t smiconductor and (3) tat t majority carg carrirs ar lctrons wit carg q. T lctrons mov to t lft wit som avrag (or drift) spd v. Sinc t lctrons ar moving troug a magntic w fild, tn t Lorntz forc, q[122], dflcts tm upwards towards t top dg of t smiconductor. Tus t top dg bcoms mor ngativ tan t bottom dg. As t cargs build up, t potntial diffrnc across t smiconductor riss. Evntually, t lctric fild tat rsults from t collction of sparatd cargs balancs t ffct of t Lorntz forc and t potntial diffrnc racs a maximum. Tis maximum potntial diffrnc dnotd V H is calld t Hall voltag. Analysis sows tat V H as t following form V H = γibsinθ, [125] wr γ is calld t Hall cofficint and θ is t angl btwn t dirction of currnt and t dirction of magntic fild. A numbr of variabls ar involvd in q[125]. Tis mans tat if t numrical valus of all of t variabls ar known xcpt on tn tat unknown can b calculatd. For xampl, lt us suppos in t abov xampl tat t dirction of I and B ar known but t idntity of t carg carrir is not. If t voltag of t top of t smiconductor rlativ to t bottom is found to b ngativ tn it can b infrrd tat t carg carrirs ar lctrons. If t voltag is positiv tn it can b infrrd tat t carg carrirs ar positiv ntitis, or ols (w sall av mor to say about ols latr in tis captr). If w know tat t carg carrirs ar lctrons and if w know wat t magnitud and dirction of t currnt ar tn w can calculat from t sign and magnitud of V H t dirction and magnitud of B. Tus a Hall Effct dvic maks an idal magntic fild snsor. Tis far in tis rviw w av considrd som of t pysics tat rsults from t flow of a DC currnt. In t nxt sction w bgin our rviw of t ffcts tat rsult wn t currnt is mad to altrnat. T pysics now bcoms mor matmatical. 113

14 Elmnts of Pysics Altrnating Currnt Elctricity Wn a rfrnc is mad to a signal in t scincs, wat is usually mant is an AC signal. AC stands for altrnating currnt. T major diffrnc btwn AC and DC is tat AC is currnt tat is mad to flow for a tim in on dirction around a circuit loop, and tn is rvrsd and for an qual tim is mad to flow in t opposit dirction and so on rvrsing continuously. T audio tat drivs a spakr in a stro systm is an AC signal, as is t voltag supplid from wall powr sockts in t om and laboratory. AC Circuits An AC wavform can b simpl or complx. T simplst AC signal is a sinusoidal on; it as a simpl wavform tat can b studid in dtail wn displayd on t scrn of an oscilloscop (Figur 124 top). T instantanous voltag v(t) of suc a signal swings continuously btwn a positiv pak voltag of +V pak and a ngativ pak voltag of V pak. Just as a DC circuit must b complt in ordr to sustain a DC currnt, so also must an AC circuit b complt in ordr to sustain an AC currnt. T sourc of an AC signal v(t) is calld a signal gnrator. In Figur 124 cntr and bottom a signal gnrator is sown in scmatic form connctd to a load wit two wirs, on out and on rturn. In t first alf cycl of t voltag a currnt i(t) flows clockwis around t loop. In t scond alf cycl t currnt flow rvrss. T currnt may b canging wit tim but at any instant of tim and at any point in t circuit t currnt as t sam instantanous valu and t sam clockwis or countrclockwis dirction. An Analog Wavform T voltag wavform in Figur 124 is an analog wavform in t sns of bing a continuous function of tim. W ar trfor justifid in rprsnting it by t continuous function ovr on cycl is 1.51 V. W sall dfin ts trms in du cours. v(t) i(t) 1st alf cycl (b) load v(t) = V pak sin(ωt) = V pak sin(2πft), [126] v(t) i(t) 2nd alf cycl load wr ω = 2πf is t angular frquncy in radians pr scond (rad.s 1 ) and f is t linar frquncy in cycls pr scond (c.s 1 ). On cycl pr scond is givn t spcial nam rtz (Hz). 1 c.s 1 = 1 Hz. It can b sn from t masurmnts prformd by t oscilloscop (look closly down t rigt and sid of Figur 124 top) tat in tis xampl t frquncy is MHz, t paktopak voltag is 4.28 V and t rms valu Figur 124. A sinusoidal analog AC voltag v(t) takn from t scrn of an oscilloscop (top). v(t) producd by a signal gnrator rsults in an AC currnt i(t) (middl and bottom). (c) 114

