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PhyncaUb.
A TREATISE
OH
ELECTRICITY AND MAGNETISM
MAXWELL
VOL. II,
'Sontron
HENRY FROWDE
OXTOBD TTNIVIiBBITT FBBBS 'VPABZIHOUSS
1 PATB&NOBTSa ROW
'^^
eiarenlion ^vts& ^stit&
A TREATISE
ON
ELECTRICITY AND MAGNETISM
BT
JAMES CLERK MAXWELL. M.A.
LL.D. BDIN., D.C.L.. F.R.SS. LONDON AND EDINBUKGH
HONORARY FELLOW OF THIHITV COLLEGE,
AND PR0FC5«0K OF EXPESIUENTAI. PHYSICS IN THE USIVEUSm OP CAMBSIDOE
VOL. II
SECOND EDITION
©xfotlr
AT THE CLARENDON PRESS
1881
[ All rigiU rMtrvei J
>}«.4.'^ UZi
MBS.C W,
ClFT
«».&«. PATTESSOM
COXTENTS.
PART III.
UACNBTISU.
CHAPTER I.
■LKMSitTXBT THEORY OF MAOKSTIItM.
Properlies of » iiinf[>i«t wbcu ttcled on liy ttic earth .. I
D(£nitii>n of Uio axU of the niiignot nnd uf (h« directioil of
magDBtic force I
Action of luagneU on ODD ftQotber. Lew of magnetic force .. 2
DttGnition of nugnetio anita ntid tliotr iliuiciutiojia S
Naturw of the ovidcocc for the law of magnetic force .. .. 4
Hagnetiem aa a mathematical quautity 4
TIm qnttntitieo of the opposite kiiiU» uf miigiictum in n Du^ct
Mw alweyi exactly ciiiiil , i
Effects of breaking a DiAgnct 5
A niagiiet it built up of parliclM each of irhicli ia n maf^iet .. R
Tltoory of mu^ctic 'nmttcr' fi
U^ietiiBtioD i« of the nature of a vector 7
Ueaning oftbe t«nn 'Mo^uelic Poluriution'.. ..' .. 8
Propcrtiaii of A magnetic piirticic 6
Definittona of Magnetic Moment, Intensity of IfagnettiatioD,
and CompouGDta of MaijuetJiatiou S
PoicDtial of a mugnetiswl dement of Tolumc 9
Potontial of a rongnct of finito stu. Tvo cxpredHona fur tliia
potential, eoiT«spODding respectively to llie theory of polar!
xation, and to that of mnguetic 'mutter' tO
Invet^tigatioD of the action of one magiiclic jxtrliule on luwtber 10
Particular casea 13
Potinlial euN]g} of a inug;n«tinsny Held of force 14
On the miignetic moment and axis of a mrgnet 15
vi
COKTESTS.
391. Ivs[mn»iun of the pntcntiul of h mognel m *]>)i<rical barmonic*
392. Th« ccntro of » nutgoot oad tli* primary and BeotHulary axM
through the centre .. .. .. ., I"
393. The aorth ciul of u niugnet iu tluN IrettiKe in that which poiDba
north, aod thv loutli encl that which points aonth. Boreal
magDDtiam i» that which is mpposcil to exist near th« north
pola i>f the earth and the south eud of a niUjpMt. Auflrul
nutgDClijiiH ia that whiuli bbloDga to the Noiith pole of tiic
earth and the north end of a magnot. Austral nuLguetifia
i% considered poailive 19
394. The <lir«<lion of ma^uotic force is tliat iii which aiutnil mi^
nctifn) tl^^d• to move, that ia, from aouth to north, and ihia
u the positive direction of magnetic lines of force. A magnet
l» mud to be magncLited from ita south eud tomrds ita north
end 19i
CHAPTER II.
MABXBTIC rOBCK AXB aAOXKnO IKDUCTIOBC.
395. Magnetic forco defined with reference to the magnetie potential 21
396. Magnetic force in a cjlindrie cavity in a magnet untlonDly
magnetized parallel to the axis of the cylinder 22
397. ApplicatioD to any magnet 22
398. An elongated cylinder. — Miignctia force 23
393. A thin disit, — Ma^etic iududiun 23
100. Relation between magnetic force, mu^ctic indnction, and mag
netizatioD 21
401. Ltneint€^ral of magnetic force, ornagnetic >ot«ntiaI ., 24
402. Surfnc*'iot<STnl of DwgDClic induction 25
403. Solenoidal difctriWtion of maguetic indnolion 2t;
404. 8ur&cea aud tubca of magnetic iuduGtion £7
405. Vectorpntential of magnetic indnclion 27
106. Itelations between tiio tcalar and the Tectorpoteiitlal 29
1
CHAPTER m.
UAavwnc souetoitw axd shuls.
407. Definition of a magnolio aoleiioid .. .. .i » ... i. 3]
408. Drfinittou of a complex ftolcnoHl and cxpnwton for tti ]»tential
at any point 33]
VII
400. Tlie potonlial uf a BUgnetlc sliell at any poict is tli« product of
its FtrcDgth raulliplwd hy tlie iiolid augle iiti butimlary Hib
tendfl at (be point 32
410. Anullier mi'thud of proof 33
411. Tbc potcntiiil at a point on the positire (d<l« of a sbcll uf
titmi;^b <t> exceeds that on tbo nonrcMt point on t]i« nrgativn
W(I«by4lF* 34
412. lAincllur iJiKtribiition of nia^uititim 34
413. 0>inplex lamollar distribution 34
1414. Potential of a Buleiioidal magnet 35
415. Potcotiul of u lumrllnr muj^ii't .. 3S
416. Vcctwrpotculial of a laniclUr tnfignct 36
'417. On tbe solid angle subtended at a given point by a dosed curve 36
418. The aotid Mxglc cxpmm^l bj tbr kti;lb of a curv« ou tlie s]ilier« 37
419. Solid angle found hy tnu line integral ions .. 38
420. n expreaied aa a determiuant 39
421. The Mlid ntigte in u cyclic rniictiaii 40
432. Theory of tlie Toctorpolential of a closed curv« 41
1 423. Potential energy of a niaguetie slidl placed in a ua^ietic field 42
CHAPTER IV.
ixbt'CKD lunxrnzATiox.
' 424. When a body under the action of magnetic force becomes itself
mafpietixed llie pbenomMioD is called maj^iietio induction .. 4 1
425. Uu{{iM>tiu induction in different (ulMtaiicu 4A
■ 42fi. Delinition of the cocHicicnt of induced ma^etiaiticin .. .. 47
427. Ustbonmtical tlieor)' of majjucUc inductiou. Poisson's method 47
1 428. Faraday'a Doetbod 49
420. CitM of a ImmIjt Nurruiindcd by a ini^pttic mfdium Al
430. Pouson'a pfaygic«l tbeoiy of the cauM of induced mflgnotiBm .. £3
CHAPTER V.
PABTICOLAB PIIOBI.BM1I IN MAOKETIC tSDUCnOK.
131. Tbeoryof a hollow Bphcricol abell S6
^432. Caae when K ui large 58
433. When i" = I 58
134. Corresponding case in two dimeBuous. Fig. XV 09
<t3S. Case of a solid spbere, Ibe ooediuients uf magnet iutioi) being
diflcreiit in difl'crcnt dircction> 00
Tiii COITTBSTS. ^^^^^^
Art. no*
436. The nine ooefBcieuts rcdaced to six. Fig. XYl 6
4S7. Theory of »n rllipAoid BClod on ]>y « uoiform mu(pi«tic foKC ..
438. CiteMOf TifylUt Mid of vcT) long c11ii>ii(ii'1it
iS'X HlatenKut of problcois solved by Neumann, KirclthofT. Mid
Groan
4'I(X Uclhod oFKpfwvximAtion to n nolntion of the gcnenl >robIi
when K is vcary atuM. Mng^ctic twditot ttnd towards
of most iulense magoHio force, aiul diamfi^ndic bodies teud
to places of maltott foroe
4il. Ou ship's magnotism
CHAPTER VI.
WEBKR*8 TBEOBT OF INDtTCKD MAUKKTISM.
442. ExpeiiineutA iudicatiuiT a maxiinain of moffiwtisaliOD ..
41,'!, Wrbcr'* muthcmiktiad theory nf temitorary uioiiuiTtiznltoii
4'I4. MotHftcation of the tlioory to account for residunl ini^fnetieatton
446. £xilanation of phenomeua by the modified theorj
4 16. UiiRDL'tijuitiou, denw^ctization, and ivmsfpieliiation .. .. 6^
447. EfTocts of nngnetlEatioii on Uio diiii«n*ios> of the lungnet
448. £xi>erim«nt8 of Joule
449.
4. to.
451.
452.
453.
454.
455.
456.
457.
4. (ft.
459.
4«'>,
4ill.
CHATTER vn.
MAOKETIC XKASl'RKlHaTe.
SoBpeniioD of Ibe ma^et
McUuhIk of otworvntion hy mirror and ecalo. Photographic
m«lbod
IViocipIc of oollimatioo mnployed In the Kew muj^ietomeleT ..
DetemiinatioD of Uie axti of a nu>gnut uid iif Uic diroction of
the borintDtal oooipouent of the ma^etio foreo *
Meaaorament of the morociit of a nngont and of tlM intoimty of
the liorinonlal oomponeiit of mofpictic force
Obwrvntioiu of tlollnxtun ,, ' iqq
Uetbod of tang«iUi Mid method of ainen .. „ „ ,. 102
OhaemtMO of ribwtiona I03
miminattOQ of the effects of mafriiettc indnition tU^H
Statical mMhod of tneaBuriug the bon»>iii*l force I0^
Biftlar su«pen»ioti loJB
Syiitifin of oWmliona in an ob*erv«toTj .. „ .. .. 113
OtxonrnliuD of the dipeirclc .. ., „ it;
C08TENT5. ^^^m iX
AH. f«*
^_ 402. J. A. Broun'* in«fl)btl of ourr«ct)oa .. 11t>
^M63. Joatc's nupeoBJon 116
464. Balance v«rtii.'iU force magDetometer .. .. 118
460.
466.
»467.
4G8.
469.
471.
472.
473.
474.
CHAPTER VIII.
ox TKRBESTBIAl. MAOXensU.
ElemcntsofUio magndic force 121
ComliiDatioD of tUe r«sulU of tie tiiajtnetic survey of & country 1 22
Deduction of f1i<! rxpuni^ioii of tlic mngiivtic potcutinl of tlic
earth in f^herical liaimonics 124
D^fiiiitiim of tlie eHitii* m«giietiii poles. They are uot at the
extritniticit of the ningnctic nxia. t'alte potcK. I^iey do not
oxUc on tbe earth's surface 124
Gaun' calculation of tlic 24 oocfRcieDta of Ifac firftt four lior
mouica 12S
Sepojution of external horn internal cauaea of mafpietic force .. 125
The solar mid luuur viuiutioiia 126
The pir iodic varint long ,. 120
The dislurtioDcea and Ibcir period of 11 yean » 127
Ktdlexioiia on magnetic iuvestig«tiona 127
PAKT IV.
RI.ECTROMAOKETISM.
CHAPTER I.
BLCCTROMAONETIC FORCS.
Int Ontef* tlisco^'cry of the action of an rlcctric cuircnt on &
BUgnet .. 129
i7n. The apace near an electric current ia a uugnctic field .. .. 129
^477. Action of averiicsl cnrrent i>n a onagDct 130
ire. Proof that tlw force due to a straight cnrrenl of luileHuitely
ETMt lengtb vurica inier>ely «• tlie dinlunoe 1 30
l79. KlMtronugnctic mcseure of the current 131
CONTESTS.
Ah.
480l Potential lunetiuu tlu« to n atmiglit canviit. It u a fuactioo
of iDnn; vnlues II
481. The MtioD of this ewreiit compared with that of « magnetic
Bh«ll baling ail infinite draiglit edgti and extending on one
Mi(l( or ihiK cdgo to infinity
482. A nmall circuit acts at a gtvat diiilanco lilcc a magnet ..
48$. Deduction from lluN of the action of a cloiul circuit of any form
aitd Hixe on any point nnt in tlic current iUnlf
484. Couiparifoii botweeD the circuit nud a muKnelic efaell ..
485. Uagnetic poti'utltil uf a clniMid circuit 134
486. Conditions of continuoim rotation of a mognel nl>nnt a currout
487. Foriu of ibc magnetic etjuipoleatial iiurfaoee due to a cloiiid
drcBiL Kg. X\TII .. .. ,
488. Untual action I»ct«eeii uij syafem of magneta and a dosed
current
489. BeHctioi) on the circuit
490. Force acting on a wira carrying a current and placed in th»
m^ietic lield
401. Th«orj of clectroinnguclic rotation*
192. Action of one electric circuit OD the whole or any i>ortioa of
another
493. Our method of invcftigatioo is that of Fnraduy
494. Illustration of the method apilied to parallel currvnta ..
496. IMnienHone of the Quit of current
496. Tlie wire ia urKvd fh>m the sido on wldcb its magnetic action
Kircngllienii the magnetic force and tovard* the side on which
itoppOMxit
497. ActioiU of an iufiulle straight current on any coitcnt in ita
plane ..
498. Rlateineut of the hwa of electromagnetic force. Magnetic force
dne to n current ..
499. Gnterality nf thne laws
500. Forc« aaing on a dnmit >Uccd in tlie magnetic field .. ..
801 . Qectromagnolie force is a mochanical force acting on the con
doctor, not on the rloclrie earrsnt itaelf I4Q
CIIAPTKR n.
amide's avxsnoxjKUt op the MtnuAt AcnOH op BEBcrtl
CDBKENia.
503. ARipfcrv'iiiBT«Eligatiouof tbe law of force twtireea tlw elemrnU
of electric cunvula
Art.
503.
SOI.
805.
toe.
'607.
mkw.
...
612.
SI 3.
S14.
fil5.
16.
17.
.618.
tl9.
520.
521.
522.
823.
521.
523.
OOKTEXTS. ^^^p XI
Pu*
His method of «x)ierim«titiDg 147
Ani>irc'M Iinluico 118
Amptrc'fl dm exporimoiit. Equal aod opposilc ciitreiits nen
tnlise each oUmt 148
Second exwriincnt. A cronkfitt conductor is tiquivulmt ta a
rtrnigbt onr earn,in» the mme current H9
Third GXpcrimenl. The oction of a closed current a* an d«
nient of luaoLher current in [lerpeudiculor to that element .. 149
Four^i cspcrinK'iil. GijuhI cnrreDts in sfEt«ius gcomctricall}'
simitar produce etjunl forccH 150
In all of tti«tie ex)crimcnlii the ncling cnnvnt iit a cloned one.. 192
Botb circuitti may, however, for mathcmntical purpoRRK W con
ceived as cooaistiuK of ilciucntury porliODB, xnd the action
of tbe oireailit u.> tbn rcjiultnnt of the action of tlicM elemento 1 S2
NeceKfliry form of the relations betwoen two «lemcntniT por
ttoiM of lines 162
The {j^mctricnl quantities which determine their rclatire posi
tion 153
Fonn of the components of their mutuul ucttou .. .. .. 154
Ke§o)uttou uf iheiN! in three directiontc, pni'nllcl, n«[icctivcl}', lo
the line joining them and to the elements themselves .. .. 153
a«i>eral expression for the ucttun of a finite current on the ele
ment of nnothcr 155
Condition funuahed by Amp^'s third case of equilibrium .. 166
Theory of the directrix and tbe <]ctenniniui(j< of elcutrody mimic
iKtion 157
KxprcMion of the delermtDant^ iu terms of tbe CMiiponcnt*
of the vectorpotential of the cnmMit.. ^ •• •• •• "^^
Tlio port of th« forc«i which U indntcrminntc can he expresse<l
as the space* variation of a potential 158
Complete exprcKMon for tbu octjuii between two finite ciurront* 1 59
MntunI potential of two closoil current* 169
Appropriateness of quaternions in this inrcdtigation .. .. 153
DeterminalioD of tlie form of the functions hy Amptrc'a fourth
cue of equilibrium IdO
Tbe eloctrodynamic and electromagnetic units of currcnta .. 160
Final expremon* for cteclromugncliu force between two elc
mentJi Ifll
Four different admisaible forms of the theory Itil
Oftbow Anp^'sb tobepreJerrod 162
■liili
^^^^xii ^^^^f CSOKTBIITS^^^^^^^^^B
M
^^^B. CHAPTER ni.
^^^^^^ OK TRB mnvcnoji op elcotrio ctmssKn.
^^^V
i^
ifi)
^^^H 629. Tbe luuthwl of thin trcntii« fDunilLil on thnt of Fnruilay ..
164
165
167
167
^^^^1 533. Induction hj llic motinn <>f the onrtk
168
^^^B 634. Tbv ilrctromotiva force due lo iuducUou do6B not depend on
ICd
les 1
1«»
^^^^ &37. Un of tli« galniDometer U determine the timeintegral of the
171
172
^^^H S39. Matbctnatical eipressioD for the total enirent of induction
173
174
^^^H Sll. Hi« mrtlKwl of fttnting the lnw> of induction vritii reference to
175
^^^H 542. Tlie luw of Lenx, and Neumauu't tlieaiy of indoetion
177
^^^H 513. Hc^mlioltz'tt dninetion of induction fiYim th» Bi«^ntcal action
^^^^P of curratta by the prii>ci]>le of oonseiratioa of energy ,.
177
179
179
^^^H CHAPTER IV.
1
^^^^^^ OK niK mnicnoK or a ccokkxt ox mscF.
1
1«
isfl
^^^^ 518. Oiflervui.« 1i«t««en tlu« caae and that of a tube eoDtaimng a
fl
isfl
^^^H 549. If there is motnentum it is not tbat of tbe maving et«ctricitf ..
i^B
^^^^ 350. NerenlldnH llie pbentmienK ani exactly tukalogout to Omm oI
fl
i^M
^^^H Ul. An eleelric rurrent luta rner^, which nw; \m calW diwtro
1
IBS '
^^^^1 fM?> Thu Trait* lu In fnrm a djnaaueal tbenrv ufntortrtc carrebta..
J
C05TESTS.
xB
"Alt.
$53.
Hl56.
557.
058.
559.
560.
561.
662.
563.
664.
565.
5«6.
r
668.
569.
570.
71.
ri
72.
673.
671.
I
76.
76.
77.
CHAPTKR V.
OK IBB BQUATIOKS OF HOTION OF A COXSROnO STWKH.
LsfiTaiiKe'a metbod furaUbe§ ap]>ropriate idcu lor tho rtady of
Uie bigbcr ilvniuDicnl adeatts 183
Tbcw ideas tonet be tmubted from tnaUKiiiatka) into Ay
Bunical luiigoaga •• •. •> 166
DegnMarfrMdom ofBeoniiMtod Nptom I66
0(n«raUz«d meaning of rclocity 187
Oenenliited meontDg of force 187
Qracndixed mcanuig of momentum snd impulM I87
Work done by « *m«II impvlM 188
Kiu«tio«i>ergy in tonoB of momenta, (7*,) 189
Hamillou't oqiutioiis of mutiou 190
Kinetic energy in t«rm* of lb« velocities and momenta, {T^j) •■ 191
Kinetic energy in terms of velocities, (T^) 192
Itdntiont lietweeu T^ and T^, p und q 192
Momenta ami product* of inertia nnd moliility 193
NecesMry conditions irliich these ooellicientfi must satisfy . 191
Relation betTr««u ro«tli«mttt4cal, dyuuiuico], luid cltictricid idcus 195
C IT AFTER VI.
DTKAUICAL TIIKOHX Or KLKCTKOMAOKCTIBII.
The cle«tnc cnrrcnt (iMiBMaM energy 196
The current is a kinetic phonomnDon 196
WoHc done by electromotive force 197
The mart general esprewjoo fur tlie kiuetic energy of a i^tem
iiidading electric eumtnta 198
The devUicil variables do not ag^iear in this expremioa .. 199
Mevhiuiicnl force acting on a uvitduutor 199
Tbt part depending on. products of ordiuniy relocttica «ad
ttreugtbd of currents does not exitt 201
Another cxprriinentnl teit 203
Piscufsion of tho cleclronioilim forOe 20S
If terms iuvolviug products of Telocitics and cnrrtnts existed
they would iutrodace electromotive forces, which are not ob
•erred ,. S06
CHAPTER VII.
TH80HT or ELBCnuO CiBCCITS.
The eWlrokinetic eoetgy of a sj'stvm of linear circuita ..
SbctramotiTv force in cnch circuit
.. 207
. 208
Xiv ^^^V OOlTTETrTS.
Art.
880. Eloctromngnetic force
581. OiMof two drcuito 209
582. TLoory of induced currents 210
583. U<«bHiiicii) action between the circuits 211
684. All tho pliuiinmcnn of tlm mtitiinl ticttnn of two cjrcuitit dopcnd
on a single quantit}', th» iratentinl of tbc tvo circuits .. .. 211
cuAPTEft vm.
eXPLORATlOX or TIl» nBLD lir HEAKS OF THE SECOKUAKT CIICVIT.
683. Tbfl electrokin«tic Riom«nt<im of tlic KOCODdnz}' circuit .. .. 213
686. Esprctaed AS a liueiuH^ral 212
fiST. Any ^jfitcni of iontiguuuK vtrcuita ia etjuivaleut Ut tbc ctrcait
formed by Uicir extorior bounding .. .. ,, „ 213
688. Electrokiiietic monivntTUii expressed na a surfiH:eiiiteg;ral .. 213
689. A cronkod portion of a circuit ((luivalifiit to n ntnugbt [lortivn 214
S90. £teotn>kiuetic momentani at a point expressed as a vector, 'SI .. 2 ] S
691. Itjt relation to the magnetic inductioD, tB. £<juiiUi(iu (A) .. 215
592. JostifiMtion of thcac naiDcM 31S
593. Cowveulions vrith respect to the signs of tranalations and rota
tions 217
591. Theory of a eliding piece 218
695. R1ec!truuiotive force dac to tbc motion of a conductor ,. .. 219
596. FJcctraniignetic force on tbe sliding piece 219
597, Four d^rfinition* of a line of magnt'lic inilaction 220
698. (leueml e<uations of electromotive force, (II) .. .. ,. ., 220
699. Aualj'iiis of (lie electromotive force 233
600. Tbo g«ncral cquutions referred to moving axes 224
601. Tbe ntotion of tbe axes cbanges notbiug but tbe apparent valiM
of tbe dcotric puleotitd .. .. 225
603. ElcctiMiMgnetic forc« oo n condnclor ,. 325
603. Electromagnetic force on an dooMDt of a ooDduding body.
£<uatioitt (C) 227
CHAPTER IX.
OKXS&AL BQCATIOKB OP THE KLXCTROHAOICSIIC FtEU),
R04. Recapitulation ..> » .
G05. E<)uaUonii of magaetixalioD, (D) .. .. 93
60$. llotatioD l>el«TDn inngaetir force and electric current* .. .. 231
607. Equations of electrk curreoto, (B) , .. . 232
608. Eqnationa of electric di>iilacciDeut, (F) S3j
Xti ^^^^^ COSTEXTSI.
JUt. p^
689. Tlie torn tding on a ptrticlo of « rahdmoce diw to its laagqct
inliiM) 353
040. ElcctmowgDctic force dao to an el«ctric current {Mstog Uiroagb
U „ 254
641. EipluMtion of tbete tonen bj Iho liypotluus of itran in •
mediun 253
642. QoMral character of the itreis required to proJnee iIm: [^coo
raeua 2S7
S43. WliuD there ti do magnotJuttion the stress is » tension in the
direction of the lines of magnetic force, oomlMued with a
[ireauire ia all directiuua at right angles to these lines, the
magnitude of tlie teiwtou and pruMuro betBg —• &', where 4
is the iDAgnetic force 2fl8
6H. Korce acting on a conductor oarrjing a ourreut 2S9
645. TlKory of stKwt in a tuediuin as xtated bif Forada/ .. .. 2S9
646. Kumcrical viduc of magnetic tcimion 260
A<Hnd)i I 2CI
A))i]Ciidix II 2G2
CH.\PTEK XII.
cvaMBKreatsa.
647. Daflbntioti of aotureiitslieet 263
618. Curreatfunclion 263
649. Electrto potential 264
&IH). Theofy of stead) ciimnia 264
est. 0ms of UDiform CDDductirity 2G4
652. UsgnatM aetton of a curreDtaltect with ctoeod ctimnta .. .. 261
663. MagnMlc potential duo to a citrreiittib<.'>:t
664. Inductioti of currents in a sheet of infinite eoodnctivity ..
626. Such a sheet i« i(ui>erTiou« to mafpietic actwu .. .. „ Ml
666. Thcorjr of a iilnne camntsliMt •
667. The atagnetie fnnclions vxpressod as derivattros uf a uugle
fonetioa 2G
658. Action of a TariaUe tnsgnette ^uleni on tiie nliMrt 27^
669. \^'hcn tiMre b no external action the carrenta d«ci9, and their
nagtMtie actioo dimiiushc« as if the sboel had awved off with
MBrtaut vslocdty A 271
'B60. The oumnts^ excited bjr the iastantaneoiu intrrwIuetiaQ of a
DBgnHic qrstent, prvdace an eflect eqninlriit to an image of
that systera 37i
^^^^^^^^^^^ COSTBNTa. ^^^^^^^ xvii
^Rk ^^^^ Vmtn
^HC61. This imago moves avny from its original positioo with v«lo
~ dly H 273
»$62. Tnil of imugea fonnccl hf a ma^olic Bjnit«m in coDtiuuooa
motion 272
663. UatfaematioAl ei]>resaion for the tffcot of ibc iiiduiNxl ctirrenlM 273
664. CaM of tlw niiifDrm motion uf ft magnetic pnio „ „ .. 273
665. Value of the force acting on the magnetic pole •• >. .. 274
666. Cbm of curvilinear mutiuu 275
667. Com of motion ncnr tii<^ i^gn of the nhnot 278
663. TheoT} of Arago's rotating disk 275
669 Ttuil of imajtei tu Iho form of a helix 278
670. Splierioil cumntiilici'tii 279
671. Tli« vectorpotential 280
t672. To produce a field of eonat«Dt Diaguetic force within a spherical
%Un 281
673. To produce a constant force on n suspended coil 282
674. Cun«iit» iiarulld to a plane 282
678. A {done electric circuit. A xpherical Kbell. An i:llipiiuidal
ieheil 283
76. A (olenoid 284
77. A long •olcnoid 28S
678. Force near the ends 286
679. .\ iinir of induction ooil* 286
680. Proper thick new of wire 287
tendlcu solenoid 288
CHAPTER XIIL
PAKAU.K1. CCBHKlrtS.
