VI MAGNETIC EFFECTS OF CURRENTS

Transcription

1 V MAGNETC EFFECTS OF CURRENTS 6.1 Ampère s investigtions t ws Ampère who first estblishe n quntifie the force tht occurs between two currentcrrying conuctors. This is not quite s simple s the Coulomb lw for the force between chrges s there re more irections to consier. As well s the seprtion of the forceproucing objects (the current elements), in this cse the irections of the two currents re importnt. 1 loop 1 F r 1 loop l Here F 1 is the contribution to the force on the element l 1 of loop 1 from the element l of loop. f θ 1 is the ngle between l 1 n the norml to the plne spnne by r 1 n l, n θ is the ngle between l n r 1, then the conclusion of Ampère ws tht the mgnitue of the force F 1 behve s: l 1 F 1 F l l sinθ sinθ r1 The observtion is tht the mgnetic force between two current-crrying elements l 1 n l vries inversely s the squre of their seprtion once gin we hve n inverse squre lw. ntroucing the constnt of proportionlity, n tking ccount of the irection of the force by writing things in vector form we hve ( ) l l r F = (6.1) r1 n the expression for F is obtine by swpping the 1 n in the expression. be The constnt is referre to s the permebility of free spce. ts vlue is efine to = 1 newtons / mpere or henries / metre. 7 (6.) t might seem surprising tht the vlue of is efine rther thn being etermine experimentlly. The reson is tht the mgnetic force formul is ctully use s the efinition of the mpère, the unit in terms of which is mesure. (An in fct the coulomb then follows s the chrge trnsferre when current of one mpère flows for one secon). The numericl vlue chosen for llows the tritionl electricl units (ctully preting the metric system) to be incorporte into the S scheme. PH4 / Cown 1 6.1

2 V Mgnetic Effects of Currents You might think tht more strightforwr wy of introucing mgnetic forces woul be to strt immeitely with moving point chrges inste of elements of current-crrying conuctors. nee the results of Ampère quote bove coul be re-expresse by sying tht chrge Q moving with velocity v will exert force F 1 on chrge Q 1 moving with velocity v 1 given by QQ 1 F1 = v 3 1 ( v r 1). r1 This force (which only occurs when both chrges re moving) is in ition to the Coulomb force, which pplies even when the chrges re sttionry. There is istinction between this pproch n tht of Ampère. Ampère s results pply to elements of conuctors crrying currents. These re electriclly neutrl since the bckgroun positive sttionry chrge cncels the negtive chrge of the crriers. n tht cse there is no Coulomb force n only the mgnetic force is observe. Although this pproch might seem more strightforwr, we hve preferre to follow the historicl rgumenttion of Ampère. t is lso of interest to note tht the bove eqution for the (extr) force between moving chrges coul be erive from the Coulomb force using specil reltivity on the ssumption tht electric chrge is invrint. However, gin we hve opte to follow the historicl route whereby the Coulomb force together with the Ampère formul my be regre s seprte experimentl observtions leing, ultimtely, to specil reltivity. 6. Mgnetic fiel n n nlogous mnner to our introuction of the electric fiel, we shll brek the symmetry of the mgnetic force lw n interpret it s one current proucing mgnetic fiel n the other current responing by experiencing force in the fiel. Accoringly, we split Eqution (6.1) s l r = (6.3) π r1 n F = l (6.4) where is the mgnetic fiel prouce by loop. Thus Eqution (6.3) escribes the ctive spect of current element, while Eqution (6.4) escribes the pssive spect. Active spect of current Pssive spect of current An electric current prouces mgnetic fiel. l r = F = i l π r 1 is the source of. An electric current i respons to mgnetic fiel. i respons to. PH4 / PC 6.