15 Avrag W ar oftn intrstd in t avrag of an AC voltag or currnt calculatd ovr on priod. T instantanous currnt i(t) avragd ovr on priod is by dfinition givn by I = 1 2π 2π i(t)dt. 0 W can dfin t avrag voltag in a similar fasion. You can trfor s tat t avrag of an AC quantity may not b t rsult you xpct. For xampl, t avrag of a pur sin wav signal (Figur 124 top) is zro! Howvr, t avrag may in fact b t masurmnt you want wn t AC signal contains a DC componnt. W sall rturn to tis subjct in Captr 2. RootManSquar It is usful to av a way of comparing a sinwav AC currnt wit a DC currnt. It is rasonabl to dfin a sinwav AC currnt and a DC currnt as aving qual ffctiv valus if ty produc t sam ating in t sam rsistor. W can find t ating ffct of an AC currnt by avraging t powr losss ovr on complt cycl: P = 1 T T 0 i(t)2 Rdt = i 2 PakR T = I 2 PakR 2 T 0 = I 2 ff R, sin 2 (ωt)dt [127] Elmnts of Pysics Spcifications of dvics and circuits ar somtims givn in trms of rms, otr tims in pak valus. It is important to undrstand t diffrnc btwn t two dfinitions so tat a dvic is usd as intndd. Pas Diffrnc Eq[126] is not t most gnral form of a sinwav function. A mor gnral form is v(t) = V pak sin( ωt + φ), [129] wr t additional factor φ is t pas diffrnc. φ dtrmins t valu of v(0), tat is, t instantanous voltag at a clocktim of 0 sconds. Indd, v(0) = V pak sinφ, so if φ = 0 tn v(0)=0. But φ is in gnral not zro as w sall s. T concpt of pas diffrnc bcoms important wn two AC signals coxist at t sam point in a circuit or wn an AC signal passs troug a systm lik a filtr. T instantanous voltag of t signals may b zro at diffrnt clocktims. If ty av t sam frquncy, ty migt appar individually on t scrn of an oscilloscop as sown in Figur 125. Sinc t signals ar zro at diffrnt clocktims ty av a tim diffrnc of, say, t sconds tat can actually b masurd wit a digital oscilloscop. Sinc ty av a tim diffrnc, ty also av a pas diffrnc. wr I ff is t ffctiv valu of t AC currnt. I ff is commonly calld t rootmansquar or rms valu and is dnotd I rms. Tus I ff = I rms = I pak 2. [128] Tus t rms valu of a sinwav currnt (or a sinwav voltag) is its pak valu dividd by t squar root of two. Praps surprisingly, t mains, wic supplis a sinwav voltag of 60 Hz at an amplitud of 115 volts rms, as a pak voltag of V pak = = 163 volts! Figur 125. Two signals aving t sam frquncy but diffring pass pass troug zro at diffrnt clocktims

16 Elmnts of Pysics W can calculat t pas diffrnc from masurmnts of t tim diffrnc and t priod. Sinc t signals av t sam frquncy f ty also av t sam priod T (wic w can masur wit cursors btwn altrnat zro crossings on t oscilloscop scrn). Lt us suppos t tim diffrnc btwn t signals (as masurd btwn t sam zro crossing of t two signals) is t. Tn t following rlationsip applis would nd to dfin t currnt, voltag and impdanc as complx numbrs, a task tat would tak us byond t intndd scop of tis basic rviw. For t momnt w sall stick wit dfining t absolut valus of impdanc. For xampl, w sall writ t absolut valus of rsistiv impdanc Z R, capacitiv impdanc Z C and inductiv impdanc Z L as: Z R = R t T = ϕ 2π, [130] Z C = 1 ωc [132a, b, c] wr φ is t pas diffrnc in radians. W av pointd out r a scond major diffrnc btwn an AC currnt and a DC currnt; an AC currnt as a pas as wll as a magnitud. Tis