6S2. Cylindrical conductors 290
683. The cstenial imkgnetic action of a cyliudric wire depends ouly
on the whole ciirrrnt through it 391
684. The vectorpotential 292
6S6. Kinetic energy of the current 292
686. Ttepulftion between tW ilirevt mid the n^tum current .. .. 293
6B7. TbiiMou of the wiree. Amp^rv's «sperim«ut 293
6B8. Selfinduclion of a win doubled on itaelf 294
689. OuTsnta of vM)ing inteoMtj' in acylindric wii^e 295
6fW. BeUtion between the electromotive force and the total current 296
691. Geometrical mean di»tuace of two figures in a plane .. .. 298
^692. Particular <WM 2»9
^B93. Applicatwo of the method to a coil of insuhkl«d wires .. .. 301
H TDL. II. b
covnsn.
CHAPTER XIV.
Art. P»»
Ilt4. I'aUititial Ana to « xpWicnl bowl 303
llDfi. M>liil uiiKl«mbt«Dilcil 1i,v a circloat any point 30a
ttt>i). I'oiuLiliul energy of iwo droukrcunvaU 306
U97. MoiTimit of tho CQuplo ndiDg botwom ttvo coila 307
008. VftUiw of />; SO?
tW\. Allrodioo liHumcu two]MnJI«l oireulftr vtureuta SOS
T(lt), CnlvMlalion of iW cM>IBctrobi for n coil of finite tiction .. .. 308
Till. I'liUntinl of two Mmllol circles cxprvned b; elliptic intt^nis 3d9
703. IJuonfrom' rtMiiwI n ciifuUr coinMit. Fig. XVIII .. .. 311
rOS. [lilTerauliiJ «>)iitttion of tbo >oleiitittl of two droll* 312
TOI. Api'mxiriMtiou wlim tli»cir<^araT«iT Dcftf OMMMtber .. 313
Pt\ fcVilticr u>]>FvxinAlion 3H
riMl. (\iil uf luiuiiuum pvlfiwluclion 31G
AptwioUx I 317
ApiMUtHxIl ... .. 320
A>pw<lii. UI ' •• •■ S21
CIIAITER XV.
KuvntoMAoxmc ncsimriatxis.
TpT. StMid*rd gklnMOMton dad «nihiv« gthtnaaal
TVA. i.\MlntatiMoffti«MMkideua
n% lUMMMlftMl thpoi; 4f Uw panwoMtn
riix. rHMJ^ib «t U* iMfMrt giinMaitar aa4 the
Tit. 0*K«MiM«w«Mkal^««il .. „ „
n». i a wiKAS »B w i fc few Mr<M i i« .. „ ..
nx otMiBiiii wfci^»twa... .» „ ^
)9««^ rh<r»tkMMi»«< Iks «*••<> I
COKTBNTS. ^^^^
724. Weber's elect rod jTiaiuotneter 337
72fi. JouIc'h cuireiitwiiirbcr 341
727. Suction of nolwioiit? 342
T28. TTniforni tone uonnal to Ku^peiid«<d coil 342
729. EtectrodjmunumiTtcr with ttmiuniirm 343
CHAPTER XVI.
■LBOTBOMAOSETIC OBSEBVATtOXS,
OWrvation of vibnitioiDi 344
Motiou iu a lognriUimic npirul 34S
RMtiUncnroeciltfttioas ID a rceUtingmediuia 346
^'aluefi of Buccesaive cEoiigatious 347
PatA aud quieiita 317
PonilKin of equilibrium determined from tliroe sitcccssi^c eloD*
^liom 347
Dctermiuution of tliu lognritbDiJi: decrement . , . , , . . . S4S
Wlien to ttn[> tJir cxporimont , . ,. .. .. ... .. .. 348
Dntcrmi nation of the time of vibratioD from three transits , . 346
Two series of observationa 349
Com^ioti for nm}ilitud« and for dumping SSO
Dead brat galranometcr 330
To measare a coostaul current wilb tbc Kalvatiometer ,, .. 331
Best an^c of ilFflcxioii of a, inn^itnt gnlvnnometa 353
Beat metJiod of introducing the current 352
McaMirem«Rt of a currcut by tlie firiit elun^aliou 333
Tu malce a »i.H<.!i of iilwiTvutiiiiiH uu u constant current . . . . 354
Uctbod of multiplicutinn for fdcbtc currentH 3M
Ut^mirrmcnt of a transient current by tint elongation , . . , 3SS
Oarioction for dauipiiijf 366
Series of obw.rTatii>nK. Ztiriiekvitrfwvii method^ 357
UlcImkI of nioltipli cation 359
t
i
CHAPTER WW.
COUPABISOX OF COIU.
752. Elocbtiml mi'Aiuremcnt fooictiraea moro accurate tbon direct
measurement 301
753. Deteniiination of 0, 363
764. Delvmiination of 9, 363
755. Determination of ll>c mutual induction of two c<h1s .. .. 3R3
756. XXtermi nation of tlie self induction of a coil 365
57. CoinporiaoQ o( the selfinduction of two coila 367
ba
XX
C0STKNT3.
CHA1>TER XVm.
ELECTBOUAflXimC UKIT OY KKSISTAXCK.
Iri
758. Definition of resistance
759. Kirchhoff'ii mHhod 368~
760. Weber's inctiioil by tniiMiicot cuiTcnta 37(
701. His method of ohscrvatioD , ,. ., 37J
7B2. liVeWs niBlliod by (lumpinf; STI
763. TliiimiKiii'H mi^tboil hy « rcrolviti^ coil 37
7G4. Mnthcninlical theory of tlie rcrolring c»il
765. C'ttleululioD of the ruristonce
7flii. Correi'tioOii 871
7K7. Joule's calorimctric method , . . . 87
CHAPTER XIX.
COUPAMBOK or TRX KLSCTBOSTATIC WITH TIIE KLBCntOMAOKBIIC
CHITS.
768. Nature nad imporUiic« of the invMti^tion _37d
769. Tlic nitig of the iinito i« » velocity 379
770. Current by convpction 380
771. Weber aiid KoliIniusc!i'*melbod 380
772. Tboniaou'a metliMl liy tepanit« eleolroroeM and elcdrodyn»
inomrler 38^
773. MaxwcII's method by eombiued elcclromdrr and cledrodyoa
niometer 381
774. Etectromi>fp«tic mea.iarcni«iit of the eupucity of a coudcDMr.
Jcnkin* method .. .,
775. Uclhod by an intermittent current . .
776. Condcmer and AVippc ae aii una of ^VIieatstoae^B bridf« . . 38fl
777. Correction wtien tlie action in too mpid 3S
778. CnpAeity of a eondeosor comporwl with th« sclfiuductioD of a
•oil 387
779. Coil and Gondcnaer coubioed . , 38^
780. Electrootatie nioamre ot reaiiriance aMnjinml wilh itt electro*
magnetic mtuunt S9l
CIIAPTEH XX.
ELECTXOlUnKBTIC THEOHy OP UflHT.
781. Compariiwn at tim imygxTli** of the «ti!ctr«iBBgn*tio mMium
witi tbcM of the medium ia the uudulatory tbeofy of light
Art.
782.
78a.
■ 78*.
786.
786.
787.
788.
789.
790.
;791.
793.
793.
794.
795.
796.
797.
798.
799.
800.
801.
802.
S03.
804.
805.
I
k
ft'
806.
807.
80S.
809.
810.
811.
812.
C0XTRKT3. ^^^^^H XXI
Eoei'gX or light during its propBgation .. .. 394
Eqniilton (if propagutiuti uf nn vltxiromOiftKtic <]iHturt>auoe . . 391
Solution wlicn tliu mrdiiim IK H nonconductor .. ., .. 396
CharacleriBtica of travepropagfitioD 396
Velocity of propngution of ploi'lroiiiMfttiftiL ilialurliances . , 397
CompiinRon of thie velocity witli tliiit uf lt)j;1it , . . . , . 397
The q>ecific iii(liictir<> ca>ncity of n <licl«<ctric is tho sqasn of
its index of rufmclJou 398
Compnrisoti of tlu^xc qunntilicn in tlio cane of jMiiulSa . . . . 398
Tlicorjr of plane waves 399
Tbe electric diHpliiociueul and the magnetic dislurtiitncfl are iu
thv plutii! of lliti wii\ifroni, nud piTpuudiuuIur to eacb otli«r 400
Encrgjr and 6tre»8 during nulistiim 401
PreMurtexertt^l liy li;{)it 402
Kquationit of niutina in a cryNlulltiscd meditiiu 402
Propagation of plnnc waves 403
Only two waves are prup»^t«d 403
The iJieory agree* wilU that of Freuiel 404
B«lfttion bctwctn electric conductivity nnd opncity. . . , . . 401
Oompurison with fact« 405
Tranapaineiil melab 406
Svluliuu of thi ciiuntionii when llin tnnHuni is a conductor . . 406
Cnsv of an infinite medium, ttie initial slate tniug given . . 406
diaracteristics uf diiTatiiou 407
Dixlnrliaucc oftlic eUvtromngnctic field when a current be^ios
to flow 407
Jtapid approxiuialion to an ultimate stati! 408
CHAPTER XXI.
MAOSKnC ACTIOS OX UOBT.
PoMitile forma of tlie ridiition between ma^etiBm and light . . 410
The rotation of tlic! pUnc tif iioUrixaliou b)' magnetic uutioii. . 41 1
The lawx of the pbciiomeua 411
Venlct's discovery of ncghtive rututiou Ui ferromngnctic media 411
Rotation produced by quartx, turpentine, &c., iudepeudeully of
mn^netiiin 413
Kinenial)C«l .innljfiB of the phenomeiia 413
'Die velocity of « circularly polurixed ray is diflcireut accordin[
tif its direction of Mtotioa 413
Ri^hl and leflltHndcd rays , 414
In iiK^ia which of IhcniMlveB have the rotatory property the
velocity ia dificrcnt for right and lefthanded eonfij^iatious 414
coirrEKTS.
lit. FH«f
815. In media Acted od by magnetisiu (be velocity is diflermt for
oppiiBtt« <lireoliom of robitUia 41SS
616. Tliti lumitiii'crouii dinlvirbaiicc, mnthomnticall; conMdercd, u •
vector 415
817. Kiiiemutic ci]uatiuiui of ctrcuUrljpolnrixed liglkt 41BH
81B. Kindic piiid p[)ti:ntinl pncrgy of the modinm 417fl
819. Couditioii of w a V6 propagation 417H
820. The action of im^Tietiaui munt depend on tt reul rotation nlieut H
die dirtvliiui of Uic tnAgnctic fuive nn an nxix 418^
821. 81itt«rin«nt of tlic results of the annlysis of tbc plieaomenoa . . 418
832. Hypothc§ia of iuo)«ciihir rortiees 419
823. Variation of the vortices aeconJinR to HeluilioU/a low . . . . 420
824. Vnriiition of the kinetic energy iti the disturbed medium . . 420
Expresaiou in terms of the current and the velocity , . . , 421
The kinetic onerg} in the GUM of phme w&vcB ^^^ b
The equations of motion 4231
Vidoiity of n circulajlypolurixod my 422
The magnetic rotation 423
lt«3carches of Verdet 424
Note on a mechanical th«ory of mulcvular vorttMu . . . . 42fi
CfUPTEU XXIL
rBRBOMAUXKTISM AXD DIAMAOXUTISU KXPLAIXBC SY KOLBCtTLiS
CURUKNT*.
832. Mngnetiim i* a pWnomcnon of nioleculci it
833. The pbenomcna of magnetic molecules nuiy be imitolecE hy
eh:ctric currents 43
834. DiJftT«uce Ijctween tho olcmentary theory of continuous tnagncte
and the theory nf motccuUr cuneota 43(
836. I^iniplieity of the electric theory 431
836. Tbeoiy of a current in a perfectly conducting circuit ., .. 431
837. Cu« in which the current is entirely due to indoctioa . . . . 433
838. U'vImt'm theory of diatiwgnetiam H
B39. Mngnccrystalli^: induction ii
840. Theory of a perfiBct ooDdndor 433
841. A medium containing perfectly conducting *>licncBl molrcides 434
812. SlechaBtiad NC4iou of DLaftuelic force on th« cunmt irhich it
cxcrtM 43^
843. Theory of a molccuk with a priu)iliv« ourreut 433
811. Modificaliooa of Weber's tfavoiy 43G
8tlj. ConBeqacncoH of the tbeoi; .. 4SftH
CONTENTS.
XXlll
CHAPTER XXIII.
:oKiK8 or ACTioir at a putasoe.
lit Fkae
846. QnnnlitJeit wliicli enter into AinjiiTd's furmutft 437
847. B«btive motion of two electric pnrticlcii 437
848. B«lntivi! moltou of four electric iwrtivlcvs. Fethoer's llieory . . 438
849. Two new foroK <if Amjibrn'ii formuln 439
850. Two difl'erciit cxprcsftioQs for the forec tictireon two electric
particles in mulioQ 439
831. Tlic9c Arc ilue to Guum und to \Ve))Cr resjKctivel)' . , . . 440
8S2. All forcrs muet be coaKixtciit with ttic pnnciplo of the oou
»ena(ioD uf energy 440
'863. Wclicr'* fomiuta ia vouMatent with tlus principle liut tlint of
Onniis it not 440
854. Hdufaoltx'a dt'iluctiona from Wobor'B formula 411
85fl. Poteuti*] of two currents. 442
866. Wcbcr'n tlwory of the induction of electric currents . . . , 44 J
867. Segrogxling force iu a conuluctor 443
658. Oue of moviog condudora 444
869. Tlie formula (if flnniu IciuIh to on erroiicouM rcmtit 445
860. That of Wchcr Bgrrrs with the phenomena 445
861. LetKr of CauMio Weber 446
662. Tboiffy of RicnuuiD 416
863. Tlwwry of C, Npumatiit 446
864. Theory of Iletti 447
863. Repugnance to Ike idcu of a medium 448
, 866. TIic idea of a medium cannot he got riil of . , 448
PART in.
MxVQNETISM,
CHAPTER r.
ELEMBNTART THEORT OP HAGNBTISH.
871.] Certatk bmlios, as, for instance, the iron ore railed load
stone, the rarth itM>If, and pitices nf Bl«d witicli have been «uh<
jcctcd to <!crt<un treattnont, are found to posdeas the fullowiug
properti«0, and arc called Mo^uetd.
If, near any part of the earth's surface except the Magnetic
Polra, 8 magnet be suspended so as to turn &eely ahout a vertical
BKis, it will in general tend to set itself in a certain aeimuth, and
if disturbed from tliis position it will oBcillate al>ont it. An nn
nugnetized body haa no iuch tendency, but is in equilibrium in
all aKimulhs alike.
372.] It is fuund that the force which acts on the body tends
to cause a certain line in the body, called the Axis of tlie Magnet,
to become psialle) to a certain line iu space, called the Direction
of the MagDctie Force.
Let OS suppose the magnet suspended so as to be free to turn
in all directions about a fixed point. 1^ eliminate the actiun of
its weight we may suppose this point to be its centre of gnvitf.
Let it come to a position of equilibrium. Mark two points on
the magnet, and note their positions in space. Then let the
magnet be placed in a new position of njiiilihrium, and note the
positions in space of the tvo marked points on the magnet.
Since the axis of the magnet coiiicidts with the direction of
Iiagnetic force in both poiiitionit, we have to find that line in
he magnet nhich oc'Cttiics the nme position in space before and
VOL. II. B
after the motion. It apiM^arr, Troni tlic tlioory of the motion c
bodies of iDvariabli! Tom), that such a Uiw iiWayn vx'kIs, aud tbafl
a motion Mjtiivnlviit to the kctuul motion Hii{;ht< linvv takcQ placq
by nmple rotiitioii round thU line. I
To linJ the tine, join the first «n<i last portions of «i«;h of the
marked point*, niid draw pbn^ hisecting ibcuf linoK at rifiht
anglcn. 1^0 intersection of these planes will he the line roquirvd,
which iudiciitew the direction of the axis of the magnet and th«
dinctioD of the magnetic force io apace.
The method just doBoribed is not oonvenient for the pmctic
determination of those directions. We shal) return to tJiis aubje
when we treat of Magnetic Meaaiiromente,
The direction of the mag:nctic force is foand to be different
different parts of the earth's surface. If the end of the axtti of
the ma^et which points in a northerly direction be marked, it
has been found that the dinoctioo in nhich it sets itself in ^ueial
deviates from the true meridian to a considerable extent, and that
the marked end points on the whole downwards id the nortJMrn^
hemisphere and upwards in the southern. ^M
The azimuth of the directioa of the ma^etic force, measured
frym tlii? irue north in a westerly direction, is called the Variation,
or tlie Thliij^iietic Ueclination. The angle between the direction of_
the magnetic force and the horizontal plane is called the Ma^c
Dip. These two angles determine the direction of the mag^e
foive, aud, when the magnetic inteusity is also known, Itio magnetl
force is complete.'!) determined. The determination of tlw vale
of these three elements at different parts of the earth's surface,
the discuwion of the manner in which they vary aceonling to the
place and time of observation, and (be investigation of the caus
of the magnetic force and its variatioos, constitute the science
Tevre«trial AUgnetism. ^d
3~3.] Let us now suppose that the axes of several magnets Iiav^J
been determined, and the end of each which points north marked.
Then, if one of these be freely suspended and anoUier brought
near it, it is found that two marked ends repel each other, thi
a marked and an onmarked end attract each oUier, and that ti
unmarked ends repel each other.
If the magnets are in tJie form of long rods or wires, uniformlj
and longiludiaally magnetized, (see below, Art. 361,) it is fo
tfiat the greatest a^uifentatiun of force occurs when the end o
one magtkci ia held near the end of the other, and tlial
UV OF UAOMTIC FOBCB.
■.
pbenoiiKia can be ncconnted for by supposing that like dulft of
the magnets ivpol rach othor, that unlike eD<U attract each otiicr,
ftoi] that tlia iuUrmcdifttc pnrt« of the nin^cte Iiave no wnsible
nuttial aetion.
Tbo onds of a loDg thin ningnt^t arc commoolj called its Polea.
the can of an indelinitely lliin magnet, uniformly maifnetiEed
throughout its IvngUi, Uie c:xtreinitio» act as centr«e oF force, and
the niA 6i' the niugn^t apjK'iirs devoid of inii(>notic action. Id
mil actual nui^eUi the magnctiitation dcviut^if from uuiforrnity. so
that no single poi»t« can be lakea an the pole*. Coulomb, how
ever, by i»ing long thin rods ningnntiEcd will) aire, succeeded in
ottobluibing tiw \aw of force between Ivro magnetic pokv*.
Tie re/,u/iion heltcrfn Urn n<ii/nelic i>qU» h in He slral^St Cmejoiatttg
^_ (Mem, and i» nummtaily c^iia/ fc (it product ^ ike atreitgtks <if
^H tiie poie* d'mdvd dy the i^uare ^the duUtnte betwecH (hem.
^B 874.] This law, of course, assumes that the strength of each
^^lole is mefljiured in terms of a certain unit, tie magnitude of which
^^may be deduced from the terms of the law.
^t The unitpole is a pole which points north, and is such that,
Tr^irhcn placed at unit distance from another unitpole, it rcjiels it
with unit of forc«, Uic unit of force being dcGued as in ^VrU <j. A
pole which points couth is reckoned negative.
If OB, and Mj arc tlic strengths of two magnetic pol(«, I Uio
distance Iwtwcvu them, and/ the force of repulnion, lU expressed
numcrieally, then _ w,Wg
/ fi
But if [«»], fi] and [f] be the concrete units of magnetic pol^
I and fo[G«, then
it follows tliat
or [».] = [£Br'J/*].
The dimensions of the unit pole arc tlwrcfore J as rejpirds length,
(—1) as rc^rds time, and } as rcgnnls moss. These dimensions
are tlie same as those of the eleetrostatic unit of electricity, which
is specified in exactly the sanM way in Arts. 41, 43.
* H» KtpuniiiiinU CM BuipuUHn aitb t]lc ToraUm Bgkluicc are oonUtnvil la
the Xtvtoirt »/ tkt AeuAmg oj Pkrrtf, 1760 9, an>l la Unt'* Traiu dt eiftlfm,
IoulUL
ft
B 3
tt
Er.EMIWT.VIlY THEORT OF ilAONETISM.
[375
S7S.] Tlie ncctimoy of this law may be considered to lure
been eatatiliiihed hy tJie expcrimcntB of Conlomb with the Toraion
BaUnce, and cotifirim'd by Uio experiments of Gauss and Weber,
aod of all ob.tervera in mi^nRtic obnerviiforics, who ani every day
making n]ea8Ui«raents of magnetic i]tmiitities, and who obtain nsulls
which wculd be incontiuit«nt with each other if the Uw of force
had been erroneously assumed. It derives additional MiipK>rt fro m „
it« conBist«ncy with the laws of electromagnetic phenomena. ^H
876.] The quautity which we have hitherto called the stlrenijtl^^
of n polo may also be called a quantity of ' MagiKtiam,' iirwvjdcd h
we attribut« no properties to 'Ma^etism' except those obaerved^
in the poles of ma^ets. ^^
Since the expresgion of the law of force between given quantities
of 'Afannetism' has exactly the same mathematical form as the
law of force between quantities of ' Electricity ' of equal numerical
value, much of the muthiimatieal treatment of magnetism mustt b«
similar to that of electricity. There are, however, other properties
of iiiii<^cts which mtisl he bonic in mind, and whicb may thruw
sonic light on the electrical properties of bodies.
4
377.] The quantity of magnetism at one pole of a magnet is
always equal and opposite to that at llie other, or more generally
thus : —
In ee^ry Magnet Ikt Mai quantity of Magneiitat (reckoned al,
braically) U ten,
Heooe in a field offeree which is tmiform and parallel Uirotigbout
tliC space occupied by the magnet, the force acting on tlie marki
end ofihe magnet is exactly cqmd, opposite aud parallel to tluit
the nnmarked end, so that the revultant of the forces is a 8lati<
couple, tending to place the axbi of the magnet in a determinw
direction, but not to move the magnet aa a whole in any direetion.
This may he easily proved by putting the magnet into a small
TPMcI and floating it in water. The vessel will turn in a eer
direction, »» as to bring the axis of the magnet as near ae poesibl
to the direvtion of the curtli's magnetic forc<>, but there will be
motion of tlMj vesael as a whole in any direction ; «> that tberc cai
he no exMcs of the force towards the north over that towanlti i
■outh, or the reverse. It may also be shewn from the fact Ihai
magnetizing n piece of steel docs not ultcr iln weight. It doe« al
the apparent position of its centre of gravity, causing it in
4
IIAOSETIC * MATTER.'
Iatibi<ic3 lo sUifl along lUe ucis towHnls llic nortb. T\>e centre
of iaertia, as detennined hy the ])li«n»in«tiii of rotation, remainn
Dnaltered.
378.] If the middle of * long tliin miignH be vxamined, it is
found to possess no magnetic [rroperticit, but if tfa« magnet be
broken at that point, each of the pieces is found to Uiivc u nut^oetic
pole at the place of fracture, and this new pole ik cxiictly equal
nod opposite to the other pole belonging to Unit piece. It is
impotfiiblc, either by magnetizntioti. or liy breakitig inBgnits, or
by any other means, to procure a magnet whose jmjUs arc un
Initial.
If vrc break the long tbin mngnet into a number of nbort pict^'8
^^vc shall obtain a series of sboit maj^ete, each of which hiui [xiUs
^n" nearly the same strength as those of the original long magnet,
' I'his mnlliplicatioD of poles is not nwessarrly a creation of energy,
^^or we miut remember that after brcukiu^ the magnet we have to
^^o work to foparate the parte, in conecriucnce of their attraction
for One another.
379.] liCt OB now put all tbc pieces of the mngnet together
I** "t first At each point of junction there will bo two poles
^^zactly equal and of opposite kinds, placed in eontac;t, so that (heir
^Binilttl action on any other pole will be null. Tlic magneto thna
^ffeebaill, has tbcrefon; the same projicrtics a» at first, namely two
polca, ODc at each end, »)ual and opposite to each other, and the
part between these pole* exhibit* no lungnelic action.
Since, in tbiK mw, we know the long mngnet to be mode up
of little short niogneta, and ainoe the phenomena are the same
a* in th<' ca»e of the unbroken magnet, we may regard the magnet,
^^ven before lieing lirokcu, as made up of small particles, each of
^Kihich haa two equal and opposite poles. If we suppose all maguela
^Ro be made up of such parlides, it is evident that since the
Vmlgebraieal quantity of magnetism in each particle is zero, the
qoantity in tbe whole magnet will also bu zero, or in other words,
its poles will be of equal stren^h but of opposite kind.
^H Tieory ^ ifaynetU • Mailer'
^B 380.] Since the form of the law of magnetic action is identical
^Hrith that of electric action, the nnie rca^ons which can be given
^Bbt attributing clrctric phcnomeaa to the action of one '0uid'
' ■ or two ' fluids' c»n alw Ix; used in favour of the cxi'^tcnoc of a
BMgnclic nuitt«r, or of two kinds of mngnetic matt<T, lluid or
6
T,I,EMI!START TIIEOBT OP UAGVETISM.
vm 1
Otherwise. In fact, a theory of mi^etio matter, if need in a
purely niath«mjitic«l senw, cnnnot fail to explain the phenomena,
provided ncir laws arc freely introduced to account for thv nctiul
fact*.