3 V Mgnetic Effects of Currents 6.3 Fiel of long wire We shll clculte the mgnetic fiel prouce by long stright wire crrying current of mperes. Let us first consier the irection of the mgnetic fiel t the point P. From Eqution (6.3) we see tht the irection informtion is contine in l r, which points into the pge. Thus the lines of form concentric circles roun the wire. Since l ll contributions to the mgnetic fiel t P point in the sme T r irection, only this component nee be consiere. We then fin the vlue of by integrting the contributions from long the l length of the wire. P The contribution, from the element l of the wire is given, from Eqution (6.3), by l r lsinθ = = 3 r r t is convenient to express everything in terms of the ngle φ n the perpeniculr istnce : l = tnφ l = sec φφ r = cos φ, from which we obtin π sec φφcosφcos φ = = n on performing the integrl π π π cosφφ π =. (6.5) 6.4 Force between two long prllel wires The mgnetic force between two long prllel wires ech crrying current coul be clculte irectly from the force expression, Eqution (6.1). However, since this cse hs prticulrly simple geometry, it is more convenient to obtin the force from the mgnetic fiel clculte in the previous section. The mgnetic fiel t wire ue to wire 1 is given, from Eqution (6.5) by 1 π =. Thus the force on unit length of wire is given by F =. (6.6) π PH4 / PC 6.3

4 V Mgnetic Effects of Currents This expression is the bsis of the efinition of the Ampère. Recll tht the numericl vlue of is 1 7. The force between two long wires, one metre prt, crrying one Ampère, will be 1 7 Newtons per metre of length; this efines the Ampère. We now consier the irection of the force. From the previous section we sw tht the fiel t pointe into the pge. Now the irection of the force goes s l, which points to the left. Thus the force between wires crrying current in the sme irection pushes them together. One conclues: like currents ttrct, unlike currents repel. This is the converse of the rule for chrges. 6.5 The Lorentz force Thus fr, in treting mgnetic effects, we hve consiere the force on n electric current. Now we shll exmine the force on single electric chrge moving in mgnetic fiel. l The force on the element of wire ue to the pplie mgnetic fiel is, from Eqution (6.4) F= l. then re Now we cn write the current s j. or, from the expression for j, Nqv.. The force F is F = Nq(v. ) l. Note tht v n l re prllel, so tht they my swppe in the bove eqution, giving F = Nq(l. ) v. ut l. is the volume of the element of wire, n N is the number of chrges per unit volume n since q is the mgnitue of ech chrge, it follows tht Nq(l. ) is the totl chrge Q in the element. Thus we conclue tht chrge Q moving with velocity v in mgnetic fiel experiences force F given by F = Q v. (6.7) We know tht chrge in n electric fiel E experiences force F = Q E, n now we hve seen tht if the chrge is moving n if there is mgnetic fiel, then there will be mgnetic force given by Eqution (6.7). n the presence of both n electric fiel n mgnetic fiel, it follows tht chrge q moving with velocity v will experience totl force given by F = q{e + v }. (6.8) This force is clle the Lorentz force. PH4 / PC 6.4

5 V Mgnetic Effects of Currents 6.6 Ampère s lw Ampère s lw els with the line integrl of roun close loop. There re essentilly two resons for being intereste in this. Firstly it will provie mens for clculting mgnetic fiels, lthough only in cses of high symmetry. Seconly, from the line integrl of we will be ble to evlute its curl: prt of our progrmme of obtining Mxwell s equtions. Let us evlute the line integrl of roun the inicte circulr loop. n Eqution (6.5) we sw tht the mgnitue of the fiel istnce from wire crrying current is π =. by An we sw tht the irection of is long the loop. Since is constnt roun the circulr loop, whose length is π, the line integrl is given v. l. (6.9) close loop Observe tht the, the rius of the loop, hs vnishe. This inictes tht the rius of the circle for evluting the line integrl is not importnt. ut more significnt thn this, we cn eform the contour of the integrl just s we i in evluting the curl of E from which we conclue tht the result of Eqution (6.9) hols for close loop of rbitrry shpe. The bove result is known s Ampère s lw, which my be stte s: The line integrl of roun n rbitrry close loop is given by times the totl current through the enclose re. 6.7 The curl of The curl of follows immeitely from Eqution (6.9) bove, using the efinition of the curl: 1 curl = re v. l close loop s the re shrinks to zero. From Eqution (6.9) the curl of is thus given by /re, n the irection is long the current flow. Now let us consier istribute istribution of current, escribe by current ensity j. We know tht the totl current threing (perpeniculrly oriente) loop of re is given by j, from which it follows tht curl is given by curl = j. (6.1) This is lmost Mxwell eqution. t ws Mxwell s genius tht le him to relise tht this eqution ws incomplete, n he me crucil moifiction, which we will encounter shortly. PH4 / PC 6.5