fact as profound consquncs as w sall attmpt to sow in t nxt sction. Impdanc Rcall from our discussion of DC circuits arlir in tis captr tat t attribut of a load tat limits t flow of a DC currnt troug it is rsistanc. A similar concpt applis in AC circuits. In an AC circuit t attribut of a load tat limits t flow of an AC currnt is calld impdanc. But impdanc (dnotd Z) is a mor complicatd concpt tan is rsistanc bcaus of t issu of pas. For t momnt, w can crtainly dfin t magnitud of impdanc, Z, as t ratio of pak or rms valus of AC voltag and currnt: Z = V pak I pak = V rms I rms, [131] so tat Z as t unit om (Ω), t sam unit as rsistanc. But tis dos not man tat w can writ a maningful xprssion of t form of q[131] using instantanous valus of voltag and currnt, v(t) and i(t). If w try tis w nd up wit a function of tim tat is ssntially maninglss. 11 T factor wic complicats t us of instantanous valus is t pas angl. 12 To dscrib tis ffct of pas most lgantly w and Z L = ωl You can s tat t rsistiv impdanc is t sam as rsistanc. T absolut valus of t capacitiv and inductiv impdanc bot dpnd on frquncy capacitiv impdanc dpnds invrsly on frquncy wil inductiv impdanc dpnds dirctly on frquncy. A grap of q[132b] and [132c] for C=1 F and L=1 H is drawn in Figur 126. T data is displayd r in a loglog grap sinc frquncy can vary ovr many ordrs of magnitud. Impdanc Impdanc Capacitiv Inductiv Frquncy (Hz) Figur 126. A loglog grap of impdanc vs frquncy for capacitiv and inductiv impdancs. Muc of wat tis cours is about concrns t working of snsors. Most snsors ar mad from smiconductor matrials, and so t topic of conduction in matrials is important. W bgin our study of tis subjct in t nxt sction. 116

17 Elmnts of Pysics Conduction of Elctricity in a Solid Immdiatly aftr tis sction w will bgin our discussion of smiconductors. Bfor doing so w nd to discuss lctrical conduction in solids in gratr dtail tan w av don so far. W first considr conduction in gnral trms, bginning wit conduction in a conductor, and tn broadn our approac to apply to an arbitrary matrial. W sall av to b brif. Modifying Our Modl of a Conductor T modl of a conductor w av assumd so far in tis captr is calld in pysics t Fr Elctron Gas Modl. Tis nam is drivd from t fact tat t conductor is picturd in t simplst of trms: as a matrial wos intrnal structur is a rgular gomtrical lattic of stationary positiv ions wit many billions of fr lctrons intrsprsd amongst tm. T lctrons ar assumd to b compltly nonintracting, intracting nitr wit on anotr nor wit t ions in t lattic. Tis modl nabls on to dfin currnt, but it fails to xplain t diffrncs obsrvd in t conductivitis of conductors, smiconductors and insulators. T modl nds to b rvisd. As a ratr obvious first stp w can assum som masur of intraction: w can includ t collisions tat t lctrons invariably mak wit t ions as ty ar drivn along t wir by t intrnal lctric fild (Figur 127). Ts collisions ffctivly prvnt t lctrons from acclrating indfinitly to vrigr spds. Eac lctron must now rac an avrag or drift spd v D at wic it movs along t wir. ig potntial or voltag A low potntial or voltag E I Figur 127. A sction of wir carrying a currnt I. Tis mans tat in an intrval of tim dt ac lctron movs a distanc v D dt along t wir (to t lft in Figur 127). During tis intrval t numbr of lctrons wic pass troug an ara A ar tos containd in a volum Av D dt. If t dnsity of lctrons in t wir is n tn t numbr of lctrons in tis volum is nav D dt. T currnt is tis numbr multiplid by t lctronic carg dividd by t lapsd tim: I = dq dt = nv Adt D = nv dt D A. Tis currnt is flowing to t rigt. W can dfin t vctor currnt dnsity J as t currnt flowing troug a