Onu of theeo new lavrs must be that the ma^ette fluids cunn'it
jiaiw from one molecuh) or particle of the mairnet to another, I>ul
that the proccMt of magnetization oonsixtii in eeparatia^ to a certain
«xt«nt th« two fluids within each particle, and caurin^ the one fiaid
to bo more concentrated at one end, and the other flnid to bo more
conoentrated at the other end of the particle. This is the theory of
Poisson.
A particle of a mnpnctizahle hody jb, on this theory, annlojoiis
to a email insulated conductor without char^, which on the two
fluid theory contjiins indefinildy Xar^ but exactly cjiiaJ qiumtitieK
of (he two electricitiest. When an t^lrctromotiifo fwrce acU on the
conductor, it Mparntei' (he elect riciticn. canning them U> beooi
manifest at opposite sides of the conductor. Iti u similar munnfti
according to this theorjr, the magnetizing foreo OHuses llie two
kind* of magnetism, which were originally in a neutrah'zed state,
to W separated, and to appear at opposite sides of the magnetized
particle.
In certain mbetancee, sttch as soft iron and those magnetic
substances nhich cannot bo permanently magnetized, this magnetic
condition, like the eloctrif ligation of (he conductor, disappears wfa
the inducing force is removed. In other euh<^aDcea, such as
Btcwl. tlic magnetic condition is produced with difficulty, and, w
produced, remains aOer the removal of the indncing force.
I'his is expressed by saying tliat in (he latter case there is
Coercive Force, tending to prevent alteration in the miignetizati<
which must bo overcome before the power of a nugnct enn
either increased or diminished. In the case of the electrified li
thii> would corrcjiiKUKl to a kind of electric resistance, which, imli
the renistunce observed in metala, would bo eqoiTalent (o complete
iaeulation fur electromotive foreca below a certain value.
Tilts theory of mognetUm, like the corresponding theory
electricity, is ericleiitly too large for the favta, ami requires to
restricted by artificial conditions. For it not only givos no
why one body may not dilfer fmni another on aecoiint nf \m
more of both Huids, but it enabUw as lo mty wluib would be
pn)priieB of a body containing an excess of one tnagnelin fl
ii in true that a mson is gircn why such a body cannot
tioi^
>lik«9
MAONRTTC POLARIZATION.
^
but this TMmn i* Only iotrodiici^d m an ancrtlioa^ht to explain
UiiB partjciiliir fiict. It docs nut (jrovr out of the thcon*.
S81.J Vi'c must thervfoix' Bcck for n moilu of ixpression vrhich
lull not bo atpnble of csprvMiin^ too much, and which sluill leave
room for the introduction of now idvas ua Wietc art duvolopcd from
iiRW fnctK. Tbi?, I think, iva mIiuII nhtnin if vrc Ini^in by suying
M tbul Ihc particles of a nia^not arw PoUrizid.
^^^^^B Meaning of fie Urm 'Polarization.^
^K Wlion » particle of a body poHsesMO!* properties related to a
" OHrtain line or direction in the body, and when tin' body, retaining
theae properties, i§ turned Bo that this direction is revened, then
if as regarda other bodies these prapprtiea of the i)artiole are
reversed, the particle, in reference to these properties, is said to be
jKlarized. and the properties are said to couetitute a particular
kind of polarization.
Thus we may eay that the rotation of a body about an a\i9
constitutes a kind of polarization, because if, while the rotatidn
continues, the direction of tlie axis le turned end for end, the body
will be rotatinfj in the opposite direction as regards spiico.
A oonducting particle through which there is a cnrrcnt of cloc
tricity may be said to be polarized, becyiusc if it were ttirmil round,
and if the current continued to flow in the »imc din^vtion as re<;ar<.ls
I the piirticlc, its dir«(ion in space would be rcvcrHcil,
In short, if any muthcmaticsl or physical tjiiantity is of tJic
Datur« of a vector, as defined in Art. 1 1, then any body or purtiole
lo which thiit direcU'd (jiiantity or vector belongsi may I* said to
Iw Poliiriziil *, bi'ciiiiKi' it huK opposite properties in the two opposite
directions or )>oU>k hC the direct<d <{imiitily.
The poles of Mu ciirth, for exaniph', have reference to its rotatum,
and liave aecordinyly dill'crcnt namex.
■ Tliii worl PolnHttlixii hM txifn iibhI in % wnit* ni>C cimdiil^^t with thi* In
Optica. >ili«ra » T*y of ti(rli> i* nfrl to Iw p<>laiiied vhuii il luui >r<ijiijiJa' nlatiii);
to it* •i'its. whith lur iileiiticnl (m nppontc tiHtn of the t*y. Thii kin'l <>( jn'taritiiii'Mi
tVttn W aofttktT kind f Directd Qiinallty, wbld) nuv bo «a11«I ■ IHpolar l)mnili[y,
ta opficidllnii Ui tlw )urmii kind, vbluli iuut (hi calltul tlnlpolar.
wiMa ■ illi'iUr iiut>ai(y 1> liini*<I aii^ (in i>n<! it ruiiinliu llif umc u bcroni.
TeoaiooB «nl FmanrH in wljd l>odM*, Eitmvi.iiui, <.Viii[im»i'iiii^ >uit Diitortinn*
Mill 11104 uf (hf D>ti(]kl, clactriol, (md Diagnttic propenioi of ur^ratalliUHl bodlu
an illpAltr <iM»tiliH.
kllji I , {fodvcod hv ninRn'tiinn in lmi>f«K»i IioJIm of t»i«liiii Ilia pliuw
ifif )• f lliv Inti.Wil Uf!" '' ''It* iiint:ii' lixti Itulf. It UDipulBr [rvpoty.
IW r..— •;.'.• ,'>.(Mrt7 nifHrml t« in .Irt. 303 it sbu uiiipolMr.
\
8
ELKHEITTART TnEORT OT MAONETlSir.
Meaning fjfthe term ' MaynetU Poiaritalion.'
S82.] In spcakinf? of the Rtut« of tho pnrtkW of » inftf^et as
ms^i'tic polarization, wc imply thftt «K^h of the BniiiHcst parts
into which ii mugnct may bo iliviilod has certuin properti<« related
to a d«6nite dircctJon through tlio parlicl«, eallod it* Axis of
Uagoctization, niul tliut. thn prnpertitt) n>Iat«d to one ond of tlii*
axis are opposittt to tlie propcrlitn reUt^tl to the other en^l.
Th« pTO]x>rtic« which wo attribute to the particle are of the «amo
kind tu> tho^e which we observe in the complete magnet, and in
BMstiniing that the particles possesi tliese properties, we only a«9«rt
whul we can prove by breaking the magnet up into Binall pieces,
for eacli of these is found to be a magnet.
M^
Propertiet of a Magxelized ParlicU.
383.] Let the clement ifriiyth be a particle of a magnet,
tot us assume that its magnetic properties are those of a magnet
the strength of tvhosc ponitivo potc is m, and whose lengrth is ■/«.
Then if P is any point in space distant r from the poflitire pole and
/ from the ncgativo pole, the ma^etic potential at P will
 due to the poritiTe pole, and , due to the negative pole,
r''^[''r). (I)
If di, the distance between the polcM, i« \er7 small, wo oiay put
/— r = (/»eoti€, (2\
where * is the an^le between the vector drawn from the magn
to P and the axis of the magnet, or
(
— mdi
r=^c08«.
MagHttle Sloment.
b.] The product of the length of a unifonnly and longitud
inally inagtKtized Itar magnet into the strength of its pu&itiTO pole
is called W* Magnetic Moment.
Initiuitg of Ma^lhathn.
The intensity of magnetimtion of a magnetic particle is tlie 1
uf its magnetic moment to itw volunw. ^Ve •liull denote it by
■The maifnvtization at any point of a magnet may be define
hy its intensity and ita direction. Ita direoUoD may be dennwl
itfl direcUoooosinea A, fi, r.
3851
PTTie
COMIftNKSTS OF MAGNETIZATION.
9
I.
I
C«mjionrHl4 (}f Jifa^iiefhaf4on.
the magnvtixiitioii nt a point of n inar;n<!t (boinj; a vector or
Oirc«1<Ml quantity) muy hv cxpTc»c(I in t^rms of He throe cum
poa«nto rercTTiKl to the axc« of coordtoAtos. Calling tbcse A, £, C,
A = l\. B = ItL, C= !»,
and the numerical value of / i« given by the equation (4)
I 7/« = ^ + J!» + C». (5)
885.] If the portion of the. magnet ivhicli we consider \» the
difTereDital element of volume (/xi/^</;, and if /denotes the intensity
of magnetization of this element, ita magnetic moment is Iiinl^dx.
Suhetituting this for meh in equation (3), and rememberia^ that
where ^, 1, C are the coordinates of the extremity of the vector r
drawn from the point (x, y. :). we find for the potential at the point
[it li due to the magnetized element at {x,y, t),
{^(e*)+B(ly) + C(C— )} ' 'i^'f^'l'.
(7)
To obtain the potential at the point {(, ij, () due to « mi^et of
6uite dimensions, we must find the integral of this expres.ion for
every element of volume included within the space occupied by
the magnet, or
L ''=///{'' (f') + 5 {l.y) + C(C^)} ~ da,d,dz. (8)
Int«grated by ports, this becomes
where the double integration in the first tliree terms refers to the
Buriace of the magnet, and the triple integriition in the fourth to
the space witJiin it.
If /, n, R denote the directioncoBincs of tlie normal drawn
outwards from the element of surface dS, we may write, as in
Art. 21, the sum of the first three t«rn»,
Ijk ff(lAimB+ ttC) ' dS,
where the inl^ration is to be extended over tlie whole surface of
li« augnet.
10
KLBHESTART THEORY OF MAGSETISSr.
If we now introdace two new symbols a and p, ilelinet! Iiy the
equations v = U ^mB+nC,
,dA
fIB dC\
— — r
the expreseion for the potential may be written
88(i ] Thin expnwssion is identical with that for the electrii
poti'niiiil due to a boily on the surfiici: of which there is an elc
trifientjon whose nuHiioedensity is /t, while throughout it« substance
tliorc IB a hwlily elect rificut ion whose volumedensity is p. Hence,
if vfv iixKumo a and p to lie Ihe mirfiice and vol uniedensi ties of the i
diittribtition of nn imaginary kuWUiiiix. which we have caltedfl
' inagnftic niatt^T,' the [lotentinl due to this imaginary distribution™
will lip idcRtieal with that due to the actual mn^notization of every
element of the ma^et.
The surfaoedensity a is the resolved part of the intensity
magnetization / in thcdireetion of the normal to the Kurfaec drawn
outwards, and tlie volumedensity p ia the 'oonvcrgcnoe* (bm
Art. 25) of the magnetisation at a ^iven point in the nia^et.
lliiM method of reprebenting the action of a magnet as Am
to a distribution of ' magnetic matt«r * is veiy oonvciiieol, but
most always remember that it is only an artificial method of
representing the action of a system of polarized particli's.
;ry I
on
'J
Oh fie Action <^oite Magnelic MolecuU oh atio4ier.
387.] If, as in the chapter on Spherical Harmonies Art. tS9,
d d d
we make
1/
dA~ dx'' "dg
■""^+"37'
(0
where /, m, « arc the directioncosines of the axis i, then tli^
potential due to a magnetic uiolecule at the origin, whose axis
parallel to i^, and whose magnetic moment \» Wi, is
(=
wlioiw A, is Ihc en«ine of thr angle between iJ, and r.
Again, if a M<eond magnetic molecule whoxe moment b m^, i
whose axis Is {larallel to A,, ia placed at the extremity of the radiua
vector r, thi> jiolential energy dne to the action of tl»c one magnet!
on the iithiT i«
587.1 FORCE BETTTEEN TWO MAGN"EnZED PARTICXES. 11
r=
Hit
dK
dk.
1
1B,M.^
rf"
lii.dk.
&
tit*jn.
= 4i'{*'u3A,X,),
(3)
(*)
mako with ench
7«rc fij. ia the co»ine of tho angle wliu'li the nxt^a
other, And Aj, A, are th« i^OHmca of tho auglcti which tliey make
with r.
»Let us next dotermme tlie moment. cT tin cfiuiile with which the
fint magnet tends to turn the second mund its cenlre.
I Iiet OS 6Dppo§e t]ie eecond magnet turned throu{;h an an^le
d^ in a plane perpendicular to a third axis ^3, then the work done
RgaiDSt iiw magnetic forces will be ^ ^ ilip, and the moment of the
force* 00 t)i« mngnct in this plane will be
TTie acdtal moment aclinff on the SLcond maffnet may therefore
»c considered m the resultant of two coaples, of which the fintt.
leta in a {dane iiaratld to the axes of both mnffncts, and ttMide to
mereaAe the angle between Ihcm with a force whose moment is
^«m(^.^.). (6)
while the second coupU; acts in the plane passing through r and
the axis of the second mn^^net, and ttnds tn dminish the angIe
between these directions with u force
■ _^!!!sco9(r^,)8in(rAJ, (7)
whore {riy\ (ri,), (*,i*j) denote the angles between the lines r.
To determine the force acting on the second magnet in a, direction
frallr'l to a line h^, we hare to calculate
I si's
= — MiBi, —J—, by Art. 128ci
= 3X,5^(^«5M,)+3^,,'!^^'A,+ 3^"'^'A.. (10)
If we Hupw«o tho Bcluid force compounded of three fefw«, H,
If, and //j, in the ilircctiona of r, i, and i^ respectively, then the
>roe in the direction of i. i»
AjA+fr^//, *^//j,
(")
12
ELEireSTAItY THEORY OP MAGNETISM.
Since the direction of ^, ia arbitrary, wo mufit faave
/I, — />J,
y/^=^i!^A.
The force It '\b a repulsion, tending to increase r; /T, and
act on the eecond mngnet in the directions of Uie axes of tb«
and second magnet rpspcctivoly.
Tliis analyGig of Hie forus iictinf^ between two email tnaji^ei
was first given in t^mas of the Quaternion Anslysis by Profeesor
Tiiit in the Qiiarterly Mali. Jouri. for Jan. tSGO. See obo bis
vork OD Quat^miont, Art. 414.
Particular Po»iti<mt.
S88.] (1) If A, and \ are each equal to 1, that is, if tlie
of the nm^^net'S arc in one straight Hnc and in the samo direction,
Pi^ = 1, and the force In'tween the magnets is a repulsion
ie+A+^=^
(^^i
The negative b!^ indicates that the feroQ il bd itfoution.
(2) If A, and A, are zero, and }iy^ unity, flu am of the ma^ot*
are parallel to each other and perpendicular to r, and the force
is a repulsion 5«,^
In neither of these cases is there any couple.
(3) If A, = 1 and A, = 0, then ttj^ = 0. (1
TT»o force on tlic second mafptt will be — 1i in the dircctioi
of its axis, and the couple will be — ^^ , tending to t*irn it parall
to the iirst magnet. This is equivalent to a single force — \ '
acting jiantllcl to the direction of the axis of th« second
and cutting r at u point twothirds of its length from m^.
t
r«. 1.
Tims in the ligtirc ( I ) two msgnets are made to ilokt on water, i
rOSCE BOTWEEK TWO 8KALL MAGXETS.
18
^,
bein^ in the direction of the axU of nr,, hnt hitwing its nwn axu
»t right nagiea to that of m^ , If two points. A, h, rigidly connected
with m, and m^ respectively, are connected by moans of a string T,
.he Bjstpm will be in equilibrium, provided Tcuts the line m, m^
at right angles at a point one third of the distance from m^ to m^.
(<) If we allow the second magnet to tura freely about its centre
till it comes to a position of stablo i^nilibrium, /^ will then be a
minimnm as regards 4,, and thcrirorc the resolved part of the force
due to m.j, taken in the direction of ^,, will ho a muximum. Hence,
if we wish to produce the greatest possible magnetic force at a
iTtn point in a given direction by means of magnets, the positions
of whose centres are given, then, in order to determine the proper
diTcttiuns of the axes of these magneto to produce this c9oct> we
TQ only to place a magnet in tlic given direction at the given
point, and to observe the direction of stable ctiuilibrium of tlic
Axis of a second magnet when its centre
\\» placed at each of the other given
poiolie. The magnets must then Iw
placed with thoirax(« in the dircclioiii
indicated by that of the second magnet.
Of couree, in performing tbia experi
ment we muMt lake nocount of terrestrial
magnetism, if it exists.
Let the second magnet lie in n posl
'tion of stable auilibrium nH regardit ita
direction, tlien since the couple acting
on it vanishes, the axis of the second magnet most be in the same
plane with that of tlie first. Hence
d the couple being
^ (sin (^1 ,*^— 3 cos (i, r) sin (r Aj)),
we find when this ie zero
tiUl(^,r) = 2itkU{rk^,
tan //, M, fl = 2 tan Rm^ H..
a ^ ' D K
Fig. 3.
(16)
n
(17)
(18)
(19)
When thta portion has been taken up by the aecond magnet the
value of Jtrbeoomee
rfr
or
AVq
rfi.
^vb«r
Ii«re ig ia in the direction of the line of force due to m^ at i^.
14
ELEJIENTAKY THEOBY OF MAOKETISU.
HODOC
(2oy
Hence the ^vc^nd magDct will tend to more towards [Joccs «^
greater rM(iIt«nt forre. ^
Tlic furcc on thu sicund ma^ot may be dccompoecd into a force
R, n'liirli in this caw is nlivikys attractire towards tbe first mu^ct
aud 11 forov 11^ jmrulli:! to ttiu iixts of the first magnet, where
CT lllj
A,
lOIDBj
(2l"
'^ /SAif+l
In Fig. XIV, at the end of this volume, the liuee of force and
vquipotentiiil surfaces in two dimendons are drawn. The ma^ct*
which produoe them arc suj'iiosi'd to be two lonf> cyliiidncnl rods^
the Kctioiia of which iire rcpivwinlcd by the circular blank sp.
anil these rods are magnetized transversely in the direction of t
uri'ows.
If we remember that there is a t«n3ion alonir the lines of force, i'
is eney to see that each maeiiet will t4nd to turn in tlte direvtion'
of tbe motion of the hands of a watch.
That OD the nght hand will also, as a whole, tend t« more
towards tlie top, and tbat on the left band towards the hoi
of the page.
On He Potential Ktifrpjf <^a Magnei plaevi t» a Magwtxe Titld.
8B0.] Let V be the magnetic potential dne to any sjrstem <
magnets acting on the magnet under considemtioD. Wu aliall call
^the potential of the external magnetic furou. ^M
If a small magnet whooe strength is iw, and whose leuf^th is d*^^
be placed so that its poisitive pole is at a point where the potential
18 V, and its negative pole at ii jwint whore the polentJul is ¥\ tbi
potential energy of this magnet will be w^T— f"), or, if d$
SMjosuml from the nc^tive pole to the positive,
If / is the intensity of the magnctitation, and A, ^ v its dir
tioncoeincs, we may write,
fiiiJ* = Idxdtfdt,
, dv dr dv dv
and, finally, if A, /I, C are the components of magnetization.
590]
POTESTUL KNEROy OP A MAOSET.
15
(.1*1 ». " ' ^"'\ > < •
(»)
CO tliat th« exprcKsion (1) for the [lotciitial vniTgy t^ the clement
»f the nacgiwt Ijecomv*
:
ITo obtain tlie pot«ntial encTfiy of a raagnet of finite siiw, wo
FDUst inte^rrate tliis cxpreeaion for vxety cleroeot of the inagnitt.
e tbue obtain
RS th« v»Ihc of the potentiftl cnt^rgy of the magnet witli reiipect
^^U> tbe in»gn«tic fitld in whicli it is pWcd.
^B The poleiiliai energy is hero expressed in terms of the eoniponentu
^^of millet tzutiou and of those of the magnetic force arising from
extcrnid vituse*.
By integration hy pnrts we may express it in terms of the
dtstiibntion of mugcietic mattvr and of magnetic potentiaJ
^=f}\Ai^B^^Cn) FdSjjJF {'^ + ^^ + If) ,/W^A (4)
where tt'h'' are the directioncosines of the normal at tho olumciit
of suT&ce i/^. If we substitute in this equation tlie exprossionti for
the snrEncc and volumcden«ity of magnctie nuiltor iw ^vvn in
Art. 38G, the cxprcofion becomes
H #= fJrffdS+ fjjypdxdydz. (5)
H We may write equation (3) in the fonn
H fr=~JJj(.4<t+B0iCy)dxdydi, (8)
^pvhere a^ ^ and y sae the components of the external magnetic force.
^^^V 0» lie MagMlie Moment and Axi* of a MoffMft,
390.] If throughout tbe whole space occupied by the magnet
the external mngnetic force is uniform in dirt'clion and magnitude,
the compODciitK n. j3, / will be conslaat quantities, and if we writ»
ffjAdxdydi = iK, JJjsdtdydi = mK. jjj CJjedyd: = « A', (7)
the int^rations being extended over the whole substance of the
UMignet, the value of Wmay be written
r=A'(/a + «.^ + ay). (8)
16
ELEJIESTAHt THBOBY OP MAGNETISM.
[391
In this oxprcsiion A *<< ■> sre tlic jirfctioncciBinee of tho axis o\
the maf^it, HD<1 A' is tho magnetic moment of the ma^ct. If
< is tlitt Bn^lti which the axis of the magnet makes with tlic
diivctioD of the mitgiiutic force ^, the value of ff' may be written
Jr=J.^oo9f. (9)
If the magTiet ia siiepcnded so as to be free to turn abont a
vertical asie, as in the cnsc of an onlinar]' compass needle, let
the azimuth of the nxis of the magnet be <^, and let it be inelinrd
$ to the horizontal {ilune. Let the force of terrestrial magrnctism
bo in a direction wlioi^ azimuth is 2 and dip C ^hen
a = ^ cos f COS 6, (9 = •& cos f sin 8, y = ^ sin f ; {10)
f = COS ^ cos ^, ffl = cos sin ^, n = sin tf ; ( 1 1)
whence W= — Jf^ (cos f oos tf cos (♦— 6) + sin Csin 0). () 2)
The moment of the force tending' to increase ^ hj turning the
mngnet round a vertical axis is
I
— J— =— A'^coafeo«fl8in(^— 8).
(IJ)
On He Expantion ^ tie Pofmtuilofa IfaffUffl in Soliil llamonic:
391.] Let rbo the potential due to a unit pole placed at the
point (i, jf, C), The value of V at the point x,y, e is
r= {(f«)» + (^,)« + (fr)'}*. (1
Thii expression may be expanded in term« of splierical lukrmooica,
with their centre at the origin. We have then
r=ro+r,+r,4&e., (»;
V^ts,T being the dialanee of (£ *f, () from the origin, (3
'^i pi •
n
whcTA
in
(»1
To det«nniiie the value of the potential enei^py when the magnet
is placed in the Beld of force expressed by this poteatial, wo have
to intes™'* *•"? expression for ff'in equation (3) of Art. 389 with
reapect to ^, y »n«l ". considering f. ^. fand r as crtustants.
If wc consi«ier only ibe terms introduced by f\, f\ aui] jr
result will depend on the following volaino.iti(egnilB,
392] KXPANSIOtf OF THE POTENTIAL tvZ TO A MAGNET. ■ 17
lK=jjJA<ltdyiz, mK^fjfsdxdsdi, nK=jjfcdxdydt; (6)
^L=JJfJxdxdfd3, M=ffJBsdxdydi, N=fjj(kdxdyds\{7)
P=JjJ(Si+ Cf)d^d9d^, Q =JJJ{Gf+Az)d^dydt,
K B=fff{Jj, + 3x)dxd,dz. (8)
We tUuR find tor the tbIoo of the poUintiul energy of the tnagoet
j<la«ed in [)rc«(!nce of the unit polo ftt tbv [Kiiiit ((, >}, ().
I
llf~y)+y,i(23t~?rL ) +C{iyl'M)i3iP,iC+QatRin)
+&«■
This €XprcenoD may also 1» rognrded as (he potcntinl energy of
thv unit pole in presence of the nia{>iiot:, or moio Bimply u tlie
potential iit the point f, >;, ( due to the magnet.
I On Ai« Centre of a Magnet and U« Primary and Secondary Axe».
392,] This expression maybe simplified by altering the directions
of thw ooordinatea and the position of the origin. In the first
place, ve Hhall nuke the direction of the axis of x paraU«l to the
I, a xifl of the nm^et. This is ecjaivalent to making
H '=1. m = 0, n = 0. (10)
If we change the origin of cwnlinatos to the point (*',y', s"), the
dircvtionia of th« axes rpmaining uncliang«d, the Toluraeintegrala
IK, mK and mK will remain unehangit), but the others will be
altered as follows :
V=LtK^, ir=MmK/. N'=NttKA (U)
P'=PK(m^in/). <^=QK{njr+U'). ir=R~K{{yi miT). (l2)
If we now make the direction of the axis of x parallel to thu
ftxia of the magnet, and put
('■>)
V=
(13)
2^^' 'JC' ' K'
then for the new axra M and iV have their valtics imclianged, and
lie valne of J/ becomes J (i/"+ jV). P leninins unchanged, and Q
Jt vanish. We may tltenfore write the potcntia] thus.
FT.KMKSTAnT THBOBT OF SrA0SBTrSH.
"We liavc thus fowD<l a point, fixed with respect to the magoot,
such ihitt thi; second bvrm of the potential auumes the aioKt nimpHH
form whin tlitu point ia tnken as ori^o of coordinates. Tim poi^^
we then*ror<> <Iclin« as the centre o( the niag;net, and tlie *xi«
drann throufrh it in the direction fonnerly defined as the directi<
or tlie Ris^ctic oxiii may be deliued as the principal axis of
magnet.
We may aimplify the result still more by turning the axes of jr
and s ronnd that of X through half tlie angle whose tangent
P
•^ — i,, lliis will oaaee P to 1>econie xero, and the Gnnl ft
M—JV
of the potential may bo written
H
This is the simplest form of the first two terms of the potential
of a magiiet. When Iho axes of y and x are thus placed they
he culled the Secondary axes of the magnet.