6 V Mgnetic Effects of Currents 6.8 The ivergence of There is prcticl spect n philosophicl spect to the question of the ivergence of. The mgnetic flux Φ penetrting surfce is efine s the integrl of the norml component of over the surfce: Φ=.. (6.11) surfce This efinition is similr to tht for the electric flux Φ E in Eqution (.6). The mgnetic flux oes not nee subscript; whenever Φ is written it is unerstoo tht it refers to mgnetic flux. The ivergence of is relte to the flux of through close surfce enclosing given volume. t is convenient to choose shpe tht reflects the symmetry of our system. So let us tke cyliner roun stright wire crrying current. Since the lines of form concentric circles, is norml to the enclosing surfce everywhere: on the curve sie n on the ens. Thus there is no flux of through close surfce. This rgument clerly generlises to rbitrry geometry. From the efinition of the ivergence it then follows irectly tht the ivergence of (the flux through close surfce ivie by the enclose volume) is zero: iv =. (6.1) The philosophicl question reltes to wht the source of mgnetic fiel might be. The conclusion of Eqution (6.1) bove is tht for mgnetic fiels prouce by moving electric chrges the ivergence is zero. ut re there ny other sources of mgnetic fiels? n prticulr, is it possible to hve mgnetic monopole, mgnetic chrge: n isolte north or south pole. n the erly ys of electromgnetism, there ws brnch clle mgnetosttics which prllelle electrosttics. n terms of this, if ρ m is the ensity of mgnetic chrge then there woul be n nlogue of the ive eqution stting: iv = ρ m. The existence of mgnetic monopoles is still n open question. They re require by the Grn Unifie theories, but thus fr there hs been no convincing evience of their existence. There is lso theory ue to Dirc whereby the iscrete nture of electric chrge is connecte to the existence of mgnetic chrge. f e is the funmentl entity of electric chrge n g is the funmentl entity of mgnetic chrge then Dirc foun the reltion where h is Plnck s constnt. eg = h The eqution iv =, which is one of the Mxwell equtions, is often interprete s being equivlent to the sttement tht mgnetic monopoles o not exist. Of more prcticl importnce, however, is the point tht lines of hve no beginning or en; they close on themselves. PH4 / PC 6.6

7 V Mgnetic Effects of Currents When you hve complete this chpter you shoul: know the physicl phenomen contine in Ampère s formul; know the units of the physicl quntities in Ampère s formul; be fmilir with ie of the mgnetic fiel; pprecite the ctive n pssive spects of electric currents; be ble to clculte mgnetic fiel of long stright wire n n rrngement of currents; be ble to clculte the force between two long prllel wires n relte this to the efinition of the mpere; be ble to clculte force on moving chrge in given mgnetic fiel; clculte the force on moving chrge in n rbitrry combintion of electric n mgnetic fiels; be fmilir with Ampère s lw for the line integrl of roun current-crrying wire. unerstn how Ampère s lw reltes to the curl of ; be fmilir with concept of mgnetic flux n the connection with the ivergence of ; unerstn tht iv = implies there re no mgnetic monopoles. PH4 / PC 6.7