unit ara of conductor. Its magnitud is J = I A = nv D. [133] J dfind in tis way is indpndnt of t conductor gomtry. Idally, t conductor sould support as larg a currnt dnsity as is possibl in ordr to dlivr as muc currnt to a load as is possibl witout t conductor ating up to an unmanagabl ig tmpratur. T carg drift vlocity can b masurd xprimntally. It is obsrvd to incras linarly wit t applid lctric fild, i.., v D = µe, [134] wr t constant of proportionality µ is calld t carg mobility. Carg mobility as units m.s 1.J.C 1. In gnral, µ is larg for a good conductor and small for a poor on. Substituting q[134] into q[133] and adding t vctor notation givs J = nµe. [135] Conductivity σ is dfind tortically as t ratio of t absolut valus of J and E. Tus σ = J E = nµ. [136] Tortical conductivity as units (Ω.m) 1, consistnt wit its xprimntal countrpart. Rsistivity ρ is tn ρ = σ 1. [137] Tus w now av tortical xprssions for σ and ρ 117

18 Elmnts of Pysics wic ar basd on a dpr undrstanding of a solid. W ar now in a position to prob mor dply into t natur of t carg carrirs tmslvs. Elctrons and Hols If our solid is a conductor tn w can xplain t xprimntal conductivity tat is masurd almost xactly wit our modl by assuming tat only lctrons ar prsnt as carg carrirs. Howvr, wn w turn our attntion to t masurd conductivity of a matrial of arbitrary typ w fail. It ad bn known for a long tim tat t conductivity of a matrial of any typ (wtr conductor or not) could b writtn tortically as t sum of t conductivitis of carg carrirs of positiv sign as wll as ngativ sign (Figur 128). v D E j j + + v D+ Figur 128. Witin any matrial t currnt dnsity vctors of ngativ and positiv carg carrirs bot point in t dirction of t lctric fild. Tis mans tat in principl bot carrirs contribut positivly to t total conductivity. W can sow tis by a simpl argumnt. In t lump of matrial in Figur 128 an lctric fild is sown dirctd to t rigt. Tus ngativ carg carrirs will mov to t lft wit som drift spd v D wil positiv carg carrirs will mov to t rigt wit som drift spd v D+. From q[134] t vctor currnt dnsity of t ngativ carg carrirs is dirctd opposit to v D and trfor in t sam dirction as v D+. Tus t conductivitis of t ngativ and positiv carg carrirs add constructivly: wil t otr is calld t minority carg carrir. Smiconductors lik grmanium and silicon av conductivitis tat lay somwr btwn t conductivitis of conductors and insulators. T xprimntal conductivity of a smiconductor can only b xplaind tortically by assuming tat positiv as wll as ngativ carg carrirs ar prsnt in t matrial. T ngativ carg carrirs ar, of cours, lctrons. T positiv carg carrirs ar mor difficult to imagin. Ty ar intrprtd as rgions in t crystal wr tr would normally b an lctron but wr an lctron is missing. Ts rgions, lacking an lctron, av a nt positiv carg and ar givn t dscriptiv nam ols. Intrinsic Smiconductors Smiconductors ar classifid as of intrinsic or xtrinsic typ dpnding on t sourc of tir lctrical conductivity. T sourc of t conductivity of an intrinsic smiconductor is at. An xtrinsic smiconductor is on wos conductivity is nancd by dlibratlyintroducd impuritis. W considr t intrinsic typ r. T drawing in Figur 129 is an attmpt to rprsnt t structur of a prfct crystal of a smiconductor lik silicon (Si) or grmanium (G). Bot lmnts rsid in t sam column of t priodic tabl along wit carbon and av a valnc of +4, tat is, ty av four lctrons in tir outr, or bonding, sll. Consquntly in crystallin form ac atom sars ts four lctrons wit its nigbors. T lins in t figur ar intndd to rprsnt a singl sard lctron wil t circls rprsnt t positiv ions of G or Si. T figur could b said to mor accuratly rprsnt t structur