We may u\m dutinnine the centre of a msf^et by finding the
positiun of the origin of coordinates, for which the surfaceintegral
of the SEiuarc of the second term of the potential, extended oye r ,
a Bpherc of unit radius, is a minimum. ^M
The quantity which is to be made a miuimtim is, by Art. 141, ^^
4{L' + .ir'i,V^~M.VNLr.M) + 3(F^ + g*+S'). (16)
The changes in the values of this quantity due to a change of
position of the origin may be deduced from equations (1 1^ and (li
Ilenoe the conditions of a minimum are
il (2L—MA')+ 3uQ + 3mlt « 0, ^
a«i(2JAA'£)+3/fl+3)iPiaO, }■ (l
iH{2iVL5[) + 3mPt3 I q =: 0.)
If we assume / = 1, n = 0, m=0, these eo&ditions b«ootD«
2 LM—A' = 0, e = 0, A =a 0, (I
which are the coodilioixa made vte of in the pr«TioiiB invc
gation.
Thia inTostigation may be compared with that by which
potential of a syatcm of f>nvitating matter is expanded. In
latter case, the most ooDvenieat point to issumo as the ori
is the centre of gravity of the systtm, and the most conven
axes are the prineipal axes of inertia through that point
In the case of the mai^net, the point oorreM[>ondiiig to the cent
of gravity is at an iutinitc distance in the direction of the
CONTESTION RRSPECTISO SIOSS.
I
19
Bnd the point wliicb we coll the centre of the magD«t is n point
lutvin^ (lifTerent properties from those of the centre of gravity.
The quantities L, J/, jV correspond to the moineutji of inertia,
and P, Q, A to the products of inertia of a material body, except
that L, Jtf and .Vare not necessarily positive quantitiea.
When the centre of the ma^^net is token as the origin, the
•phcricat harmonic of the second order is of the sectorial form,
having its axis coinciding with that of the magnet, and this is
irue of no other point,
^^'hen the magnet is eymmctrical on all sides of thia axis, a«
the case of a figure of revolution, the term involving the harmonic
of the sGoond order disnpp<;srs entirely.
S93.] At all part« of the earth's sarface, except some parts of
i« Polar rc^onu, one end of a magnet points towards the north,
^ftt Iciut in a northerly direction, and the other in a southerly
on. In speaking of the ends of a magnet we shall adopt the
popuhir mi'thod ol' calling the end which points to the north the
north eiKl of the magnet. When, however, we l^pcak in the
langtuige of the theory of magnetic (luids wo shall uho the words
real and Auntnkt. Boreal magnetism is an imaginary kind of
matter uipposcd to he most ahunduut iu the northern purls of
the earth, and Austral magnetism is the imaginary magnetic
tter which prevaiU in the southern regions of the earth. The
nelism of the north end of a magnet ist Austral, and that of
the south end is Uoreal. When therefore wo speak of the north
aod sooth ends of a magnet we do not compare the magnet nith
the earth as the great magnet, but merely express the pusition
which the magnet endeavours to fake up when free to move. When,
on the other hand, we wish to compare the distribution of ima
ginary magnetic Suid in the magnet with that in the earth wc shall
nw the more grandiloquent words Boreal and Austral magnetism.
894.] In Kpi»king of u field of magnetic force we shull um the
phraae Mognetie North to indicate the direction iu which the
north end of a comjiaKs neetlle would point if placed in the Held
of force.
In speaking of a lin« of magnetic force vm Rhall always KUpjiose
it to be traced from magnetic oouth to magnetic north, and ithall
otll this direction positive. In the same way the direction of
magnetization of a ma^et ia indicated by a line tirawn from the
south end of the magnet tonarda the north end, and the end of
tbe magnet which pointA north is reckoned the posibive end.
Hbo
ma
the
Bnatt
Kuagi
■ the I
20 ELEUEKTABT THEOBT OF UAQNETIBM. [394.
We Bhall consider Austral magnetism, that is, the magnetism of
that end of a magnet whiob points north, as positive. IF we denote
its numerical valne by m, then the magnetic potential
and the positive direction of a line of force is that in which V
diminishes.
CHAPTER II.
MAGNETIC yOBCB AND MAGNETIC INDUCTIOS.
I
896.] Wb h«vc already (Art. 385) determined the magnetic
tcntial at a given point <lti« to n maf^not, the tnagnetization of
which 18 given at every point of its substance, end we hare Hhenn
that the maUicmatical rcKult may W Lxpre«eed either in terms
of the actual oiitguetization of every elvmctit of the magnet, or
in terma «f an imaginary distribution of ' mngnvtic matter,' partly
condensed on the surface of the magnet and partly dillb.tcd through
out its suhstanee.
The magnetic potcati»1, as thus defined, i* found by the same
znatlieniatical proceee, whether the given point is outside the magnet
or within it The force exerted oo a unit magnetic pole placed
ttt any point out«ide the magnet is deduced from the potential by
the Ewne process of difTcrciiti^ilion as in the corresponding electrical
problem. If the com>oneuts of this force arc a, fi, y,
dF . dF 4F
H' ^=.
m •& ■ " ~ rfy ' ' ~ ./r *'*
H To detcrmino tiy experiment the magnctie force at a point witJiin
Htbe magnet we must begin by removing part of the magnetized
^uobatance, ao a» to form a cavity within which we are to place th«
awgnetic pole. The force acting on the pole will depend, in gei>enil,
on the form of this cavity, and on the inclination of the widls of
the cavity to the direction of magnetization. Uencc it is ncccMsary,
in Older to avoid ambiguity in speaking of the magnetic force
^^irithin a magnet, to Bi^cify the form and position of the cavity
^pritliin which the foroo i» to be measured. It ia monifeat that
when the form and pn«ition of the cavity is specified, the point
ffithin it nt which the magnetic pole is placed must be regarJi'd as
nt 11
BUGKETIC FORCE AKD HAONCTIC INDUCTION.
no longer witbin tlie substanco of the nui^et, nnd thcitifoK Dm
ordinaT} methods of determtDtng; the force bccom« nt once nppltotble.
396.] Let 119 now consider » portion of H miif^it in nIik'li the
direction and intensify of the niii<piftiitat.ion nrv anifonn. Witliiu
this portion let a cavity be hollowed out in the form of a cylinder,
the axis of nhieh is parallel io the direction of magntfttzatioo, unit
let a msgnEtic pole of unit stroiif^th be placed ftt the middle point
of the axis.
Sinue the gcncmtiiig line* of this cylinder arc in the dJrecUoi
of magnvti/iitiou, there will be no snpcrfieial distribution of mstg
nctinn on the curved surface, and since the circular ends of the
oylitider are perpendicular to the direction of magnetization, thertfll
will be a unitbrra Ruperfieial distribution, of which the surfaoem
density is 7 for the negati^'e end, and — / for the positive end.
Let the length of the axis of the oylinder he ■>&. and its radios a.
Tlien the force arising from this BUptrlicial distributtoQ on a
magnetic pole placed at the middle point of the axis is that dae
to the attraction of the disk on the positive dde, and the repulsion
of the disk on the ne^^tive side. These two forces are equal and
in the same direction, and tlicir sum is
From this expression it appears tiiat the force dcprnds, not on
the abwlule dimensions of the cavity, but on the ratio of the len^^tfa
to the diameter of the cylinder. Hence, however small we make the
cavity, the force arising from tJio snriiicc diBtributiun on ile wullf
will remain, in general, finite.
307.] We have hitJierto supposed the magnetization to be nuifonD
and in the same direction throughout the nhole of the portion of
the magnet from which the cylinder is hollowed out. \Vhen the
mugnetiution is not thus rcHtTict«d, there will in general be a
distribution of imaginary magnetic matter through tho Bubstance
of the magnet. The catting ont of the cylinder will remove part
of this distribution, but since in simihir solid figurrw the foreoa at
corresponding i>oint« arc proportional to the linear dimensions of
the figumt, the ultemtion of the force on the nagnetJo jmle due
to thr volumedensity of magnetic matter will diminish indeiinitoly
as the size of the cavity is diminished, white the effect due to
the surfacedcnnly on the walls of the cavity remains, in genemt,
finite.
If, tbeiefore, we Mxume the dimensions of tins cylinder so smal'
MACirrnc poiica re a cavitt.
23
E
pthi
I
that the mn^etirjitioD of the {lurt rvmovid may bo rcfrarded at*
evtirywliere parallel to the axi* of the cvliinlcr, and of constant
mafc^itdde /, the force on a mag:Dctic polo iilucwl at the middle
point of the axis of the oylindrical hollow will he comjioundoJ
f t«o forces. Tlie 6rst of tbeae is that due to the diKtrihiition
.of magnetic matter on the outer surface of the iniigmt, and
roughout itfi interior, exclusive of the ijortion hollowed oiit^ llie
oompooeots of this force are a, fi and y, derived from the {loteutinl
by equations (l). The second is the force R, acting along the axis
f the cylinder in the direction of ma^etizatiou. The value of
ibis force depends on the ratio of the len^h to the diameter of the
oylindric cavity.
398.] Cate J. Let this ratio be very jreat, or let the diameter
of the cylinder be small compared with its length. Expanding the
t>n for /! in t«rms of ^, wo find
(3)
ity which vanishes when the rati" of i to a is made infinite.
Hence, when the cavily is a very narrow cylinder with its axis parallel
to the direction of mnf^netisnatiun, the magnetic force within ihe
cavity in not afTictod by Ihe snrfucu djutrihution on the ends of llie
indcr, and the components of this force are simply a, A y, where
^ovhiii
a = —
dr
y=di
0)
shall define the force within a cavity of this form as the
magnetic force within the magnet Sir William Thomson has
When we have
shall denote it
called this the Polar definition of mngm'tie force.
to consider this force sus a vector we
occasion
by^.
S99.] Caae IT. Lot the Icn{»th of the cylinder be very small
eompand with it« diamct«r, eo that the cylinder becomes a thin
disk. Expanding the exproesion for S in terms of  , it 1)ocomcs
« = ^';nSH
(5)
E
the nltimate value of which, when the ratio of o to ^ ia made
iDttnite, is 4x1.
H«;nt«, when the cavity is in the form of a thin disk, who«e plan«
Dormal to the direction of magitetizaliou, a unit taagtuetie poU
U
MAOKETIC PORCH AKD UAOSETIC IITDUCTIOK.
placed at the middle of the axis experienoee s force 4*7 in tho
tlircclioD of magnctiiatioD arising from iWe su))er(icifll tQagnetism
on the circalar Burfaoes of the disk *.
Siooe tho componeata of / are ^, i? and C, the oomponcDts of
this force are ivA, iaB, and 4ii(7. This murt \» ooin>Qui)d«d
with the force whose oomponcnte are a, ff. y.
400.] Let the actual force on the unit pole be dcnol«d l>y Uie
vector ^9, and its compODontc by a, b and c, th«D
a = a + inA, \
b = fi + iiiB, I (6)
AVe shall doGno the force witliin it hollow disk, wlioso plane sides
are nonanl to tho direction of mn^netization, as the Magnetio
Induction within the magnet. Sir William Thomson has called
this tho Electromagnetic deJinition of magnetic force.
Tlic three vwtorti, the magn«tixntion 3, the magnntic force ^,
and the mugDCtic induction t8 are counncUtd by the vector oquuttoQ
'«
tineImkgral ^ Maffnttle Fone.
401.] Since the magnetic foro^, as defined in Art. 39d, is that
due to tho dif^tribution of free magnctiMm on the surface and through
the interior of the magnet, and is not affiictcd by the suHkofr
mogDetism of the canty, it may be derived dinxtly from the
general expression f<Hr the potential of the magnet, and (he line
integral of the magnetic force token along any carve from the
point A to the ]>oint B is
'litiere Fj and /^ denote the potentials at A am) B respectively.
* Om tht /vret ftUki* carkla o/ tiktr form*.
I. Anx owTov cTWTMss. The Uavt atvBiiK ftoca Um MrfMcnuq^Mlkni k
l«r MH * ia Uiv (bsctUHi of tile aotiiial to Uu pkaa of Iha cm*mb«, wImt* « U Uw
■ogle b«tw««a tikii uoriDkl oiul Uia dlractkn «f HtgnaUiMiga. Whan tha tnvMma
li pMalUl to the ditoelifn of luis^ni'iinitDa Uic ruro* » ^h* anfiiotu foitv ft ; vham
dio e r u T M ia la pnrpendicDUi u i^ directlui of nu^MtUMion Uin fonc b Uia
miKiutio Indncci^ !&.
H. In *a clooeiud cjliiidw, tha alia nt wikit makaa an v^U t nltti ilia
dinolluu of uiagBHixatloo. iIm &«m< ariiias rrvoi tho tmUtnvaipteti^ it itf ud *,
pitpaaiicuUr l« tbt axil ia iW iUaa ooMlaliiiqg Du Ktk and tha dknUsa uT
nu^MtindoN.
S. In • iiltn tL» faro* afUog from ■ntboomagnctlan la  a / la Uu dlioction uf
piUBallwlkM
402]
BDBFACEINTEORAL.
25
Surface littoral (^ Magnetic Iiuluctioit.
403.] Tko magnetic iiiductioa throtigh the suriaco S U deEned
the value of (h« inlnnal
«=//
S co« </5,
W
J where ® denotes tlie tnsgnitude of the magnetic indaction st th«
^•element of surface dS^ and < the angle betwoeD the direction of
, the indoction and the normal to the element of eurfaoe. and tho
iut^grition is to be extended over the whole surface, which may
bo either closed or bounded by a closed curve.
■ If a, 6, e denote the components of the magnetic indnction, and
If M, a the directioncosines of the normal, the eurface integral
may be written ^^
Q =jf{ia + mt>iuc)dS. (10)
I
If we subKtitute for the components of the magnetic induction
their values in Urn» of those of the magnetic force, and the
magnetization as given in Art. 400, we fmd
q=jj{ia + mfi^nY)^Si'iirff(lA + mB+7iC)dS. (II)
^■extends is a closed one, and we shall investi^to the value of tho
^■two Urma on the righthand side of this equation.
^1 SinoQ the mathematical form of the ri^'lation between magnetic
^■/orce and free magncli:nn is the wime as that Iwlween electric
force and fVee oloctricity, we may apply the nsult given in Art.. 77
to tho Hntt tcnn in the value of Q hy substituting a, /J, y, the
componeiitK of magnetic foroe, for X, Y, Z, the com)onent» of
electric force in Art. 7", and M, the algebraic sam of the free
ma^etism within the dosed surface, for e, the algebraic sum of
the free electricity.
We thus obtain the equation
%
jf{ta+iii^+nY)dS=^ inM.
(12)
Since evtry magnetic particle has two poles, which arc equal
in numerical magnitude but of opposite signs, tho nigvbntic sum
of tho mn>^ctism of the jiwiiciv is u^ro, Uence, those particle*
which are entirely within the clond aurtiiee S can contribute
nothing to the algcbiaic sum of the magnetism within S. The
26 MAGSETIC FOKCE ASD MAGNETIC INDUCTlOS. [4O,;.
value of 31 must therefore depeod only on those magnetic particles^
which are cut by the surface S. Hj
Consider a Bmall element of the magnet of lMi;rth t and trans^*
verse section *', magnetized in the direction of it* len^h, so Hut
the strength of its polee is m. The moment of this snoAll magne^H
will be M«, and the intensity of ita ma(*iietization. being the ratit^l
of the magnetic moment to the volume, will he
(13)
Lot this small magnet be cut by the surface S, so that the
direction of magnetization mates an angle / with the normal
drawn outwards from the surftce, then if i/<S denotes the area of
thescction. l^ = dScost'. (H)
The negative pole — m of tliis magnet lies within the snrfnce S.
Hence, if we denote by dM the part of tlie free mag;neti*ra
within S which is contributed by tbia little magnet^
dif = m = W,
= — 7co9e'rf5. (ll
To lind M, the algehnie snm of the free magnetism witliin til
closed Hiirfiicc St wc must integrate this expression over the elc
surface, ao that
H^Jflco^t'dS,
or writing J, li, C for the com{>onent>! of magiirtintion, mad I, m,k
for the direct ion cosines of the normal drown outwnrdtt,
3twiff{lJ + MB^HC)d8. (1
This ^ves ua the value of the integral in tho second t«nn
equation (11). The value of Q in that equattOQ may therefore
1>0 found in teTons of equations (12) and (16),
Q = 4v3I4v3l=0,
or, lie turfaceittlegral tjf tit nayntlie imluction tirougk any
titrfiin M sen.
403.] If wo sEsiimc as the closed surfuov that of tbv ditTcrcntiai
cli'inent of volume 4x4ydt, vre ohtiUD the equation
da M
de dg
*%'■
{"
TI1U ia the solenoidnl condition which i« always saliaiied by tl
ooinponeDts of the magnetic induction.
405]
LIKES OF MAONRTIC mnPCTION.
27
^^ Sinrc tlic di»trit>Htion of magnotic indutitton U Golotioiilikl, the
^H^^actTon tliruugli any «urrace lioutuletl by a closed curve de];)endft
6n1y on tbe Torm and {Kmition of tJio dosed carve, and not on that
of the »urriuM> itttftf.
I40(.] Surlacea at every point of nhieh
la + mb + ne = (lit)
an called Surfuccs of no iDdiiction, and the intersection of two euch
BnriaraM ix catted a Line of induction. The conditiouK thiit a curve,
9, may be a line of induction arc
H A eyetem of lines of induction drawn through every point of a
Bclosed curre forms a tubular surface called a Tube of induction.
The induction acroea any section of ench a tube is the eame.
If the induction is unity tlie tube is called a Unit tube of in
duction.
■ All that Faraday * eays about lines of Diagnetic force and niiig
netic spbondyloids is mathematically true, if understood of the
Unee and tubes of magnetic induction.
The mn^etio force and the magnetio induction are identical
outside the magnet, but witliin the substance of the magnet they
most be carefully distinguished. In a straight uniformly mag
■ netizcd bar the mngaetic force due to the magnet itself is From
the end which points north, which we call the positive pole, towards
the south end or aegatife pole, both within the magnet and in
the spAoe without.
Th« magnetic induction, on the other hand, is from the positive
pol« to the negative outside the magnet, and from the negative
pole to the iiositive within the magnet, so that the lines and tubes
of induction are r&^entering or cyclic figures.
The importance of the magnetic induction as a physiutl <[uantity
will Iw more clearly seen when we study electromagnetic phe
nomena. When the magnetic field is explored by a moving wire,
as in Fataday s Erp. Ret. 3076, it is the magnetic induction and
not the magnetic force which is directly measured.
I
Tie FechrPotdiflal qf Shffnef.ic Induetion.
405.] Since, as we have shewn in Art. 403, the magnetic in*
doclion throngh a surface bounded by a closed curvo deiMnds on
• £q^ Bet., —n«» xxtiii.
28 MAGNETIC FORCE AKU MAONBTIC INDUCTION. [4O5.
thfi closed carve, and not on the form of the snrfivce wliicb is
bounded by it, it most be possible to determine tito induoUoD
tluroug^ a closed curve by a process depcaditi;; only on Uie nature
of tliat curve, and not involving the conslrucUon of a sarlaoei
forming: a diaphragm of the curve.
This may be donv by finding a reetor $f related to 39, the maprneE
induction, in such a way that the lineinUgtal of BI, extended round
the clo«cd curve, is equal to the surfiieointcgtal of 9, extended
over a surface lionnded by the closed curve.
If, iu Art. 24, we write F, G, U for tlie componenta of 9(, and
a, b, c for the components of id, ne find for the relation bctwc«n
these compouenta
dff
dF
i=U~
^. (21)
djf ^' ' ^ m* das dy
Tlje vector 91, whose components are F, 0, II, is called the vector
potential of magnetic induction.
If a uixgiictic molecule wlioee moment is n and the direction of J
vrho«e axi» of mngnetization is (X, ft, r) bo at the origin of 00j
urdinateii, the potential at a point (ir, y, t) distant r from the origin
is, by Art. 387,
V rfu djf <lz' r
d* d'
= iw(x
didi*''d^. + ''
d^>7'
m
He
which, by Laplace's (equation, may be thrown into the form
d ^^d d^l d , d ds.1
The <juaQtJtica a, b may be dealt with in a similar manner.
From this expreesion G and // may be found by aymmeiry. We!
thus seo Uiat the vtctorpolentlal ai a given point, due to a
mi^r""*'*"^ particle placed at the origin, is nnroiirically equal to
the magnetic moment of the particle divided by tlie square of tho
radius vector and multiplied Ly the sine of the angle Ixlweeii the;
aria of magnetiziition mid tho radius vector, and the dinxtion of
the voctoriiotcntial i« perpendicular to the plane of the aiia of
I
TWrOBrOTEKTTAL.
29
nugnetization and the radius vector, and in such that to ao eye
looking in the positive direction alon^ the axis of magiictiziition
the vectorpotential is drawn in the direction of rotation of ttic
handfi of a watch.
I Hence, for a ma^ct of auv fonii in which A, B, C are the
compoDonta of magnetization at the point xy:, the com)onent»
of the vectorpotential at the point f r C, are
F=
^=m^t~^%)^^s^.\
(22)
Fwherc p in put, for coneisenesB, for the reciprocal of the distance
between Uie points (^, i), C) <"><! [x, y, 2), and the intogratious arc
cxt<^ndcd over the sjwce oceu)>ied hy the magnet.
406.] The ECahir, or ordinary, potential of magnetic force,
BAiI. 3SS, Iwcomes when cxprcsstid in the same notation,
I
Itemcmbering thai ~ = — ^, and that the integral
///
^(S+?S)^**
' has the value —iir(A) when the point {$, ij, f ) ia included within
tbe limits of integration, and is zero when it i^i not so included^
(J) being the value of A at the point {S, v, C)i ^'^ 8^*1 ''"' the value
I of the xcomponent of the magnetic induction,
(21)
The Bret term of this «3ipre<don is evidently —"Jr, or a, the
^iRiponent of the magnetic force.
7/f
30
MAOJTETIC POECE AKD MAGKBTIC IKUfCTlO^r.
The qiuwtitv' under the integral sign in the second term is zero
for every fiU'mL'nt of volume except that in which the point (f, tt, ()
is included. If th« v«ltw of ^ at the point (f, »j, f ) ia (A), the
value of Ui<; fvcond term ia easily proved to be 4t(^), when (J) ia
evidently zoro at all ])ointK outside the maffnet.
We may now write the value of the ircomponest of the mn^etii!
induction „ = a + i!i{A), (25)
an cqtiaUon which ie identicul with the first of thoee given in
Art. 400. "Hie equations for 6 and e will also a^ree with tlioee
of Art. 100.
We have already sceu thut the magnetie force {*) ia derived firom
the Bcatar magnetic {lotential I' by the application of Hamittoo'l
operator V, so that no may write, aa in Art<. 17,
*=Vr, (26)
and that this equation is true both without and wtttiin the maj^net.
It nppoars fnjm the present investigation tliat the mo^pietie
induction © is derived from the vectorpotential 91 by the appli
eution of the itame o>erator, and that the result is true within the
Biagiiet as well as without it.
The application of this o]wrator to a vectorfunction prodaceO)
in general, a M^alar quantity as well aa a vector. The scalar part^
however, which we have called the eonveigenoe of the vector
function, vanishes when the vectorfunction satisliea the saleooidi
M
condition
dF dG . dU ^
loidnU
(27™
By differentiating the expressions for F, 6, Jf in equationa (33), we
find that this equation is satisfied by these quantities
We may therefore write the relation between the magnetic
ioductioD and ita vectorpotential
which may be expressed in worda by mytng that the magnetic
induction is the curl of its vectorpotential. Sec Art. 85.
«
CHAPTEE in.
MAOyETtC 80I.BKO1D3 AND SnZUA*.
On PaHieutar Forma of Ma^tiel*.
407.] Ir a longf narrow filament of TQagnetic matter like a nire
is ma^etized everywhere io a longitudinal directiou, then the
product of any transverse section of the filament into tlie menu
mtcnsity of the ma^etization across it is called (he strength of
the magnet at that section. If the filament were cut in two at
the section without altering the uiagnetizatiou, the two surTaecs,
when eiparated, would be found to have eqiiid anil opposit« qtiun
IbitioH of Euporficial roa^vttzation, each of which is numerically
equal to the etrength of the magnet at the section.
A filament of magnetic matter, so magnetized that its strength
to the mrnfi at every section, at whatever part of its length the
section l>c made, is called a Magnetic Solenoid.
Ifnr i« the strength of the solenoid, da an element of its length,
r tlie distance of that eleioent from a given point, and c the angle
which r makes with the axis of miignetizatiou of tlte element, the
^noteotial at iJw giveu i>oint due to the element is
^^^^tegrating tliie expression with respect to », so ss to take into
H account all the elements of the solenoid, the potential is found
r, being the difcianco of t'he positive end of the solenoid, and r,
titat of the oegutive end from the point where V exists.
Benoe the jiotentiul due to a solenoid, and coneequeuUy all ite
maglMtio eflcctn, dei)end ouly on ita strength and the position of
flitir COB c mdr ,
N «r*,
■ Sw Sir W. Hiuiuoii's ' UstbgniAtiol Tbcory of Hasnctiaa,' PkU. fraM, 1S50,
Btprint.
MAOITETIC SOr.TiSOIDS LVD eilSLTA
iU ends, and not at all on it« form, w)>i'tJi«r straight or curved,
between these pointa.
Hence the ends of a solenoid lUAjr be oallcd in a strict eeu^^
its polea. ^M
If a solenoid forms a closed curve the potential due to it is zen^^
at ereiy point, bo that such a solenoid can exert do ma^ottc
action, nor can tte magnetization be discovered n'ithoal breaking
it at some point and separating the ends. ^M
If a mai;nct can be divided into solenoids, all of which ciihc^^
form clomd curves or have their extremities in the outer surGioc
of tbe niHgnct, the magnetization is said to be solenoidal, and,
sioce tbc action of the magnet depends entirely upon that of 11
ends of tbo solenoids, the distribution of imaginaiy magnetic matt
will be cnlirel}' sniHrrScial.
Uonce the condition of the magnetizatioa being solenoidal is
^ +
dg ds
where A,S,C xsq the components of the magnetization at anjr
point of the magnet.
40B.] A longitudinally magnetized 6lament, of which thestrcngtJi
varies at different ports of its length, may be conceived to be made
lip of a bundle of solenoids of different lengths, the sum of
litrengtlis of all tlie solenoids which pass through a given aecti<
being the magnetic strength of the filament at that section. Ueai
any longitudinally magiietizod filament may be called a Complex
Solenoid.
If the strength of a complex solenoid at any section is m, then
the potential dae to it« action is
lade
'=/:
■■ da where m in variable,
r^ Tf J r d»
■.'!i
lliiii ahens that berades the action of the two ends, which
in this case be of different strengths, there is an action duo to tb
distribution of imaginary magnetic matter along the filament with
a linear density d^
ifa^uetie Shell t.
400.] If a thin sliell of magnetic matt«r is magneti»d id
SITRt.LS.
S3
direction ovt^rywliere iionns) to its EiirTnc*. the intensity of tlu!
Bin^tiL'tiKiilioTi at nny pWn multiplied l>j the thiebncsB of tJic
,th4«l at that [i1a<:o i» callotl thu Strvn;tl> vf tlie ms^etio bIicII
Uiat plsoo.
If thfl »twi»frtli rif a iih<'ll i^ cvcrywUere cqiml, it is cnlled a
Simple CDAgiictic »boll ; if it varies from ]x>int (o puiitt it may be
ctmccived to l>f mude up of n niimWr of simple Hlmllg saporpoeed
and overlapping c«cb other. It i» tb«refori; calW a Complex
Hpia^ctic Mhcll.
^R Let dS lie an clcmont of the xurftce of th« tthdl at. Q, nod ^
^Hhe strength of the Khttll, tlicn tlio potcntinl at any poiiit^ P, due
^Bo the clement of the shell, iit
dF= * ^ rf5 ooa t.
^
wlieie e is the anple between the vector QP, or r and tlio normal
drawn from the positive eide of the ehill.
»But if d» is the «itid »nf>le subtended by dS at the paint P
I'dti =dS coat,
whenee iiF= <t> dai,
and tlicrcfore in Uic caw of a simple mngnetJo hIicII
ir, fie poienlial due to a magnetic xhell at any point is tie prodtiel
it* ttrengtk inla lie loiid angle tubttnded liy ita edge at (he
given point*.
410."] TTie same result may be obtained in n different way by
■uppoRing the magnetic alicU plaoed in any ficit] of mn^netic fnrce,
and determining the potential energy due to the position of the
•belt.
If F ia the potential at the clement dS, then the energy due to
this element is rf/ ^y jr. , „
or, li« preiltiei of tie tirenglA of tie thcU into tie pari of lie
turfaccinltgral of V due to tie element dS qf tie tkell.
Henoo, integrating ivilb reM]>eet to all micli elcmentN, the energy
due to the position of the ohell tn the field ia equal to tiic product
of the Btrengtb of the shell and the Murfaceintegrol of the magnetic
iadaetioQ taken over the surface of the shell.
Since this surfaceintegral is tltc name for any two surfaces which
• Tbia UiiKtfCBi ii iliM loUkiM. GrNfmf Theory ^ Tnrttlriat M<ign<iUki.\iZ ..
VOU n. D
3«
UAOKEnC SOLEJTOIDS AND SUEU^
have the same bounding edge and do not include between them ,
any centre of force, the action of the msg&etic shi^l depends on^ri
on the form of its edge. ^^
Xow rap>oae the 6old of force to be that due to • msgnetic
pole of gtrengtJi m. We have seen (Art. 76, Cor.) tliat the surfaoe
int«fn^ over a eurface bounded by a giren edgv is tho product
of tho strength of the pole and the solid angle subtended by tb*
edge at the polo. Honce the energy due to tho mutool actio
of the pole and the shell ts
<t>M(i),
and this (by Greeo's theorem. Art. lOOj is equal to the prod'
of the atrvngtb of the pole into the i>otential due to the shell
the p»lt^ The K>tfntial due to the sliell is tberoforc 4>a>.
411.] If a magnetic pole » titarttt fn>m n point on Oxe nt^ti
aurface of a magnetic shell, aud travels along any path in spiic« so
to come round the edge to u point olose to wIktd it started but on
the poiiitivc aide of the shell, the «olid angle will vary continuoas),
and will increase by 4ir during the process. Tho work done
the pole will l>e ia<t>m, sod the potent.iul nt any point on
po«itive wide of the shell will except tlial at tlic neighbouring poi
on ihc negative aide by 4ir<l>.
Jf a magnetic shell forms a closed Kurfu<.'o, the pot«atial outside
tho shell is everywhere zero, aud ttiat in tbc space witbin ia
everywhere 4«4>, being positive when Uic positive side of the shell
is inward. Honce such a shell exerts no action on any magnet
placed either outside or inside the shell.
412.] If a magnet can be divided into simple magnetic shells,
either closed or having their edges on tlie suriacc of the magnet,
the distribntioD of magnetism is called Lami'llar. If ^ is the
sum of the strengtlu of all the shells traversed by a point
pasving from a given point to a point xjfe by a line drawn witi
the magnet^ then the conditions of lamelUr magnetization ar«
^'rfp' ^Jf ^dT
The quantity, ^, which thas completely determines the inagnct
ixation at any poiut may be called the Potential of ^faguetizatioo.
It must be carefully distingui^ed from the Magnetic Potential.
413.] A magnet which can be divided into complex mag
shells is sid to have a complex lamellar distribution of
netism. The condition of such a distribution is that the lines
415.]
roTSNTIAt. DUE TO A LAMELLAR MAQ!fET.
35
I
magndimtioii niuat be such that » systotn of siirruoos citu be drana
cutting tliem at right ungLs. Thk condition is 03tprcs>e<l by tlie
rellkiiowti m{ nation
Pormt of the Potffaiiafa of Solfno'idai atitl Lamellar Manned.
414.] Tbo g«oeral expression for tho scalar potential of a Dtagnrt
fwbere p denotes the potential at {3),y, i) due to a unit magnetic
pol« plnvcd at ((, it, 0, or in othir worde, the rcciprooti! of the
distaum betwran {(, ij, (), the point at which the pot«ntid is
■neasuFcd, and (x, j, s), the jKisition of the element of the niiignvt
.to which it is due.
This quantity mny be iuttgr»t«Ml by parts, as in Arte. 96, 386<
r=fjp {Al^ Bu + Cn) dS ffjp (g + ^^ + ^) ./.<fy^.
*
»:
wbere /, m, » are tlie directioncosines of the normal drawn out
wards from dS, an element of the surface of the m^net.
When the nui^ct iif solcnoidal tho expression under the integral
aign ia the second tvrin is aero for every point within the magnet,
M that the triple integral is zero, and the scalar potential at any
p<Aiit, whether outside or inside the magnet, is given by the soriacc*
integral in the first term.
I The Ecuitu potential of a solcnoJdal magnet ts therefore com
pletely determined when the normal component of the miignet
ization at every point of the surface is known, and it is iudeiienilenl
of the form of the xolciKiids within the ma^'net.
415.] Id the case of a lamellar magnet the mugnetizatJon is
determined by ^, tlie potential of magncLizatioii, so that
dg
He expression for V may therefore he written
Integrating this expression by parts, wc fiutl
dp
A = 5—1
09
L = 3
k
//*(' 1)«///* (0^ S)^*^'
DS
^A
»e
MAGNETIC SOI.EITOlDa IKD SHILW.
I4I6.
Th« «fCoii<l term is stiro iitiiois Uic point (f, ij, () is included in
the mxffno^ '" whicli cnse it licwimeji 4 57(0) where {4.) ia the «d»e
of 4 at tho point (f, rj, (). Th« smrfaceiiitcgral may be expressed in
terms off, the line dniwn from {j.j, r) to ((, ij, f), and $ the angle
whi<;)i tluK line makea witli the normal drawn outwards from JS,
so that the potential may be written
r=y//?j*cosflrf5+4Ti(^),
itAuiK the xefond term is of course zero when the point {(, 7, f)l
ntft inclixlcd in tltc vubntanee of the magnet.
The pot«ntial, f', expressed by this equation, is continuous ei
at the Ktirfuce of the magnet, witere ^ becomes suddenly zero, m^
if wo write
a s: jj ^^ eoi lis,
and if ilf is the value of i2 at a point just within the surface,
flj that at a )>oint close to the first but outside the sor&oe;
or Fg = r,.
Tlic quantity fi is not continuous at the surfiiee of the magnet.
The component* of magnetic induetiou are related to £1 by
is]ttntions
« =~
J=
rfQ
</a
da
dx ' ' ~ rfjf ' * ~ dt
416.] In the case of a lamellar distribution of magnetism we
may aI>o simplify the vectorpotential of magnetie induetton.
Its xcooipoiieiit may be written
m/iri)''^'^'
By integnttion by ports we may put this in the fonn of
surface • in tegral
or
^"//'("SffV
ITie other components of tlii> Tectnri)oieiitial may be
down from three exprasaioiut by making the proper suLstatutiona.
On Bond jHfftet.
417.1 We have already proved Uiat at any point P ifao [•oli^nti
'Htl^^J
SOLIO ANOLBS,
37
dae to s msfpietic slicll iis «qiiiil to tliu eolid »ngh siilttcnduil by
the e<\ge of the elipll tnultipllvd by tlic >>lnD^th of tfav slii;Il. As
we eball have occasioa to rtfcr to Bolid an^^leti in thv theory of
electric currents, n'c Khali now explain how they nrny Iju mui^iieJ.
KDefiiution. Tha mlid anisic eiibtondvd at a jg\\c(t point by n
osed curv« is imtUiurGd by the area of a splutTiciil surfaco whow
(Dtre is thu ipvcn ioiiit and whose raditis is unity, the outline
of which is tnu'LYl by the intereuctloR of the radius viTctor with the
sphere as it traces the closed curve. This area is to be rcckoaed
positive or negative (ioeordiii>f as it lies on the luft or the right
hand of th«! path of the radius vector a« sc«q from the ^vvn point.
J^ (f> 1) he the (^iven point, and let {ji, y, s) \w w point on
the closed curve. The eoonlinates .r, _^, z uro functions of t, the
len{*th of tli« curve rockonvd from a given point. They are periodic
fuDction« of ', rtwurrin;; wheiwver » is increased by the wholo Un^lt
of the elOKol curv4;.
We tnny caleulatc the solid aug'Ie m directly from the dcliiiitioa
iJius. Usin^ iqihcriciil coordioati's with centre at (f, t), ()• ^^^
patting
x—i = f siuflcoa^, j>— ij = rsialJsin^, i—^= rco«fl,
we find the area of any carve on the sphere by infegrattng
I w = f(lf:os0)di>, I
or, OBUi^ the rectangular coordiuateit,
^vtt intention being extended rouni) the curve «.
II If the axis of ; j>assc8 oncu thnrugh the el<ised curve the first
Bterm ifl 2t. If Uic axis of e does not jugs through it this term
T^iszero.
418.] This method of calculating & solid angle involves a cboi<.«
of axes which ta to sotne extent arbitrary, and it does not (Upviid
solely on the oloeed curve. Hence the following method, in which
no snrlace is mpposed to be constructed, may be statod for the suke
of geometrical propriety.
Aa the radios vector from the given point traces out the closiil
curve, let a plane passin>; through the given point roll on the
closed curve so as to be a tan';cnt plane at each point of the curve
^Bn sueoosston. Let a line of unitlength be drawn frotn the given
^Boint [wrpendicular to this plane. As the plane rolU round the
38
IIAQSETIC SOLESOIDS AKD SHBLLB.
[4'9
closed curve the ntTCmit}' of the pcrpcndicnUr will tnce s second
dosed curve. IjcI the loDgDi of the second cloved curve be v, Uten
tlie oolid angle Mubtcnded l>j the lint closed cun'o is
This follows fron> tliu wellknown theorem tli&t the am (^ s
dosed curve on a ^hcre of unit nwliuK, togt'thiT with the circnm
ferenoe of the polar curve, is numerically equal to the circuDifermce
*f ■ grent drclc of the si>here,
This constnictton is somttimos convenient for calculatinff the
»>lid angle sul>teinlnl hyx nctilin«^iir figure. For our own purjKise,
which is to form clear ideas of ithymciii phcoomena, the rollowiop
mcfliod is to be pnArred, as it employs no conitructions which do
not flow from the physical datdi of tJic problem.
419.] A dose<l curve » is given in spam, ani) we have to find
the solid angle vulitvniled hy < at a given poiot P.
If wo consider t)ie solid aiiglc us the potential of a magnetic shell
of imit strength whose edge ooincidec with the dosed eurv«, we
iDHst define it as the work done 1>y n unit magnetic pole against
the magnetic force while it moves from an infinite dislancd to tJie
point P. Hence, if « is the path of the )>ole as it approaches the
point P, the potential must bo the result of a lineintegnition alon^
this path. It must also be the result of a lineint^gnlton along
the dosed curve i. The pro])er form of the cjtprcssion for the solid
angle must therefore I>e thai of a double integration with rusiect
to the two cur^'cs » and <r.
Wlieii P ii at an inllnite dixtnnce, the solid angle is evidently
zero. As the point P approaches, the clowd curve, as seen from
the moving point, appears to open out, awl the whol« solid angle
mny Iw conceired to be generated by the op<«nm( nK>lton qS the^j
diHereiit dements of the closed curve as the moving point op>^
proadm. ^^
Ah Uio point P moves from P Ui P' over the dement da, the
element QQ' of the closed carve, which wo dciiolc by d; wiU ^^
change its position relntivdy to /', and the line on llie unit .^phera
oonvsponding to QQ' will sweep over an area on the Kplivriuul
surfara, whiuh wk may write
r/H> ri'/x/.T. (I)j
Til (litd II lul IIS <itii]i«iiu> P flxi<d while the olomd curve ia movtdl
parallel (o itwilf ttiroiigli a dlalunou dv etpwl to PP" tMit in titc
uppiMilr dimilinn. lint hlaUve nii>ti«iu of tlio {loint P will be thi^H
sunie H» ii> I li« roul uaw, ^^
I
420.]
GESERATIOK OF A SOLID AITOLB.
BQ
I
During th» motion the olcmcnt (^Q' will generate itn urea in
the form of a paralliflogram whose sides are paralUl Rail equal
to Q(^ MMJ Pi**. If we construct a pjramid on thin pHralldogram
u base with its vertex at P, the solid angle of this jiyramid will
te the increment (/» whioli we are in search of.
To detennine the value of this solid
angle, let and ^ be the angles wliich
ih and d<r make with PQ reupcct
ively, and let /}> be the ang^le between
the planes of these two anglvM, then
the ares of the projection of the
psrallelo^ni <fj .da on a plane per
T to PQ or r will he
dtda sin 9 Bin ^ sin <^,
eioce this is equal to f^rfw, we find
du = ndsda = ^ sin eia ^ sin <)itU da.
n = ^ Biu$ sin ^sin tp.
(8)
(3)
420.] We may express the angles C, 6', and ^ in termH i.f r,
and its diflVreatial coefiieientB with roxpect to » aud a, for
Dcnoc
OOS0 =
dr
„ dr
coa0'= T.
d^r
(^)
 , . ™v = i. and Binflsinfl'ooa* = r rT
tat dn ^ did
We thus find the fullowing value for 11',
"■4['()']['&']iO' (^>
A third expresBioQ for ri in terms of rcctangulur eoordinute*
may be deduced from the consideration that the volume of Hie
pyramid whose solid angle is du> and whose axis is r iit
Bat the volume of this pyramid may also be expressed in terms
of the projectiotta of r, d*, and da on the axis of a, y and f, as
a determinant formed by these nine projections, of which vre must,
take the third part. We thus find as the value of 11,
f», n», (',
n = A
^. ^. i^,
da da da
da dif dz
dt dt d«
(«>
*0
MAOKETIC SOLENOIDS ATfD SIIELIS.
ityoT
Iose«
in
lio,
indfl
'Flits «xpre«Kion giv«jt th« yaloe of n frefi from th« ambiguity of
ei)^i iiitniiliK'ol by ctiiAtii>n (5).
421.] TIk' viilui;! of w, tlic solid angle subtended by the «Io
arvp ut Uic point /', may now be vrriiun
wkcro the intrarfi^ion tnlli rus{M.'ct to « is to be extended ooinpl«lely
round tlie cloMfd cur^'o, ood that with rv^pcct to a from A a lix«l
M)iul (in tbr ctirro to the point P. The constant Uq is tbe value
of tht' colid Kiiglit at thv point ^. It is zoro if J is at an iwti uite
distance from iiw clu«ed ourvv.
The vnhio of u at any jtoint P is inilependvnt of (In* form'
tlie curve between J and P prov(di.d tlint it dws not [<asi> Ihrnii^
the niiiKnctie shell it««lf. If the shell be sup>osed iiifinit^'Iy Ihio,
and if P «i>d ^aro two ]>oiuts close tofrrther, but Poo the ixMcilivai
and /** on th« nefjative surface of the shell, then the curves JP and]
JP" must lie on opposite sides of the edge of the shell, so that PJi
in a lint which with the infinitely short line P'P forms a closed
circuit vmbrseiuff the edj^. The nine of a* at i' exceeds tlutt nlP' ,
by it, that is, by the surfaee of a sphere of radios unity.
llenoi, if a closed curve be drawn so as to pass once througli
the shell, or in other words, if it be linked ooce with the
of tl»e shell, the value of the integral fffUtitr exteiid«d
both eunics will U' Ir,
lilts intr}^) theivftnv, eoosidered as depeadin^ only on the
«lowd curve t and the aibitniy curve AP, k as instance ot
ftutetton iif multipU* x'uluos, nnce, if we pom &om A \jt> P\
diSkrvnt <ath« the intc^n^l will have dlflereiit values
to th« number of tiuura which the corre XP » tviaed rouad the
c«rt« a.
U eM ftm of th* ranrv betw««a J a»d ? on he ttaasCamed
nito,aw4be« fey «Mmnt>m notkat without iBtns«vtiiiy the carve
^ tht iatagnl will h«n lh« wm vahn for both coma, hot if
dvuir the tBHMfcnBklmt it ia l ewaeto lb dMnl cnrv a txans tlx^
valuMof iheibhxnlwiUJi^by 4t«. fl
ir « aiKl • ai« aay tv)\^ ekot^ <«rm> n ^acn tka, tf Ihif air^
««* KeVmI h><wlhM. lh<> i«le«nJ Mtmilei eaaa immI Wtft
If thcT an tatuvtwiOHU • \.p\^ la t).v ssntr ttTr^tk^a. th«
of the uMig«l i» *•> h » pMwhK huwewr. fcv two
I
VECTOBPOTEimAL OP A CLOSED COBTE.
41
to bo intcrtwineid alt^Tnately in op[)oiiit« tlirectionB, mo tlmt th«y
mro inMpuruMy linked together though the value of tliv itiU'gruI
La zero. Sw Fig. 4.
It was tlie divcovery by Gauss of this very integral, oxiiresflitig
the work don« on a magitetio pole while dc
acribii^ a iilosed curve in prt>sence of a closed
deetric current, and indicating the geometrical
ooniMsion between the two closed curves, that
led liini to l»mtnt the small progress made iu the
(icomt'try <ii' PnHition since the Lime ui LiiLuitz,
Euler and Vandermondc. We have now, bow ^' *
ever, some progress to repori, chiefly due to Eieuiaun, Helmholtz
and Listing.
423.] Let Ds now investigate the result of integrating with
nspect to a round the closed curve.
One of the terms of n iu equation (7) is
If wc now write for bre«ty
"'Hi' "'Z;!^. "=51%^ w
tile iot^TiilB being taken once round the closed curve t, this term
j of n may be written rf^ (/sy/
the correspondtti^ term of f TlJs will be
dnilff
Collecting all the tvnoE of Xi, we may now write
■ tit}
'IGyn rdf
diUo
\j,
rff
di
>fO <IF. d^
Thin quantity is e\idently the rate of decrement of w, the
miigtulj<! jwtcntial. in passing along the curve a, or in other words,
it \» the ma^etic force in the direction of da.
By a»«umiii^ da stKceesively in the direction of tho bx« of
, y and ;, we obtnin for the values of tlie compouents of (ho
lAiagnetic force
42
UAOXETIC B01.K?fO]1>3 AiTO SHELI^
rf« dU tiG^
_ ^/w _ ^ _ J/f
^ ~ f/i, ~ rff rff '
_ rf<.> _ rfO _ ///*
'' ^ tic di dn
Tho quantities F, 0, //»re the components of the vectoriiotffntiil
of ttio mafrnetic shell whose etrength i» xtnity, aad whoae ^ge it
the curve a. Tbey are not, like the scalar potential •, function*
having a series of raluee, but sre perfectlj detcnDioatc for every
point in space.
The vectorpotential at a point Piae to a ma^etic shell bounded
hy H closed curvo may be foand by the following geoatetrkalj
oonstniclion ;
Let a point Q tiavcl round the closed curve vrith a velocity
nHmericnlly equal to its distance from P, and let a second point
R start from a fixed point A and travel with a velocity the direction
of which is alvrays parallel to that of Q, but whose magnitude is
unity, ^Micn Q has travelled once round the closed curve join
JH. then the tine JJt represents in direction and in numaiieal
mngnitnde the veelorpotenlial due to the closed curve at P.
PftrmtUit Eatrgg tfa ihfnttk SietlpUetil ta « AfofMiit Fte!4.
438.] We hare almdy shewn, in Art. 410, that the potential
energy of a shell of strength ^ pbced in ft magnetic Seld whose
potential is T, is
where f, », ■ arv the directtoacoeuMS of the nonul to the
drawn ftviu the ]M<»itivo side, aad the sorftoeiuteyiiil u
OTtir the shell.
N'ow this swfitee^talwgnl nwy bs tzwlomed into a linein
by nMau of lite Teetoc>pet«aiti«l ef tlw m^petio Geid, and m
xuKf write
i
J'^^/Cs+elfffJjA (»;
w1h«* (be inteptntiM is exIanM «m* nMoi the chwed enrre »
which f^>ni>« the tOite ef IIm H ii ^ iw tiu sMl the dirsetiQa of di
being oppoeit* Xa that «t the hawls of n wntch whc« vwwid tram
the p.iMlit>>i«Wf>f tbs>WU.
If we iMW Mpv«e thni the B Myw He S«U is tfel Am bi
«
423]
POTESTUL OF TWO CLOSED CURVES.
43
^^wtii
second roogUDlio slioU wIioni> Btrong^h is ifi', we rnny det«nniDe the
value of /"iliroctly from the rccalto of Art. 416 or from Art. 405.
If f, m', «' W the Oirectioii«OKineii of tli<! nomiKl to tlie element dS'
of tiie •ecoix) (hell, we have
^=*JhfA''f,>
\vTe r ig the distance between the clement ilS' and a point on th«
boondary of the first shell.
Now this BurfaceiDt«i;ra1 msy he converted into a lineintegral
ronnd tlie boundary of the second shell ; viz. it is
//
iW.
(14)
Id like manner
=*/i
A'
d»'.
Snbetituting those raluea in Uie expression for Jf we find
'whtrft the Intef^tion is cx(<nili.d once round * and once round /,
Thia exprcKKion g'lvcf the ]<ot^ntiul energy due to the mutual actit>a
of tite two shellK, and is, ajt it ought to be, the same when t and /
are interchanged. 'Diiit expression with ita sign reversed, when the
strength of each shell in unity, is called the potential of the two
closed curves t and s'. It is a quantity of great importance in the
(heory of elcyitric currents. If we write « for the angle between
the direclionH of the elements dt and ds\ the potential of » and t
nay be written
(16)
JJ^dsJ^.
It b evidently a qnantity of the dimension of a line.
CHAPTER IV.
IKDCCED lUCKETIZATIOy.
424.] Wk have hitlierto considered Hw actual (Jwtribution of
BUkgnetiEatioH in u m&^et w given explicitly amon^ th« dita
or the invtwtigxtion. We b»ve not made soy usttimptioD w to
wtiPttitT lliiti iiugtKtinition is pcnnanout or t«Rii>orary, except in
thoeo \axia of ottP Teuonin^ in which we havo £U)>iM3cfd the magnet
broken np into Gmall purtions, or small portions remoTMl frona
ttie inti;^itrt in such a way as not to alter the ma^uetixatioQ c^
•oy part.
NW havo now to coonder the magnetization of bodies with
mpact (o tltK node in which it tuj Im prodaoinl and changed.
A bar n( irou It^ parallel to the diicetion of the earth's magnetio
fovet is found to l)««onM n^netac. with its poles tamed lb« np
poaita way IVmu thov^ of the rartb, or the aame war a> those of
ft compan Modh" in ^tallp c^uilitiriura.
Any )«)cr<p nf snfl ircoi placml in a aufiMtae SeU is found to rshilni
nayneli. pTi^ivNiMa. IT it be phwvd in a part of the fidd wber«
the MMKnaic fi<n« it gnai,, as bekw«ea the pnles of a honseshoc
— gw rt . the numwftisnii of the iron b ecppw inteiwp. If the iran
if T<rmi^\<^ frxMn th* t w n o t t io fwM. its mapMlM properties an
Iffwally WYfckwM".! n* iMw uy ar eatiiv^r. If tha nagnetic properliM
(/ the i^m A^^wi Mtlurty oft the mugMtic tmt of tk ti.ld
nWh tt ' and \'anuh wk<« il » noKMml &<aa thr
it M l,N»1l•.^■ >  ■'•^. livMS «hi<h i* aoA in th* BM^nriif)
M ahu miA i« thr hiM«l MMMVL II w n^ to bm^ n jq^
il a pr n wawe*: .«l. Vu
the wo*S^ SWih
•
I
425]
SOFT AKD HABD STBKI,.
I
»
up the tnagndic state eo rcaility ne soft iron. TIjp operation of
Uamiiicnitif, or nny oihi.T kind of vibriition, allows liari! iron uudvr
the intiiinK'i* of magnetic force to assumo Ihe magnetic s1at« wore
readily, and to part witli it more readily wben the taagoetixxng
for(H4 is rcmovcf!. Iron which is nin<^ftica11y liard is also more
•tiS'to biiul and more apt to break.