at a tmpratur of absolut zro wn t lattic is compltly witout vibrational motion of any kind, in a stat of zro nrgy wn no ions or lctrons ar out of plac. σ = n + µ + + n µ. Howvr, t two trms in tis rsult ar not usually of t sam magnitud sinc t conductivity (or numbr) of on typ of carg carrir usually xcds t otr. In an insulator lik glass bot n + and n ar vry small. But in a mtal lik coppr n is larg wil n + is narly zro. T carrir tat contributs t most to t conductivity is calld t majority carg carrir Figur 129. A twodimnsional rprsntation of t structur of a smiconductor crystal at zro K. 118

19 At a nonzro tmpratur, owvr, t crystallin lattic will possss small trmal vibrations tat rsult from t continuous mission and absorption of trmal potons. In a smiconductor tis vibrational nrgy just appns to b of t ordr of t binding nrgy of t valnc lctrons, wic ar only modratly bound to t atoms. It is trfor possibl for on of ts lctrons to absorb noug trmal nrgy to scap from its parnt atom and go wandring about t crystal (Figur 130). Lft bind in t crystal is a plac wr tr would normally b an lctron. Tis vacancy, wic as a nt positiv carg, is a ol. Fr lctrons and ols ar producd in tis way in pairs for ac fr lctron tr is a corrsponding ol. Figur 130. A smiconductor crystal at som nonzro tmpratur sowing a fr lctron and a ol. If w apply an lctric fild to t crystal (say to t rigt in Figur 130) a currnt will flow. Tis currnt rsults from t movmnt of fr lctrons to t lft. In t nigborood of t ol it is possibl for a bound adjacnt lctron to fall into t ol, tus anniilating t ol or making it go away, laving bind a nw ol wr t lctron cam from. Tis as t ffct of making a ol mov to t rigt, so ols, too, tak part in conduction. Tus a small conductivity ariss from at alon. Sinc tis conductivity is inducd by at it is calld natural or intrinsic conductivity. Tis kind of conductivity is of only scondary importanc in t functioning of smiconductor dvics. As w sall sow in t nxt sction t mor important typ is t xtrinsic typ. Elmnts of Pysics Extrinsic Smiconductors In xplaining ow lctrical conduction taks plac in a smiconductor dvic lik a diod (in t nxt sction), w sall s tat xtrinsic conductivity is mor important tan intrinsic conductivity. Extrinsic conductivity rsults from a procss calld doping, wrby a trac amount of a spcial impurity atom is introducd into t smiconductor mix wil still moltn. Tis procss rsults in a socalld Ntyp or a Ptyp matrial dpnding on t typ of impurity introducd. T word xtrinsic rfrs to t fact tat tis typ of conductivity is t rsult of an xtrnal procss. NTyp Smiconductor An Ntyp smiconductor is mad from a batc of pur moltn matrial in t following procdur. Lt us assum to bgin tat t matrial is silicon wos valnc is +4. Trac amounts of a similar sizd atom, say posporus wos valnc is +5, ar introducd to t silicon mlt to rplac som of t silicon atoms. Sinc t atoms of silicon and posporus ar of about t sam siz t lattic is not unduly disturbd by t substitution. Four of t fiv valnc lctrons of t posporus atom pairbond wit t nigboring silicon atoms, laving t fift lctron unpaird and trfor only loosly bound (Figur 1 31). Tus t nrgy of tis fift lctron is only sligtly lss tan t nrgy of a fr lctron, so at or an applid lctric fild can asily supply t lctron wit sufficint nrgy to nabl it to brak fr and tak part in conduction (Figur 132). Figur 131. A crystal of a P or donor typ smiconductor sowing t donor lctron (diagonal lin) unpaird and only loosly bound to t positiv impurity ion (in black). If w apply an lctric fild to t crystal (Figur 132) to t rigt, tn fr lctrons will mov wit som drift spd to t lft. Sinc t positiv impurity ions 119