The prooMMw of hammering, rolling, wiredramn^, and snddea
cooling tend to hardvn iron, and that of annealing' tends to
ioRcn it.
The miLf^etic as well as tlio mechanical dideroncee between steel
of hard and »oft tomptr arc much greater than those between hard
and mfb iron. Soft sUel is almost as easily ma;;nctizi?d and de
niAgiietiz4.d as iron, while the hardest Et«el is the beet material
for magnets which we wish to bo permanent.
Cuct iron, though it contains more carbon than steel, is not
CO retentive of ma^etization.
K a magnet conlil be constntctvd so that the distribution of its
magnetization is not altered by any mugnctic force brought to
act upon it, it might be called u rigidly mn^etized body. The
only known body which fullils this condition is a conducting oircuit
round which a constant electric current is mjide to flow,
Such a circuit exhibits magnetic properties, and mu}' therefore be
called an elect romiignct, bat tbeso magnetic properties are oot
nffeetcd by the other magnetic force* in the 6cld. We shall return
to this subject in Part IV.
All actual magnets, whethor mode of hardened steel or of load
stone, are found to be affected by any nugnetic force which is
brought to bear upon them.
It is convenient, for scientific purposes, to make a distinetJOB
iMtwccn the permanent and the temporary magnetization, dclining
the permanent magnetization as that which cxist« independently
of the rangoetic force, and the temj>orury iniigneli/.attou as that
wliich depends on this force. We mu«t obterve, however, that
this diittinction ia not founded on n knowledge of the intimate
nature of magnetizable subetances : it is only tJie expresiiion of
an hypothe«i8 introduced for the sake of bringing caloalation to
bear on the plienometui. Wo shall return to tlie physical theory
of mngnetizution in Clmpter VI.
42iS.] At present we shall inn'stigate the temporary magnet
ization on thr nwiimption that the magnetization of any particle
of the mibstitnee deiwnds wicly on the magnetic force acting on
INDUCED HAONBTIZATlOIf.
4
«
are fouB^^
ttiat particle. This magnotio force may arise partly from esteroal
cnum'ii, and partly from the temporary niugnetizatioa of neigh
bourinj^ partklcs.
A liody tlius roagnctized in virtue of the aotioa of magneli'
foroc, is wiid to be ma^otlzed by induction, and the magnetizatioiii
ii anid to bo induced by the niagnetiziug foroe.
'Hie inajtiiiHization induced by a given magnetizing force diflvrs
in difli>rt^nt suhstaDoea, It is greatest in the purvet and sofleet
iron, in which the ratio of the nuignetlzatioD to the magneUc forottj
may reach the v&luc 32, or even 45*^.
Other subaUuices, anoh as the metals nickd and cobalt, are
cu)>able of an inferior degree of magnetization, and all sabstancea
whfu subjected to a suffieieDtly tAnog magnetic force, are found
to give indicatione of polarity.
When the magnetization ts Id th« same direction astfae
fi>ne, m in iron, nickel, cobslt, &c., the eubetauce k tailed
mitgnciio, t'erromagnctic, or more simply Magnetie. When the
indnced magnetizatioo is in the direction opposite to the magoetie
foioe, as in bismuth, ke^ the Eub«t»nc« is said to b« Diamagnetic. ^u
In all these iRib«taoces the ntio of the magnetization to the^
magitetio force which produce* it id exceedingly entail, being only ^'
about —tWvn >» ^^^ '^*^ ^^ bismnth, which it the most highly ^y
diatnagnetio aubetanco known. ^M
In ery«t«llized, »tmincd, and organiEed substances tlie directloD
of the magnrtixalion dc<i< not always coincide with that of the
nwguatie forw which produces it The nJation between the 000
ponenU of magnettaation, referred to axes fixed in the body, and
tht««> of the ntagnetto force, may be expnseed by a system of thm
tin<«r equations. Of the nine coefficietito involved in these equa*
tions «« dull shew that only six are independent The phenomena
afbcdiaaof th» kind are classed uadcr the name of Slagnecrystallio
lltKOOSttHI^
Whea placed ia a field of u^inetao fbttn, nyatals ind to set
ihewMtlvn* so that the axis «t s;ntAt^ parami^niitiii. or of kwt
4iMas«netM. iA^KtM i« parslM la th* l»e« tf
See Alt 4SS
In spft irMt the directkta of tW mnailiialiiM  u
that of the aucavlw foR« at tha poiu. and ttt ^mB
Iha BUgnstia foiw U« niafiw^ualiea ia aawly 
• naUa, .Vw« Jrt« «qk .s,« jK C«L. 11
I
I
PBOBLEM OF DtDUCED MAOSETIZATIOK.
47
IAs ihe magnetic force increases, however, the □ugitL'tizatioa in
etoucs nore slowly, aod it woald appear rrom experimeiitti dctcribcd
ia Cbap. VI, that there is ft limitiiij> valao of the mii^nrtuuitioa,
bejond which it cannot p«s, whalvrcr b« the raltie of Mm
mognt'tic force.
In the following outline of the tbeory of inducnl magi>^i^i''>>
we shall be^n by EUpposing the magnetization profKirtion*! to the
magnetic force, and in tbo same line with it,
IJkfinitum of the Cbefieteni ^ Inductd MagntiiMiion,
426.] Let ^ be the ma^^netic force, defined as in Art 398, at
KOj point of the body, and let 3 be the mafrnetization at that
point, then the ratio of 3 to ^ is called the Coefficient of Induced
Ibgnetization.
Deooting this coefficient by «, the fundamoatnl tsjuation of
indQCcd magnetism is
3 = .«. (1)
■ The coeRinent k \» positive for iron and {)arainagnetiu subKtancvri,
and negative for bivniuth and diamagnt'tie substances. It reacbe*
the value 32 in iron, and it \» »aid to be large in the case of nickel
and cobalt, bnt in all other casefl it ia a very email quantitv, not
greater than O.OOOOl.
tThe force ^ arises partly from the action of magnets external
to the )>ody magnetized by induction, and partly from the iiiduoed
magnetization of the body itself. Both parts satisfy the condition
of having a potential.
^7.] Let y be the potential dae to magnetism external to the
body, let H be that due to the induced magnetization, then if
V is the actual potential due to both causes
»u=r+a. (2)
Let Uie compooenls of the magnetic force ■^, resoUcd in the
directions of x,g,t,\>o a, fi, y, and let those of the magnctizatioa
3 be J,.d, C, then by equation (1),
Multiplying these equations by dx, dy, dz teepectively, and
adding, we find
^ AdxiSdjr+Ctl: = *{ad4S+fidjr+yJt).
48
INDCCED MAOKETIZATIOK.
[427
Bnt sinoR a, and y are derived &oni the poteotial U, w« maty
writv tlie second memljer —iciiU,
Hciive, if M is constant tbrou^hant the substanee, the Rni meml
tniiDt ftlao be a eompl«tc difTerontial of a functioD of jt, y ukl
which we shall call ^, and thu oqiintion bi^comcs
vb«n
ate
A = '^, B =
34,
(5)
The magnet ization is therefore lamellar, as defined in Art. 412. 1
It «'«» shewn in Art. 386 that if p is the volumedensity of &ce
mn'metisni.
which becomes in virtue of oquation* {3),
"^ydi
But, by Art. 77,
''='W
fda d0
4iy
dy ^ is
f£
Henee (l + 4«)p = 0,
vrbeooe ^ = f6)
ihrougliout tJie substance, and the mognotiiation is therefore sc4e
noidal as well as Unxllar, Sve Art. 407.
There is therefore no free ■Qaguetism except on the boandi
stirfnce of tlie body. If r be the normal drawn inwards from tl
surface, the magiKtic BnrfRoodensi^ ia
"% (')
The po(<ntial U due to this nagnptization at any point may
therelorv be found (Wm (he mriaeeinte^nl
The value of Q will be finite and continuoas everrwbere, and
will Mti»fy lAplan's pqaatioit at emry point both within and
without (h«< ■nrfiii'e. If wv di9tin{;ni:»b by «a aetvat the value
of n iwt»i*le the MiHace. and if »' be Um nonoal drawn ootwnnU,
ire have at llir suHikv
(8P
tt'^u:
(!W
428.]
POiasOlrS METROD.
49
^ + 3^ = 4ir<r. byArt.78,
We may therefore write the surGk«ccoudttioa
IciMic tlie determination of tliu magnetism induced !u a homo
geneous iMUttpic body, bounded by a surface S, and acted upon by
external nui^nutiu forces whow potential is ?', may be reduced to
tlie following umtliomutiuil problem.
We mu»t find two funotioni; il and il' mitisfyingf the following^
conditjonn :
Within the surface S, Cl musi be finite and continuous, and must
Kstisfy XiHplact^'s eipiation.
Ontxido the surfa^'e 8, Cf must be iinite and continuous, it must
vnuiHli nt an inlinite distance^ and must satisfy Laplace's equation.
At <f\tiy point of the surface it«iIf, H = il', and the dcrivalives
of A, Q.' and F with respect to the normal must satisfy equation
ThiM method of treating the problem of induced magnetism is
due to Poit«OD. The quantity i which he uses in his mcmoiTs is nut
tlie nme m «, hut is related to it as follows :
*BK{il) + 3i = 0. (11)
The coefficient r which we have here used was introduced by
J. Neumann.
428.] The jiroblem of induced magnetism may be lrc44t<^ in a
different manner by introducing the quantity which wo have L'slled,
with Faraday, the ilagnetio Induction.
tThe relation between ?*, the maguetic induction, ^, the magnetic
Ibzc^ and 3, the magnetization, ie exprceeed by tiic cijuattoa
« = ^ + 4ir3. (12)
The equation which expresses the induced magnetization in terms
of tlie magnetic force i»
I
I
I
'*.
(18)
Tqt,n.
]Ieace, etiminatiDg 3, we Bod
© = (1 + 4.,)^ (U)
as the relation between the mftgnetio indactioD and the magnotic
force IB Bubatanees whose magaetization is indaced by mognvtic
force.
In the rooiit general caw « may be a function, not only of the
position of tbe point in the substance, but of the direction of tbc
vector ^, bat in the case which we are now consideriag jc is a
numerical (uantity,
If we next write fi = l + 4ir«, (18)
we may define n as tJie ratio of the magoetio induction to the
magnetic force, and we may call this ratio the magnetic indoctiTo
cajnoity of tbe snbetanee, thus distinguishing it from k, tbe co
efficient of induced magnetization.
If we write U for the total magnetic potential coropouoded of T.
the potential due to exti'mnl csuecs, and Q for that due to the
induced msignetization, wo may express a, 6, e, the components of
magnotic induction, and a, fi, y, the components of magnetic force,
as follows : jf/
JU
' = ">' = ** 35" * J
The components a,&,e satisfy the Bolonoidal conditioa
4a d6 de
(»•)
(17)
Hence, the potential U must satisfy Laplace's equation
d*U iPU d^V , .
at every point where ft la oonstant, tlint is, at every point within
the homogenoouH substance, or in empty q»w.
At the surfiice itself, if i is a normal tlmwn towards tbe magnetic
euWance, and v" on« drawn outwards, and if the symbols of qnan*
titiiMi outside the sub«tance are distjngaished by accents, the con>
ilition of continaity of the magnetic induction ia
dv dp dv df Ju dv
0;
(19)
439
^Kor, by eqaatioDB (16),
FARADAY'S THKOBY OP HAOSETrC ISDCCTION.
0.
6t
(20)
do "^ dv'
II, the coefficient of indtiotion outside the maici^et, will be uuit^
unless the enrrounding' medium be mttgoetic or disma^etio.
h If we stibslitate for U its vnlue in terma of Tand ii, and for
H fi its value in terms.of k, we obtain the same eijuatioD (10) as we
arrived at by Poiason's method.
ITbe problem of induced ma^ctism, when consideri'd wi(h respect
to the relation bctwcrcn miignctic induction and ma^ettc force,
corresponds exactly with the problem of the conduction of electric
enrrente thToa;fh hctero^neoue media, ae giv<n in Art. 310.
The magnetic force h derived from the magnetic potential, pre
cisely as the electric force is derived from the electric potential.
The mafpietic induction is a quantity of the nature of » fli:s,
and satieGes the same conditions of continuity as the electric
current does.
In isotropic media the matfnetic induction depends on the mag
netic force in a manner which exactly corretiponds with that in
vrliicb the electric cnrrent depends on the electromotive force.
^B The specific magnetic inductive capacity in the one problem corre*
^^ xponds to the specific conductivity in the otlior, Hcnoc Thomson,
in his Ti</>ry o/ Indnetd Miiffnntign (Itepmif, 1872, p. 48<), has called
this quantity the permea&i/ify of the medium.
I We are now prepared to consider the theory of induced ma^etiam
fnmi what I conceive to be Faraday's point of view.
When magnetic force acts on any medium, whether magnetic or
■liaiiingnetic, or n u(ml, it produces within it a phenomenon cjilltnl
Magnetic Induction.
Magnetic induction ia a directed quantity of the natnni of a Bus,
and it satisfies the same conditions of continuity as electric curreuta
and other lltuces do.
In isotropic media the magnetic force and the magnetic induction
arc in the same direction, and the magnetic induction is the product
of tie magnetic force into a quaiitity called the coefficient of
imluctjon, which wo have expressed by i*.
In empty space the coefficient of induction is unity. Tn bo<lii'ft
tapable of indnced magnetization the coefficient of induction is
l + 47R = /t, where k is the i]nantity already defined as the co*
efficient of induced magnetization.
Im 4S9.] Let f^ ^4' be the values of fi on opposite udes of a surface
52
INDDCBD MA0NET12AT10N.
ccpAriiling two media, then if F, f" are the potentiiUs in tin tvio
media, tlie iiiflgnciio forces towards the surface Id the two media
dv . <ir
are r »nd rr
av an
Tb« qimntitie« of magnetic induction through the element of;
rarface dS utb njdS and fi' jy dS in the two media rupcct
ivcly reolcoiii'd towards dS.
Since the total flux towards dS is zero,
dF ,Ar
Bill by the theory of Uie potential near a surface of density tr,
I
'»^+M'tt = 0.
Hence
dV dV ,
If X, is the ratio of the Bujicrficinl magnet ization to tite normal
force in tlic first medium whose oocfGcient \t ft, we have
4
Hence V, will be positive or negative BCondingf «$ it i» gTcat«r
or le«$ than ft'. If we put n t= ivx+l and fi'= 4iric'+ >.
In thin exprowion k and k' arc the coefficients of induoed BaagJ
nitiKatioii of the fimt and second media deduced from expennentoj
made in air, and k, is the coefficient of induced mitgnetixatioo of
tlie tint medium when surrounded by the second moilium.
If k' itt greater than «, theu k, is m^^tive, or the apjiareufcl
RULgnitiTntion of the 5ni medium is tn the oppositv direction from
the nisgiii'ttzing foice.
lliun, if a veKHv! oontainiug a weak aqueous solution of a para
magnetic saii of iron is su^ndcd iu a strouger solution of the
same salt, and a«ted on by a magnet, the vetiwl moves bx if it
vnn magnetixed in tlto opposite direction from that in which »^^
magnet would set it«clf if suKpcnded in the same [i1m«. ^M
This may be explained by tiw by^thesiM lluit the solution in '"
tlic vo&scl is really maguetixed in tiie lainc dirwtioR aa the mag
nelao force, but that the solution which surrnunds the resatil is
mi^nctizcd more slrongly in the same direction. Uenoe the vessel
ta like a weak mngnet placed between two strong ones all m.*"!
1
430.] POISSOS'S TUEORY OP MAGNETIC INDPCTrOS.
Kd
»
netizcti in the same direction, so that op])OKi(>^ poles aro in contact.
The north pole of tlie weak taigvet p»tiit« in the same direction
as thow of the strong ones, but since it is in contact with (he south
pole of u ittronj^r tna^ct, there is an excess of soutli tnii^iieti8in
in the neighbourhood of its north pole, whicli cau»>ii tlic small
magnet to appear oppositely magmetiieti.
In Komc suhslances, however, the ajiparent magnetization ia
ne^tiv« even when they are sufi])ended in what is called a vnctium.
If we atcume « = for a vacuum, if will lie negative Tor thcte
sut»tances. No substance, however, has been discovered for whieli
« ha» a negative valne numerically gi^eat«r that — , and therefore
for nil known Habstances fi is positive.
Subit«nO(« for whitb k is negative, and therefor.) ju le«s than
unity, arc culkd Diama^etie eubgtanoes. Those for which k \*
poaitivc, and ft greater than unity, are culled Paramagnetic, Ferro
mngnotic, or «'mpiy magnetic, substances.
Wc shall conndcr the physical theory of the diamagnetic and
jnramagnctic properties when we come to electromagnctism. Arts.
831815.
430.] The mathematical theory of magnetic induction was flret
givvn by Poitison *. The physical hypothesis on which he foundtNl
his theory was that of two magnetic lluid>«, an hypothesis which
lias the same mathematical advantages and physical difBculties
as the theory of two electric fluids. In order, however, to explain
the fact that, though a piece of soft iron can be magnetized by
todudion, it cannot be charged with unctpial quantities of tlic
two kinds of magnetism, he suppoees that the BuhstAUc< in general
is a nonconductor of these fluids, and that only certain Kinall
portions of the substance contain the fluids under circumHtanccft
in which tbcy arc fiv* to obey the forces which act on them.
Thwc nnall magnetic elements of the substance contain each pre
cisely equal quantities of the two fluid", and within each element
the fluids move with perfect fr<'edwm, hut the fluids can never pa»»
from one magnetic clement to another.
Tlie [iroblem thenfore in of the Winn kind as that relating to
a number of small coniluctom of eli'(;tricity disaemiuated through
a dielectric in>uhiting medium. The conductors may be of any
fomi provided they are timall and do not touch each other.
If ibvy arc elongated bodies all turned in the same general
• aiORMm tU rituttiut, IBM.
04
IFDUCED UAOKETIZATIOK.
L430
** = f»i
dirwUon, or if lliey arc erowded more in od« dtrectioc than another,
th« mixUuia, as Poisson himself eli«we, will not be isotropic. Poiseon
thcrefure, to avoid usoteas inlri«acy, examinee the cose in which
L>Hi.'h mugiHitic element U spherical, and the elements are dibsem
ioatid without regard to axes. Ue supposes that the whole volome
of at) the magnetic elemente in anit of volume of the eubstanoe
ia t.
We have already considered in Art. 314 the eleotrio oondoctivity
nf a medium iu which small spheres of another medium are dis
tributed.
If the conduotivit; of the moditim is /i,, and that of the spheres
p,, wo have found tliat the conductivity of the composite syetetn le
Futting p, = I and Hj = x, this becomes
l + 2it
This <]mintity fi ii the electric condoctivity of n median eon
Hitrtin^ of porrectiy conducting spheres disseminated throug'h a
m«iUum of ootidiKiivity unity, tiie aggrtgute volume of the spheres
in unit of volume Uing It.
The *yiubol fi uhni rvpmvnt* the oocfficieot of magnetic iodnctien
of a milium, counititing of it{)herM for which the pcrmeabtltty it
infinite, disst'miiiaUd ihrou^ffa a mMliam fur which it is nnity.
Hi« itjubol i. which W0 shall call INiiMon's Ma>ncttc Coefficient,
rtfumaata thv ratio of th« volume of the ma^etic elements to the
whola Tolunt of tlw substance.
Th« symbol K i» kai>wn as Neumann's Cuefficicot of Magnet
tsattun by IndiHtion. It ir morv coovmiint than I\>moo'b.
The «yub><l >i we shall call th« OwtScicnt of Ma^faetie IndactiotL
Ifa advanti^ is that it fiKilitatca the tnuuformati.iu of magBctJe
probkua int» problem* retattaf to eWtridty sad h«au
The rclaticawuf thewthrtvaymbolsarvaafBDows:
i=:
U
If M ptll k  M. Ow mIm ffivM hy tUUa a* KpsuHots M
4
4
4
d
43a] poisson's thboet of magnetic isopctiok. 55
soft iron, we find i = \^. This, aecordiBg to Poisson's theory,
IB the ratio of the volume of the magnetic molecules to the whole
volume of the iron. It is impossible to pack a space with equal
epheres so that the ratio of their volume to the whole space shall
be ao nearly unity, and it is exceedingly improbable that so large
a proportion of the volame of iron is occupied by solid molecules,
whatever be their form. This is one reason why we most abandon
Foieaon's hypothesis. Others will be stated in Chapter VI. Of
coarse the value of Poisson's mathematical investigations remains
imimpaired, as they do not rest on his hypothesis, but on the
experimental &ct of induced magnetization.
CHAPTER V.
PABTICULAR PROBLEMS IK MAOSETIC HTDDCTKHT.
A Jfolhte SpHerieat Siett.
431.] TuK firet cxwnple of tli* complete Kolution of a problem
in mnpiftic iii^ductioD wan tliat given hy Poisson for the com of
« hollow ephi.'ri<al «hell act«d on by any magnetic force* whatever.
For Mitnplii'ily w« Bball suppose tlw origin of the nuif^netic forcee
tn bo in tho epftM oat6)d« the ehdi.
If f <leuot«8 the potential <luc to the external magnetie ayat«m,
we may expand T in a series of solid harmonics of the form
where r is the distance from the centre of the sbdl, S( in a sarface
b«TmAnio of order i, and C, is a coifficicnt. ^1
This Bcne« will be convcrgrnl pTx>\idL'd r is lees than the dtntauee^^
nf (he nvnrMt magnet of the sjnt^m which produce* tbi« pot^ntiaJ.
Hence. lor the hollow spherical shell and tl»e tpoee within it, &i»
expansioii is eonTFTgmit.
Let the external mdiue of the ehell be a. and the inner rwdins «j ,
and let the {>otential due to it« induced niagoetian bell. The form
of llie f\inet)oa Q will in general !•« diflrrrat in the hollow vpnce,
tn tfam substance of th* shell, and in tito space beyond. If we
expand these ninctions in harmooie eeriea, then, confining oar
ittmtioB to those tcnns which inralre the ntbre hannonie St,^^
thiill find that if il, t* that whieb comspottds to the hollti*^^
s)iA«> within the shell, the expnnnon tt Q, mmt be in pc«itjVB har
nuwic* nf ' '' ..> .f, .^ >*. becnuse the potvMal moA not beeone
infinite n .; >)ihvTv whive ntdius iE«.
lo the mbaUnoo of the shell, when: r^ hm l<etw«ea «, nnd «,, 
the wrtN BUtjr cwatnin K^th [v«iti>e wd MgMiTe po«*» oT r,
t'f t)M> form J,^f*f i£j5,re'i\
Outside the oWll, «Imi* • is gnmWt than ■,, auH* U» aavs
43 1 J
BOUOW SrnERICAL SI1BI.L.
6i
I
must b« ooDVCTgcnt however great r may be, wc mmt have only
nogntivo i<owew of r. of the fonn
The condition* wlitcli umxt be satisfiod by the function Q arc;
rt niuttt he (I) linit«, nod (2) continuous, and (3) must vaoiwh at
it« dixtAnre, and it must {*) cTuywliere nitisfy Lnplace's
On norouDt of (l) ^i = 0.
On iKxount of (Z) when r = u, ,
and when f = it,,
On aooount nf (3) A, = n, nad the condition (I) \» KatixGcd
everywhere, wnce the fun<tions are hannonie.
But, bcsidcH the«e. there are other condilioim to be iiatisficd at
the inner and outer curfaee in virtue of eqiiatioD (10), Art. 127.
At Die inner aurface where r t=. <7,,
da, da, dF
ilr dr dr
atxl at the outer vurfnce where r = a^,
dV
(,+4„)'^_— . + .,,, =0,
(>^)'^'^
B K T = 0.
dr
(3)
From these conditions we obtain the equations
and if we pot
^. = ^ : 7irrr. («)
(i + 4,.)(2'+i)*4(4«<fi('> >)('© )
we find
. MH.
A, = ~{A^^f ,(i+ I)(l  (?1) ).V, C. (9)
^«=4,«.[2, + l+4ir«(.+l)(l(^')"*')]A'<C.. (10)
B^ m <«i(2i+ 1) V'*'A; C. (I I)
S,=ii,Ki{2i+i+i^M(i+l)){a»'*^a,"^^)N,C,. (12)
TheNe qnanlilie* being euWituLed in the liarmonic cxpunstons
j>ive thi> part of the potential due to tho magnetization of the shell.
Tlie quantity Iff '" alwayit positive, since lf 4irK can never be
B^ativc. Hence A, \» always nogative) or in other wordts th«
UAaKETIC PBOBLBHS.
I
kction of the magnetized »l>el) ou a point within it is alwayi
poHcd to that of the cxtvnial magnetic force, whether the she
HiramagTietic or diumiigiKtic. Tlie actual value of tJie resultant
putootiul u'ithin the »hull in
or (l+4w«)(2»+»)*jVC!S.'' (H)'
433.] When k is a large number, as it is in the cut of soft: iron,
then, unlvsH tlie shell is very thin, the magnetic foree within it^i
is l>ul a omall fraction of the external force. ^^
In thiK way Sir W, Tliomsoa has reudercd his marine galvano^^
met«r independent of external magnetic foroe hy enchwing it in^
a t'Uhe of oofl iron.
4S8.] Tlie cAne of grefltest pra<!tica1 importanee is that in wlik
(Ml. Is this cofe
(")
9(l+4,r«) + 2(U«)'(l{^f)'
^,— 80«)'(iQ)').v,r^.
J»,= I2wita,'>',C,.
5j = 4»«(3 + 8ir.)(VV)'V,Ci.
The magnetic foioo within the hollow shell is in this nase
and equal to
C.J 3('H") «
9(l + ««) + 2(<w.)»(l{a))
If we msh (o determine « by measnring the magnetic
within a hollow shdl and nuuparin^ it with the external
fortx, the limt value of tite tJiicknMs of tke ibeH nay bo Iband
(Vom the mjuattaon
Ihlin^Tlttfir fcnrt ia^de the shell is tbes half of its nke ontadt.
Siui'c, in ths nan of mw, ■ i» a number between SO aad 30,
ihiikuMM of lb» ■btU OHfbt to l>o sUmt 1^ boiMlradth part of r
ladius. Tlii> awtlvMl i» a)>ptiakt>le naly whui the value of «
bry*. When it is v««> omall the value U A^ Umnn UKnnbJc'
Moe H di^icDila on tiM *>{uar» t^ «.
idt.
1
d
4M0
srnERrcAt, snBtL.
SS
For a nearljr solid epkere with a very small spherical hollow,
llTK
3 + 4ir«
4gic
3+4 vx
(18)
I fron
I
I
The whole of this inreBtigntion mig4tt have 1»en jf^aoed directly
from that of oondnctioii through u Hjilierioal stiell, as given ia
.312, by putting ;fj = {^\^Avi)k^\a the exprassiona there given,
remembering that //, and A. in thi' [iToblem of conduction are eqni
ralent to f^ f ^, and C^ f A., in the problem of magnetic induction.
434.] The corrcHpomling solution iu two diinensions is graphically
represented in I'ig. XV, at the end of this volume. The lines of
tiuluction, which at a distauoc from the centre of the figure are
nearly horizontal, are ri^jirescnti^d as disturlx^ by a cyllndric rod
magnetized tran)iven>ely and placed in its position of stable equi
Ubrium. The lines nbich cut this system at right nngliM represent
tbe eqnipotential tiiirfacm, one of which is a cylinder. The targe
dotted circle represent* tl)i> Kcctiou of a cylinder of a paramagnetic
eubstance, and the dotted horizontal stmight lines within it, which
are continuous with the external lintrs of induction, represent the
lines of induction within the subsbiuce. The dotted vertical tines
represent the internal eiuipoli>nIinI snrOiccs, iind are c<^ntinuous
with the external syst^'m. It wilt be obeervtd tbul tlic lines of
iodoction are drawu nearer t^tgethi^r within the substance, and the
equipoteatial surlaees are vepurutcd farther apart by the paramag
netic cyliuder, which, in Ihc language of Faraday, conducts thtt
'Uoee of indnctioD better than t)ie xurrouiiding medium.
Tf we consider the system of verlifjil liaes as lines of induction,
and the horizontal syitem as ei)ui[iot<:ntiul surfaces, we have, in
the first place, the caac of a cylinder magnetized transversely and
placed in the poKition of unstable ei^uilibrium among the lines of
force, which it causes t^ diverge. In the second place, considering
tbe large dotted circle as the section of a diamagnetic cylinder,
t}i« dotted straight lines within it, together with the lines external
to it> represent the eflect of a diamaguetic substance in separate
iog the lines of induction and drawing together the e(ui potential
sar&ccs, such a substance being a worse conductor of ma^etic
iulaction than the surrounding medium.
60
MAOSETIC PROBLEMS.
[43
Catt of a 8pher« in teiUi (Ae G*e^!euft of MagntUMtiim are
Dijfereni in Different Birevtitm*.
435.] Let a, fi, y be the compDnentd oi magnetic force, and A, B,
C those of the nia^iu'tiKatioii at a.ay point, then tite moet geoerai
lincitr relation between theac ({uantitics \s given by the equations
A = '■,a+;:^/3+ j,y,j
where the eoeffieicnt* r, /», q are the nine coefficient of magnet
ization.
Irft lis now Bi])i»iee that theec arc the conditions of magnet
ization within a iiphere of ladiHti a, and that tlic magnetization at
every point of the Niilwtanoo w uniform and in the aame directiim,
having the components A, H, C.
L«t ti8 aUo vu)>poK4> that the external magnetizing force ie also
tinifonn and puralkl to one direction, and has for its componcote
I. y, z.
The ralne of T is therefore
r = {.Vi+f> + ^r), (2)
and tliat of 0.', the potential of the magnetisation outside the ^herc,
i«, by Art. 391, At a^
^ ^ cr=^^{^*+5^+c4 (3)
The value of XI, the potential of the ma^etixution withia the
Thu actua] potential within the qihore W F+ti, to that we ohall
hnvc for tbs compoDents of the mag^otic force within the sphere
y=ZivC. )
Hence
(l+J»rOJ+ iwp,B+ jTftC.r,X4ftr+y,2;
Stilving then eqiuUiona, w» find
i
I
CRYSTALtIXE SPHEBBi
I
I
(8)
^
where Ifr{^ ''1 + Ji'('»'ift?j + f,r3ft?,) + (8'')'0.
^P( = ft  i " (£« ?.ft n ).
to.,
wlwMe i> is the detvrmiuant of ihe ccx^fficients on the right nd« of
c(aittions (6), and 1/ that of the coefficients on the left^
Tho new system of copfBci«nU p. if, / will be symmetrical only
wh«a the system p, q, r a ttymmetrical, that is, wlien the co
rfficienta of the fonu p aro vquui to the corresponding ones of
tb« form q.
436.] ^Tbemotnentof tbecoaplet«DdiDgtottin) the sphere about
tlie axis of j from 1/ towards z is found by eonsidcrin" the foroea 011
ton elementary volume und tidiing their sum for the whole sphere.
The result is
= %.anP,'ii'9:y^Hft'r:)TZ+J{q,'Z~n;Y)). (9)
If we make
X = 0, r=Fcos0, Z=Fmn0,
thi« corresponds to a magnetic force F in the plane of y:, and
inclined to y at an angle 0. If we now tnrn the sphere white this
force remains constant the work done in turning the sjihere will
f"
be / Ld0 in each compIet« revolution. But this is equal to
j;r««'?«(/V?.') (10)
Hence, in order that the revolving vphorc may not become an
ioexbaustible source of energy, /),'= j,', and similarly p/= y/ and
These conditions hIicw that in the origiual iHuntioDs the coeffi
eieot of .B in the third equation in equal to that of C tu the second.
> Ilk aanalitT of tbe cwBi':! hiIh p uul f limy be hIiowd oi fi'Uuwa : Lst the fonu
Mto; oa tnn •lilioro turn it ationt » iliunrtvr irlim (ilrwtfuuvodinai arc K f, '
thfoogh Ml Nistc >f : Uicn. If W douotc tin mngj <;>f \ht iphorc, wo L>vu, by
An. 430,
811I if tlw KIM of cwrdlribUi bn filed In tbe iplitn wa htv« in <gii««queiioc of tbo
raUtiiMi
Bjr(rV2/i)>«,«tc.
B«M« v« nwv pnt
Tlikt tlw nrolrinc «ph«n niaj' ni)t bocam* k »ource of enoiuy. tbo ciprtarion cm the
light hmul o< llMlMt vqtutiun iiiuat b« » pnrflKt Jiiruniiiikl. Utioue. lioM A, B, C
■fB Wninr ItndiiHu of i. Y. Z, it follow* Uot W it t> ijcuKlivCiu fanctluii uf JT, 1'. Z,
amA 1^ rcquirad mull i> iX once ijeducwi.
Sw >bv Kir W, TliuiiiNo'i R«iiiiiit of Paptn tm Ettdriettjf and Majnaism,
pp. 4S0 461.]
92
SUQXETrC PBOBLKHS.
and 80 on. Ueaco, the ayetem of eqnations is eymmclrioil, iind Uto
equations become wbeo referred to the priDvipnl »xt» of m«{p
Detizatioo,
A =
S =
i + J"'.
1+5'"*
X.1
Y.
C =
z.
(uy
llic moment of tlio couple tending to turn t)i« sphere round th«
axis of fit
i= JM»
f.— f.
XZ.
In most cases the diS^encos between the cocSicient« of magne
ixatioa in different direction* are very snuU, so th»t ne may pat
(
1
1 0^1
lei
I
Tliio is the force tending to turn a crystalline sphere about
asiii iif X from / toiriirdis /. It aliraj* tends to plac« ttie axis
greatest magnetic eoefiieient (or least diamagnetie coelRcientJ parallel
to Ute line of magnetic forec.
Tlie oorrosponding case in two dimensioiu Lt represented
I'ig. X\l.
If we ynppose tlie upi>er aide of the figure to be towards the
north, till' figure reprewut^ the lines of force and equipotential
wirraeea as dinlnrbed by a trau9verM>iy magnetiied cylinder placed
with the north side eastwards. Tlte re»uIUnt force tends to turn
tlw cylinder from eaat to north. I'he large dotted circle repretieiits
a aection of a cylinder of a crystalline enbstaiMe wliicli has a larger
corfllcient of indoelioB aloof; an axis from north«a»t to southwest
tltau along an axw fnun northwatt to •ontheuL The dotted lines
within lli« (Side npnM^nt the lines oTindootiofi aad the e^uipotential
cur&cea, which in this cbod are not at right angles to each other.
The T«»uttaBt (bn» on the cylinder ia avideiiUy to turn it from east
t*t nortli.
497. 1 Tlte eace of an e)tips«Mit placed in a field of mufnrm and
l«t«Uel Bta^Betic forea hm booi nlvnl in a nnr ingcniooB manner
U r is the piitcati^ at the pmt (»,/, s^ ^M to the gnritntioii
«i a My of My form of luufotm <hnly a
a,^„
£:»the
ELLIPSOID.
>
poUittinl of tlie magnetidm of the same body if aniformly m^^
tuAiv^ in the direotion of « with the intennty 1= p.
dV
For the value oi — jhx at any point ie the excess of the value
of r, the potentinl of the luidy, ahow l", the toIhc of the potential
when the body \» moved —his in tho direction of «.
If we sup>o«ed tht! Imdy »iuftc<i through the di»Unee — d;r, and
ita density chang<>d from p to — p (that is to tay, made of repulsive
instead of attractive matter,) then — j~hi would bi' tin i>otentiul
doe to the two Iiodies.
Now consider nny rlomotidirj portion of the hody fontainin^ a
volume hr. \\» (jiHintity is plv, and corresponding to it there is
an element of Ihn ithilVd Wly who^e quantity ia ~ptv at a
distance —hx. Tlic cHect of theeo two fltratnta ia equivalent to
that of a magnvt of strength pBr and length hx. The intensity
of magnetization is found by dividing the magnetic moment of an
element by its volume. Tlie rcEnlt i* phx.
Hence — t 8* ia the magnetic potential of the body magnetized
with the intctuity phx in the direction of z, and — XT '* *^' ^^
the body magnetized with intensity p.
Tliis potential may he also considered in another light. The
body was shifted thri>ngh the ditttance —hx and made of density
— p. Throughout that [lart of space common to the body in its
two po«itionB the density i» zoto, for, as far as attraction is oon
iU!m«Kl, tho two eqoal and oppodite dcusitteif annihilate each other.
Th«Ta remains therefore a shell i>f positive matter on one side and
of negative matter on the other, and wc may regard the rosultsnt
potential as due to these. Tlie thioknoM of the bbell at a point
where tho normal drawn outwards makes an angle < with the axis
of f is 07 COS ( and ita density is p. ITic surfacedensity is therefore
/)&« coa r, and, in the cose in which the pot4>ntial '< — >~i "i^
surface>deoMty is p eog t.
Id this way we can find tlie magnetic potential of any body
uniformly magnetised parallt't to a giv<n dircution. Now if this
uniform magnetization is due to mugnelio induction, the mag
netizing force at all points witiiin the body niiul also be uniform
ami parulleL
Tilts force conststs of two part*, one due to external causes, and
64
MAOHBTIC PROBLEMS.
[43;
^
th« otker due to the magnetisation of the l)ody. If lln^refore th«
exteniftl ma^etii^ force U iiaiform anil panillt^l, Iho magneliti force'
due to the magneti]'3tion must also l>o iinifomi and panllel fi
all {lointa within the body.
Ueoce^ in order tliat this metliod may lead to • solution of the
problem of magnetic induction, fz must bo a linear fuDct)<»i of
llie coordinate* x,y, s within the body, and therefon; F mxiai
a quadntiv function of the coordinates.
Now the only cases with which we are act^uainted in whiob
is a quadratio function of the coordinates within the body are thoM
iu which the body is bounded by a complete surface of the seed
degree, and the only case in which such a body is of finite dimen
KiouK is when it is an ellipsoid. We shall therefore apply tb'
method to the case of au ellipsoid.
Let
be the equation of the ellipsoid, and let •t>^ denote the definite integral
7. *Afl»j.A»iaai.rf,s\f^.i.Aii\ • "/
Then if wc make
inalc
M=iv^
.**«
d(i^r
X=i
'4
tlic value of the potential within the clliiwoid will be
^=_(i,« + Jfy+3V) + oon9t. (4)
If the ellipsoid is magnetized witli uniform intensity / in a
direction making angles whose cosines are /, m, m with the axes
of s, g, :, so that the components of nia^netizatioD arc ^^
A=Ii, B^tm, C^Iii, ^
the potential due to Uiis magnetization witlun the ellipeoid will be
a^I{Lh!+MmfiNK:). (a)
If the ext«rDal magnotitiag force is ^. and if it« componenta
are X, i', /, iU potential will be
■Hie oomponont* of the actual magoetiring force at any point
within the body are therefore
x+JA r+j?j/, z+cK (rj
• Sot TVuMnn luul T»H"< SMniat Fhtonfig, { C23l
I
438.]
ELI,IP30n>.
65
I
The nuwt geiiCTHl relatioiui betwifen tho mngncttzalion ani) the
nugtielixin^ force arc g^ven by thiwo linear eciuationit, involving
nine coitffii'knttt. It is neoessary, lioweror, in ordor tt> riillil tltc
condition of tho coniM^nration of energy, ttiat in the «a«e of ning'iattc
joduction tlirM of theitc shoald be equal respectively to other three,
•0 that we shoukl bave
c ^ K\(x+AL)+K\(r+Bi/)+K,{2+c.y).)
From thOM! <ffunti<in£ we may determine A, B and C in terms
of X )', Z, imd tliis will give the most general tiolution of tho
problem.
The potential outiiide the ellipsoid will then l>e that due to the
ougTieti/Ation of the ellipaoid together with that due ia the eictenial
magnetic foroe.
43B.] The only case of practical importance is that in which
«'. = «"» = "'s = 0. (9)
We have then
^ =
B =
'T^'
(10)
/l^^'
If the ellipsoid has two axes oqnal, and is of the planetary or
tKl form,
...
Ifi
in'*),
)•
(")
(12)
If the ellipsoid is of tho ovaiy or clon^^tcd form
i^j, = _a,( log )
JV
1
1'
l+e
= ..(! ,)(f...S0
In the €■« of a sphere, when e = 0,
i = Jf=iVaJi
(13)
(14)
(13)
roL. It.
66
MAOKETIC PBOBLEltS.
[43*
*ctf
%
Id th« cose of a very {lattened planetoid L becomes in Ibe limU
cquul to —in, and JTand I^ bvoome — k'*
In the case of a veiy eloa^t«d ovoid i and Jf approxim&t
to the value —2ic, while .V approximates to the form
nnd vanieliOK wlun « = 1.
It appours from these results that —
(1) Wlii'ii K, the coefficient of magnetization, is very Bniall,^
whether positive or nc^tive, tlie induced ma^netizatioD is nearly
equal to the magnetizing force multiplied by k, and is almost
independent of the form of the body.
(2) When ic is a targe poiiitive quantity, ibe magnetization depesda
principally ou the form of the body, and is almost indepvodont c^
the precise value of k, except in the caae of a lon^fitodinal for
acting ou an ovoid so elongated that A'x is a small quantity tboug
R is large.
(3) If the value of « could be negative and equal to —
ahouM have iin infinite viduo of the maguctizaiion in the case of
a miignetizing force actinjr normitlly to n flitt plnte or disk. TlHta^
abKurdity of (bin reitull confiniut what w« »aid in Art. 428. ^M
Hence, ex peri men tit to determine the valac of k may be made
on bodies of any form, provided ■ it very sniull, as it is in the CMC
of all diamognetic bodieei^ and all nuLgoetic bodies except ii
nicko), and cobalt.
If, liowercr, as in the case of iron, k is a laign number, expen
ments mode on aphcres or flattened figuru are not suitable to^i
determine k; for instance, in tiie case of a sphere the ratio of th^
magnetization to the magnetiung force is as t to 4.32 if ■ = 30^^
as it is in some kinds of iron, and if k weru inlinite the ratio would
be as 1 to 4.19, so that a rery amall error in the dcterminatioa
of the magnetization wonld introdtioe a very large one in the
value of «. ^J
But if we make nsc of a piece of iron in the form of a Tei^
dongat«d ovoid, then, as long as Ak is of modentte value com '
paml with unity, we may deduce the valno of * from u determination
of the magnetization, and the smaller tho value of ^V the more__
accuTutv will be the value of «. ^1
In fact, if Nfhe made unall enough, a snuill errw in the valo^fl
438]
CTLisraa.
67
I
»
of A' itaelf will not introduce macb error, so that ne may use any
etoDg&tcd body, hucIi as a wire or long rod, instead of an ovoid.
Vfe mnst remember, howeiier, Uiat it is only when the product
Xk ia small compared with uoity iliat this Etibstitution is allonable.
la laot the distributitm of ma^ctiem on a long cylinder with flat
cads does not reeemblc that on a long ovoid, for the free mag
netism is very much concentrated towards the ends of the cylinder,
vrb«reas it varies directly as the distaouc from the equator in tlie
caueof the ovoid.
The distribution of electricity on a cylinder, however, is really
comparable with that on ao ovoid, as we have already seen,
Art. 152.
These results also enablo us to understand why the magnetic
nuHnent of a permanent magnet mn be made so much greater when
the oui^nct lias an elongated form. If wc wore to ma<;ncti7.e a
disk with intensity / in a direction normal to its surfaie, and then
Imiv« it U> itself, the int«ri»r [larliclos would experience a constant
demagnetizing force equal to 'In/, and this, if not snRicient of
itself to d<!Klroy jmrt of the niognetixutioD, would soon do so if
aidod by vibrations or changes of tvnqHTuturc,
If we were l^i magnetize a cylinder trunsvereely the demagnet
izing force would be only 2 it I.
If the magnet were a sphere Uie demagnetizing force would
1m. s«/.
In a dixk magnctlMd transversely the demagnetizing force is
a' ' /, and in an vlongatod ovviii magnetized longitudinally it
ia least of all, being 4 n ^ 7 log ^ •
Hence an elongated magnet is less likely to lose its magnetism
than a abort thick one.
The moment of the force acting on an ellipsoid having dJITerent
na^etic coeSicienta for the three axes which tends to turn it about
the axis of x, is
Hence, if k^ and xg are smalt, this force will depend principally
on the crystalline quality of the body and not on ite shape, pro>
vided its dimensions are not very unequal, but if Cj and c, are
coasideiahle, as in the case of iron, the force will depend principally
00 the shape of the body, and it will turn so as to set it« longer
axis parallel to the linee of force.
68
UAONETIC PBOBLBUS.
[439
]
>t4
If ft gufni^icntly itrong, yet aniforoi, field of nuigTieUo force oodM
1*c obteincnl, an Glongated isotropic diaina^netic body vould also
set iUelf witk its longcitt dimecuion parallel to the lines of magiiietJo
foroc.
439.] Tlic qtientioD of tlie diMtribution of the tnagnetization of
OD ellipsoid of revoliiUon under the action of any magnetic forcee
bas been investigated by J, Neumann*. KirchhoCTt has extended,
the Dtethod to tbe «a*e of a cylinder of infinite leof^ acted on bj
any force.
(Ireen, in tlie 17tli wction of 1ii» Eisay, kas given an invest
igation of tbe distribution of magnetitim in a cylin<ler of finit
length acted on by n uaifonn external force pantllvl to its axis.
Though ftome of the vtepa of this invc«tigntion are not very^_
rigorous, it is probnble that the result repreocnts roughly tlM^
ncttml mngnetization in this moot important c*»e. It certainly
vxpress^if very fairly the transition from the case nf a cylinder
for whii'h k is a large number to that in which it is very small,
but it Ihils entirely in the cose in which k is negative, as in
diamagitetM substancee.
Oreen Bnds that tbe linear density of free magnetism at
distance x from the middle of a cylinder whose radios is a and]
whiMo length is 2A u
X = xxX/a p ►
«"■ +b"^
where pis* nomflrical quantity to be found from the eqitatioa
0.a31«l>32log.i) + 2ii»— ^.
The following are a few of the oorresponding values of p and k.
IC
P
>
f
■0
U.803
0.07
SSft.4
0.0 1
9.137
0.08
GS.Ol
a.ot
T.5I7
0.03
4«.416
0.09
G.SI9
0.10
tMTf
D.04
0.1437
1.00
90.18S
0.04
0.000]
10.W
liJ9i
0.00
0.0000
Msxno
*
imapau
• tWbkU
>sV
t iV*,M
•J'*" v*^^*L
44aj FOBCE OK PA8A ASD BIAMACSETIC BODIES.
69
I
Wbea the leo^li of tlie cyliiKlcr ia great compared with its
nditiit, tlic whole (]iuintity of free mag^nttism on either side of
the middle of Ike cylinder is, lis it ought to be,
Of this i/jjf is on the flat end of the cyliinler, and the di^tauce
of the centre of gravity of the whole quantity M from the end
of the crltiider w  •
P
When K is vexj" small p is large, and nearly the whole free
magnetism is on the ends of the cylinder. As k increases p
diminishes, ami the free magnetism ia spread over a greater distance
from the ends. When « is inflnite the free magnetism at any
point of the cylinder is simply proportioDal to its distance fron
the middle pointy the distribution being similar to thnt of free
electricity on a conductor in • 6eld of uniform force.
440.] In all substances except iron, nickel, and cobalt, the oo
cfEcicnt of magtietizjition is so small that the induced magnetisation
of the Iwdy produce* only a very slight tdtcrjition of the forces in
the magnclio field. We may therefore w^^nme, iw a first approx'
imation, that llie actual mikgnctie force within the body is tlie same
u if the body had not been there. The siijHrfii:ial magnetization
dF dV
ot the body is therefore, as a first approximation, « ti where 2—
■« the tste of increase of the magnetic potential duo to the external
magnet along a normal to tlie surface drawn inwards. If we
now calculate the potential due to this superticial distribution, we
may nse it in proceeding to a seeoncl approximation.
To find the mechanical energy due to the distribution of mag
netism OD this first approximation we must find the surfaceintegral
taken over the whole surface of the body. Now we have shewn in
Artv 100 that this in equal to the volumeiotegral
^=Kfff'(^\
4r
) A^Jydt
taken through the whole space occupied by the body, or, if R is the
resultant magnetic force,
E=\jjf^JPdxdsdt.
Xow since Uie work done by th« magncUc force on the body
ro
MAQVETIC PBOBLCMS.
during a diBp1ac(>ment fi^ is Xim where .V ui the mcchiuiciil force
in the direction of x, and «nc«
f XtxhS = constoot,
which shews that the force acting on tho body is as if every part
of it t«nded to niovi> from places wlicr« ^ is less to places where
it ia greater with a force which on every unit of votunto \»
If K is negative, as in diatnanTtcttc bodies, this force is, as Farads
first showed, from stronger to weaker parts of the magnetic
Most of the actions observed to the case of diamugnetic bodies
depend on thia properly.
J
441.] Almost every part of magnetic science fmdtt tt« use in
navigation. Tbc directive actton of the earth's magnetiKin on thej
oompnes needle is the only mctiiod of lUMerlAining the ship's cou
when the sun and stars are bid. The declination of the needle I
the true meridian seemed at lirst to be a hindrance to the appli
ration of tho eompnes to navigation, but after this drRiculty lud
been overcome by tho constniction of magnctio chart* it appeared
lively tbiit the desalination itMlf would assist Ui« nuiriner in de
termining his fhip'^t plucc.
Ilie greatest diHiculty in navigation bad always been to ascertain
tb« loQgitndo; but ainee (be declination is diflercnt iit dtlfcrent^H
points on the same parallel of latitude, an observation of the deV^
clination t<^ther with a knowledge of the latitude would enable
the mariner to find his position on the magnetic chart.
But in recent times iron is so largely used in the construction of
ships that it bas become impossible to use the compass at all without
taking into uceount tbe action of the ship, aa a magnetic body,
on the needle.
To determine tho distribution of magnetism in a mass of iron
of any form under the influence of the earth's magnetic force^
even though not subjeetv<d to mechanical strain or other distur
ances, ia, as we have aren, a very difficnlt problem.
In this ease, bovever, Uie problem is simplifwd by tho followU
ronsideratioui.
44'.]
8BIPS MAOSETTSM.
71
I
I
»
The compass is supposed to be placed with its centre at n fixed
point of the Bliip, and so far from any iron that the nmjntlisin
of tbr Dccdh^ docs not induce any perceptible magntdiiim in the
ship. The rizc of the compass needle is Bup)09ed so small that
we may regard tli« mn^ctic force at any point of the needle as
tbe same.
The iron of llie ship is supposed to l>c of two kinds only.
(1) Hard iron, mai»nrtiM'(l in n ctmstant manner.
(2) Soft iron, the magiiotiKat ion of which i» induced by the earth
or other magnets.
In Htrictnees we must admit that the hardest iron is not only
capable of induction but tiiat it may lose part of its eocalled
pemanent magnetization in variolic way«.
The sofleiit iron id capable of retaining what is called resUwal
magnetization. The actual propertifs of iron cannot bo accnmtcly
represented by fiiippoaing it compounded of the baid iron and the
soft iron above deGned. Itut it has been found tlial when n ship
ia acted on only by the earth's magnetic force, and not subjected
to any extraordinary gtrees of weather, the supposition that the
magnetism of tbe ship is due partly to permanent magnetization
and partly to induction leads to sufficiently accurate results when
applied to the correction of the compass.
The equations on which tbe theoiy of the variation of the com
paas is founded were giren by Poisaou in the fifth volume of the
MAaoirn dt VliitUuf, p. 533 (1824).
The only assumption relative to induced magnetism which is
involved in these equations is, that if a magnetic force A~ due to
external magnetism produces in the iron of tbe ship an induced
mag^netization, and if this induced mugnctlKiition excrLf on the
compass needle a disturbing force whose components are A", I", Z",
then, if the external magnetic force in altered iu a given ratio, the
components of the disturbing force will be altered in the same ratio.
It is Inic that when the magnetic force iwting on iron U vciy
great the induced magnetisation is no lunger pmportional to th<!
external magnetic force, but this want of proportionality is quite
insensible for magnetic force* of the magnitude of those due to the
earth's action.
Hence, in practice wc may assume that if a magnetic forc«
whose value is unity produee» through the intervention of the iron
of the ship a diniurhing force at the compass needle whose com
ponents arc a in the direction of «, dia that of y, and y in that of «,
72
MAGKHTIC FROBLKHS.
[443
the components of the disturbiti^ force due to « force X in the 
dir^f^tjon of a witl he a A', (/,V, and,,ffX.
If therefore we Assume axes fixed in the ship, m ihut x 'a loward*
the ship's bead, g to the starboard eide, and s toiranls the Iced,
and if X, )', X represent the componente of the earth's mag^ncttc
force in these directions, and A'', Y', Z" th« components of the com
bined magnetic force of the earth and ehip on the compose needle,
 r=r+j,T+«.i'+//+i2, [ (1)1
W z'= z+gX+ir+kz+R. )
In these equations a, 6, e, d, «,/,y. A, i are nine oonstant co 1
efficients depending on the amount, the arrangement, and Ui«l
capacity for induction of the soft iron of the ship.
F, Q, and It are constant quantities depeoding on the permanent i
moffuctization of the ship.
It is evident that these equations arc sufficiently general if ,
ron:;netic induction is a lint^ar function of magnetic force, for they i
urv neither more nor less than the most general cxi>re«>ion of a
vcvtor lU a linear function of another vedor.
It mny also be shewn that they are nut tM> general, for, b;
pro]K'r arrangement of iron, any one of the eocfficivnts may be
mmlc lo vary indo[iendently of the othcnf.
Thuii, a long thin md of iron uiMJer the action of a longitudinal
roognetie force acquirer poles, the strength of each of which ia
numerically equal to the cross aection of the rod multiplied by
the magnetizing force and by the coefficient of induced ua^et
isation. A magnetic force transverse to the rod produces a roach
feebler magnetization, the effect of which is almost insensible at
a distaoce of a few diameters.
If a long iron rod be placed fore and aft with one end at a
dUlanee x from the compass needle, measured towards the sbipV
b(^, then, if the section of the rod is J, and its coefficient of^j
ntagn<!tiratii>u k, the ctreDgtli of the pole will be JkX, and, >^
Am — , tlip force cierted by this pole on the compass ncvdlu
will be II .Y. Tlic rod may be supposed so long that tfae effect o^Hj
the other pole oa the eompim may be neglected. ^H
We have tliua oblaiuod the means of giving any inquired ralaa^
to the cieffieioni it. ^M
If we place another rnl of •eetion S with one extremity at th4^
■ome point, dislauL « Ironi the eum^wH toward the h^ad of ^i^
I
441.]
ship's maonettsm.
73
I
Tttwelj and extending to starboarcl U> sucli a disUnce Ihnt llie
distant pole produce no itonstblc ifitct od the compass, the di»
tarbin^ force doe to tliix rod will he in the dircctiou or x, and
equal to — , , or if ^ = — , the force will be jr.
This rod therefore introtiucpa the cof flioient i.
A third roil cxtiMnlin^ downwards from the some point will
introduce the coelTieitint c.
Th« coeEBwents d, e,/ may he prwlueed by th»e rodfl ertending
to bead, to starboard, and downward from a point to starboard of
Ui« compaea, and y, i, it hy three rods iu pAralk'l direotioDs from
a point below the coropans.
Henoe each of the niD« coefficients caa be ecparateiy varied by
means of iron rods properly plwwd.
The quantitie* P, Q, It ure eimply the compODeatl of the force
on the oompaai arixing from thv permanent magnetisation of the
ship logetlier with that part of the inducwi mag'ncti/ation which
is dnc to the action of this permanent mii^nrtizution.
A complete difcuiwion of the eqiiativn« (1), noil of the relation
between the tme maf>nctic courst of 1he ship and the course as
indicated by the compass, is given by Mr. Archibald Smith in the
Admindty Manual of tie Devialion of the C'lapant.
A valuable graphic method of investiffating the problem is there
giv«n. Taking a fixed point as origin, a line is drawn from this
point repntDenting in direction and magnitude the horiKotital part
of tbc actual magnetic force on the compassni.'edle. Ax the ship
is swung round so as to bring her head into difTorent azimuths
in saccosiott, the extremity of this line df«;ribcM a curve, each
point of which corresponds to a particular azimuth.
Soch a curve, by means of which the direction and magnitude of
the force on the comjuifi^ is given in terms of the magnetic course
of the »lii[>, is cstllfd a Dygogram.
There are two varieties of the Dygogram. In tlie first, the curve
is traced on a plane lixod in space as the ship turns roimd. In
the second kind, the corvo is traced on a pluue fixed with respect
to the ship.
The dygogram of the Grrt kind i* the Ltma^n of Pascal, that
ofUiK second kind is an ellipse. For the construction and use of
tbese curves, and for many thvoremn n* interesting to the mathe
niatieian sw they are important to the navigator, the reader is
referred to the Admiralty Mamiaf of (At /Mvialion t^the Compau.
CHAPTER VI.
WRBEIt's THEORY 0? IKDITCED MAOHSTISX.
44S.] Wb h»ve seen iliat Poieson mippoeeB tho magiielization ot
iron to c^onsist in « sepantion of the mafrnctic fluid* withiD e»ch
SMgDMic tnoleonle. If we wish to avoid Uiv usumptioit of tiui
psislt'iicc nf niajjniUc fluids, we may rt«t« ihc sam* tbeory inl
another fonn, by Mjing lluit racli molecule of the iroii, whiin tbel
ma^nrtiiiiti; furcv nets oa it, becomo* a magnet.
Weber's tluiiiy diffcw fntm Uii» in assnmin^ that tbc molMukal
of Lhe iron an> always nuifrneU, vvea before the application of
the magnetising forvo, but that in ordinaiy iron tiuf magnetic
axes of thf moleoulfti are turned indilfercntly in every direction,
■o tliat tho iron as a «ho1« exhibits do mnf^nrtic proporliw.
Whrn a luagnetit' force acta on the iron it tends to tarn the
L«f tliu molecule* all in one direction, and so to cause the iron,
I'whole, to become a majn>et.
If the axes of all the motcculoe were eei panllal to tmeh other,
the iron would «>xhibit the gmtan't intensity of magnetization of
whii.'h it » r«]«be. Hi'neo Weber's tb«<ory impji*^ the existenee
of a limiting iuleusity of maf^netintion, and the experimental
rvidenoe that swh a Hmil exists is tbeivfism nere«saty to the
tlworjr. K\peTtmieot» sh«Hria(; an appnrMch to a limiting valoe of
BMgnelitatnm hav* tw»n made by Joule * and by J. Muller t.
The miMcintttto nf Berta t m aleetnttjr^ iron deposited nnder
tW action of majpMkio Rhm Awuuk Um Moot ewnpleto evidem
of lbi» limit.—
A filvet wiiv w«t tmmuliMl, and « x%rj mutmt lior on the
• iMMbyn>Mk^M»,t»^m,ia»i «%&««. till.
I
445]
TH8 MOLECtlLES OP IBOS AKB SIAONEre.
75
metal mu laid \mn by making n fiDo lonj^itiKliiuil Mnitch on the
varnhih. The wire was tlicn immersed in n solution of a suit of
troD, and placed in a iiiaf^nAtiv Bcid vritli the scratch in the direction
of a line of ma^elic Force. By nutking the wire titc oatiitide of
an electric current through the aolution, iron was <lepo»it«d on
the narrow expojunl surface of the wire, mo1ecn1i> hy molecule. Th«
filament of iron thns fbrRMd wita then exHmiued maguetieally. Ila
magnetjo moment was found to be very grout for so small a manf
of iron, and when a powerful magnHizini^ force was made to act
io the same direction tlio iiioroaac of temiKirary mngnettzatton was
found to be very small, and the pcrmnncnt magnetization was not
altered. A magnetizing force in the reverse diniction at once
reduced the filatnent to the condition of iron magnetized in the
oidinarj' way.
■Weber's theory, which supposes that in this case the mn;jnetizing
force placed the axis of each molecnlc in the same direction during
the instant of ita deposition, agrees very well with what ia
obserrecl.
Beetz foond that when the electrolysis is continued under the
action of the niagnetixing force the intensity of ma^gnetiicatioQ
of the sabeequently deposite<l iron diminislie^. The axes of the
molecules are probably di'fle<^ted from the line of magnetizing
force when they are being laid down side by side with the mole
ciilea already deposited, so that an approximation to parallelism
can bo obtained only in the case of a very thin filament of iron.
If, as Weber supposes, the molecules of iron are already magnet*,
aoy magnetic force sufficient to render their axes parallel as they
are electroly tically deposited will bo sufficient to produce the highest
intensity of magtietizatton in the deposited filament.
If, on the other fuind, the molecules of iron are not magnets,
Imt are only capable of magnetization, the magnetization of tho
deposited filament will dejwnd on t!ic magnetizing force in tho
famv way tn which tliat of soft iron in general depends on
it, Tlie expcn'meut* of Bcctz learc no room for the latter hy
fothcsis.
448.] W« sliull now aanime, with Weber, that in every unit of
Tolutne of the iron them are » magnetic molecules, and that the
magnetic moment of coch is m. If the axes of all the mulecules
were placed parallel to one another, the magnotic moment of tho
unit of volume would l>e
3f= ttlM,
76
WEBER 8 THEOBT OF INDUCED MAONETISJI
and this would be tho great«)>t inUnsity of magnetization of which '
the iron is capable. i
Id Hie unmagnetizod atatc of ordinary iron Wchcr oiippmcs tbc
axes of H& iiiolei'uli;)< to bo placed indilfi^rently in all direction*.
To ezprcsR this, we may suppotte a itpbere to be described, and
a radiuM dratvn i'roiii the centre {WTallel to iJie direction of tlie axis
of eaeli of tlie n molecules. The distribution of the extiemitiee of
these radii will express that of the axes of tlie nioleculee. lo i
tlic case of ordinary iron tiiese n jioints are equally distributed
over every part of" the wirfac^ of the sphere, eo that the number
of moleculeis whose axes make an angle less than a with the axis
ofris „
(I COB a).
and the number of molecules whoso axes make angles with that
ofa^ between a and a + daia therefore
 cin a^a.
This is the arrangement of the molecules in a piece of iron which ^
has never been magnetized.
Let us now supjKxse that a magnetic force X is made to act
on the iron in the direction of the axis of x, and let as consider '
a molecule whose axis wah ori^iially incliDed a to tlio axis of x.
If thit; moltTulo is perfectly free to turn, it will place itself with
its oxiK jwrullcl lo Uic axis of x, and if all the molecules did so,
the very itlighte«t mngnetixing force would be found sufficient
to develojie the Yciy highest degree of nuignctizutiou. This, how ^
ever, is not tlte case. H
The moItH^ules do not turn with their axcn pnrnllcl to x, and ~
this is either becauae each molecule is acted on by a force tending
to pnwrve it in its original direction, or because an equivalent!
effect is produced by tlie mutual action of the entire system of]
moliculn.
Weber adopts the former of them suppositions as the simplest,
and suppoBi* that each n)ol<«nle, when doRccted, tends to retnrn
to its original ixwition with a force which is the same as that
which a majn^etio foivv D, arting in the original diivction of it«
axis, trould produce.
The {Hwition which the axis artiially assamcs is therefore in the]
direction of the ntmltaat of X and I).
Irtt JI'll ivpnwiit a nii'iliin ul'a siihere whose mdius ivprasents,
on a certain wuih), Uw force H.
I
d
443
DBFLBXIOS Of iXE9 OP HOLECCLBS.
77
Ld the niiai, OP be pamllct to the uxiit of a particnUr molcculo
in its on'giiuJ position.
Ijet SO repreMnt on the mmv iwalo the miignctizing foroct X
which i» Kuppcwed to Ml Trom S lovrurilit 0. Tb*D, if Ihc mokrule
\a aut«d OD by the forw A' in the diredion SO, »n<I by a force
I) ia » dirwtion jxirttli;! to OP, tho orifjimtl direction of it» axin,
it« axis will fit itself in tlie direction SP, that of the reiult^nt
of X and O,
Sinoc the axes of the moIeciiIeH are ori^nally in all directions,
P may be at any )M>int of the sphere indifferently. In Fi^. 5, in
whioh A' i* lt;M than 2), SP, the final poiiition of the axis, may be
in any dire«tion whatever, but not indilTerently, for more of tJie
I molecules will ha?e their axes turned towards A than towards Jf.
In Fig. 15, in whioh X is {i;rpater than i>, the axes of the molecules
FSg.S.
Kg.».
Hence there are two dtiTercnt canes scoordiiig as X is leas or
greater than Jf.
II 1/et a = AOP, the oHgioat inclination of the axis of a molecule
^B to the axis or«.
^^^K ff = JSP, the inclination of the axis when deflected by
^^^H the force
^^H ft = SPO, the angle of dtflexion.
^^^H 80 = X, the magnetizing force.
^^^H OP = Jf, the force tending towards the original position,
^^f 8P = R, the resultant of A' and G.
^K m = magnetic moment of the molecule.
^^ Then the moment of the statical couple duo to .V, tending to
diminish the angle 0, is
ml. s MA'itind,
tho moment of the couple due to D, tending to increaaa 0, ia
mL=. mi7siaj9.
78
WEBERS THRoay OF ISDCCED HAONETISM.
Equaling these values, and remetubering that J3 k a—0, we find
. . If una ,,
tan« = T> — r, iu
to determine the direction of tlie axia after dedexioa.
Vfe have next to find ihe intensity of ma^uetization produredl
in the niiiw l>y the force A', and for this puqxiae we must resolve 1
the inaynolic nionient of every molecule in tlie directiou of s,
add all tliL'se resnlved parts.
Tlie reaolvod part of the moment of a molecule in the direcfaon
of .7 is HI cos 0.
The Dumber of molecules whose original inclinations lay botv
a and a+ila is «
 »in ada.
We Uave therefore to integrate
2 = f  cos sin ai/o, (sl
rcRXtmbering that $ isa function of a.
We may express botJi $ and a In terms of R, and the cxproeion]
to be intcgtated l>ccom«8
tlie geneml int^^ml of wlncli is
Id the fimt ca§e, that in Tvhich .V iH ten Umn J), tlie limits of
integration are H = J)+X and K = It X. Id the second cosv
in which X is greater than J), the limits anj JimX+J) and
B = XB.
1= ~mn.
When X is Ices than S,
Wbcn J is equal to 2>,
Wbeo X is greater than D,
and when X becomes infinite
According to this form of the theory, which is that adopt
by Weber*', as ttu maguetixtDg force increosos from to i7,
• TImm {■ wvio nibtala In Ik* Ceramk g<nai bj Wsbcr (TVwh. Atad. Sat i.
B. in (1613), or Pen.. Jra Usavii. p. !«; (1812) m (ba noOl d (Us iatrgriUft.
^•bfaorwhUsniutllTwIvUB. Bkfcnnwlaii ^
iM>
J*^^JC'U■^.^D'
444]
LIMIT OP MAOSETIZATIOir.
79
nmgiietization increases in tlio some proportton, Wb«n Uie mag'
tuilizing forv« attains th« value J/, the magnetization is twotbirds
of ita limiting value. When tlie magnetizing force im further iit
creawd, the niagnetiiiation, instead of increasing indefinitely, tends
tonarda a finite limit.
Fig. 7.
»
Tb« Uw of magnetiiation is expressed in Fig. 7, where the mag
netuiBg force is reckoned from towards the tight, and the mitg
iwtitation is expressed by the vertical ordinatt^s. Weber's own
experiments give results in aatiafact4>ry accordance with this law.
It is probable, however, that the value of J) ix not the same for
all the molecules of the same piece of iron, so that the transition
from the straight tine from to £ to the curve beyond E may not
be 00 abrupt as is here represented.
444.] TTkj theoiy in this form gives no account of the residual
tnagn«ttzation which is found to exist atiter the magnetizing force
is removed. I have therefore thought it desirable to examine the
results of making a further assumption relating to the conditions
under which the position of equilibrium of a molecule may b«
permaneutly altered.
Let UB Buppoeo that the axis of a magnetic molecule, if dcfleelcd
throogfa any angle less than ff^, will return to its original
position nhen the delleeting force is removed, but that if the
deflexion $ exceeds ^g, then, when the deflecting force is removed,
the axis will not retom to its original position, but will be per
tnaDently deflected through an angle /3— ^g, which may be called
the permanent «<■/ of the molecule.
This assumption with respect to the law of molecular deflexion
is tM>t to bi^ regarded as founded on any oxa<^'t knowledge of the
L intimate i<truclure of bodies, but is adopted, in our ignorance of
tlie trtic state of tb« case, as an assistance to the imagioatiott in
following out the «{>eculatiiia suggestvd by VVcbcr.
Let l^JDtiafla. (9)
80
WKBEirS TnEORT OF INDITCED KAGSBTiair.
[444
then, if the moment of tli« couple ftcUof; on a molocnlo is loses tb&n
tub, thcro will be no ponnaiii'nt dtflcxion, but if it excpwla "w/f ■
there will Im! tt p(;nri»nvHt change of thi> position of i!KuiHbrium.
To tnice the results of tbis eupposition, dctrcribc n Hpbore whose
centre ifi and ratlins OL = L.
As long as A' ia less than L everything will be the same ax
in the case already considered, but at! soon as Xoxcet^K L it will
bcf,'in (o proda«e a permanint detlfxion of some of the moleculefl.
Let ua take the case of Fig. 8, in whieh A' is greater than L
hut leiia than D. Through S as verti^x draw a double oone touching
the sphere L. Let this cone meet the Kpherc D'mP and Q. Then
if the axis of a molecule in it^ original [wsition lies belweca OA
and OP, or between OJi and OQ, it will be dellected through an
angle less than ^3^, and will not be permanently difloeted. But
the axis of the molecule lie* originally between OP cud OQ, then
a couple whose moment is great«r than £ will act uiwn it and
will deflect it into the position SP, and when tlie forw X ecwea
to act it will not resume its original dir«'etion, hut will be per
manently set in the direction OP,
Let xa put
L = Xsin tfj where 0„ = PSA or QSB,
then ail those molecules wbo«e axes, on the fonner hypoti
would have values of $ between 0^ *^ '^■'o will he made to have
the ralne $^ during the action of the force X.
During the action of tJio forco X, thererore, tbom nwloeal
whose axca when de6iected lie within either xheet of tlie doubU
cone whose semivertical angle is d« will be arranged as in th«
fonn«r ease, but idl thorn whose axes on tl>e fonn^tr thts^ry wotild^^
lie outside of the*e sheet* will be pennaneiiUy deflected, so tha4H
their axes will form a dense fringe round that sheet of Uie cone
whieh lies towards d.
aw
>h\JM
d
445]
MOniPIED THEORY.
81
I
Aa X increases, the mitulicr of molecules belonging; to tli« oone
about S continually iliminUbex, and when X becomes eqiiul to J)
all the molecales have been wrenched out of their former positions
of equilibrium, and have been forced into the fringe of the cone
Toiind A, BO that when X becomes greater than D all the molecules
form pari of the eono round A or of its fringe.
When thv force A' in rcmoTcd, then in the case in which X is
IcM titan L cvcrj'thing returns to On primitive etiito. When X
is between L and J) tlien there ts a cone round A whoKc au^e
and another cone round S whose angle
liOq = 0,p^.
Within theae conca the axes of the moleculce are distributed
uniformly. But all the molecules, the original direction of wbo»e
axes Hy ouUide of both th^j^e cones, have been wrenched from their
primitive positions and form a frinjre round the cone about A.
If X is greater tlian I), then the cone round B is completely
dispersed, and all the molecules which formed it are converted into
tiie fringe round A, and are inclined nt the angle ^gf ^y.
446.] Treating this ca«e in tiie same way as before*, we find
• (Tfcp nanlta gi*Bli in th« tott may ho ohXtAoci. with one lUglit raosption, by
tile ptoeeon givon b«l»w, ihti i<l*li>iii»iit nt ihn niuUlioil thaary of Art. 414 bdng M
fglloir* : Tha &iU of ■ nia^iMtic iiiu1k»1?, if ilvllix^liat Uiroiij^li au angle l*" (bui 8,
will rMuni 1u ita uHifinkl noiritioii whea the dpfloitinij fiirvii <i niiii<iT*il ; but wlxn
th* 6tB»*lMk nomli 0, tho TorHi tending to oppOH the dpfletinu givw my »n<l
ponnila tka malcculD lo be ddlcirUd Into tbB wmc diioctiun u lliuH irhuns HcfleKJun
u A., >ad whsn llw JitlfciJnit f<ir«a la r«mi>vc4 the ni'ilocula t.\ktm up > direciioa
panllcl to tluti of the moliKuti) wImm dvflraioQ wu >>•■ Tlila dimclion ouy bo
call*] tL> peniuuiunl nt of Om mukouloi.
la (Iw our X :■ L . D. lh« c^ptoHon / (or tbu m&gnelio momont conMnUnf tiro
put*, tliv tint ef which !■ dun lo ths noEMulo* witkin tha oonsa A OF, HVQ and U Ui
be found iiiwixly lu tu Act. 443, duo r^^aril h«iuit had to tho limiui of integraUon.
BaAimag to ¥ig. 8 v* find for th* mfouiI pari, aoconKn); to ths above itatsiiuint of
1 .en Projootioa of QP on W
fMmenASPn = r^^ .
nio t«o parta i^gotliM whoa taiuc«d give tbo malt in the teit.
Whui X > /*, tha tatcttral a[aiu oiiuiiittii nf two inuli. mic of whitih (• to bo takca
OTMtliacuna A Of a* in Art. 448. The tusoii'l puiia, l>^>((C)i
.ct, rroiofitfon of BP oa BA
{tnneaaJSPx — * jjk ■
OP
The raluv of J En Uua Ca*^ ati«n reddMd. dtffon from tho valne gimi la tbft (eit
In thv thud tens, vb, : w« bar* then ^ , ». iniluad '■f ~ « v ''^'" ""^^ '^ *'^
S JC^ V A
oba^go im tlis taU« nf miinrrfal Tallies glrtn tn tlio toxt will be that ohr'ii X  S.
7, If, iku tarrerpmlhijf iatu«« uf X will ba SS9, 917, DSO. TttMo ciiugoi do not
TOU U. O
82
WEBEBS TnBORT OP ISDOCED MAGNETISM.
[445
for the intensity of the tWDpomry magnetitttion during' the iiction
of tbe forai X. which is euppoM'd to u«t on iron wbich hm Derfr
beforv b^ f n magnetized,
2 X
When A' ia lees than L, I~ Mft
' 3 D
'A'h
When A' is etjnal to L,
Wh«n X it between h and /),
When X \i etiual to Z>,
Wiien X i» greot«t than 2>,
Wluin X is infmit«. / = Jf.
6J»Z*
When A' is less thsn L the magnetization follons the former
luw, and is proportional to tho magnetizing force. As »oou as X
uxLeeds L the magnetization assumes a more npid ntbe of increase
on account of the molecules beginning to be troniferred from tbe
one cone to the other. This rapid increaue, bonercr, soon cornea^
to all cntl ns the numl>er of molecules foiming the negative coBi^
diminishes, and at laut the magnetiiatioD reacbca tbo limiting
Taliie M.
If ne vera to aaaume that the valiieR of £ and t>t D are difTerenfl
for diOiorent moleeulea, w« should obtain a rasult id vhicb tbe
different stagn; of magnetization arc not m distinctly marked.
Tlie nndaal inatniettxation, /', produced by the magnetiziog force
X, and observed after the force has be«n removed, is as follows :
When X \a leas than £, Mo residnal magnetizatioa.
When X is between h and B,
r.J^(ig)(i^.
•Iter Iha gaiMnl fbnwiOT of llw can* «f TtMBpana; Vif— finHim givta
^i* m)>H o( r la Uii MM «f rt(. S b
•/;.
n* nJiM »f r h tin cMB uf IV > BUT h* laMi la tika I
'1
llf AlTD RBSTOFAL MAOKETIZ.ITIOK.
83
'Wbeo X it) cqutti to 2>,
"When X w greater than J>,
"When X is infinite,
li we make
J/ =1000, i = 3, /I = 5,
we Bud tlie following values of the temporary and tlic residual
magnetization : —
lUatfncitldng
Tamponuy
MHgiiiiiiutlon.
X
/
1'
1
J33
3
267
3
400
i
729
260
9
837
410
6
864
485
7
882
537
8
897
574
1000 810
These reeult* arc laid dunu in Fig. 10.
. J^^^^^^Si^taiBiaiAmrlMtmMaMtm
Kg. 10.
The eurve of tempoiary maj^aetizatioii is at 6nt a straiglit line
84
WEBRR's TnEORY OP INDCCED M.VOSRTISM.
[446
t, and™
from X=0 to X=:£, It then rises mom rapidly till X = J),
nud 1L8 X iocrcAGes it uiiprotictivH it« horizontal asymplotv.
The L'urvc of rocidual iiiii);netirjit.ion bpgins when X^L, and
approaches an asymptoti »i 11 iliiilanoc = .St Jlf.
It must bo rcm(;mlj<.Tcd that tlie n.'«idit»I miif^etutn tliuB foimtl
correspondB to tbc case in whicii, when lh<t fxtcmnl Toroc is rmioved,
there is no domagiictizirijr force arising from t^v diotnbiition (it
inagnetifim in tlie hody it^lf. Tlie calculationH are therefore ap
pltciibli only to vory eloD^atid bodiat inagiivtizvd longiUidiiiHlly.
lu thu cam of itiiort, thick bodies the residual mn^ctism will be
dimini«licd by the reaction of the free magnetism in tlie m&«
way as if an external reversed magnetizing foroc were made to
act upon it.
446.] The acieuUfle value of a theory of \\\i* kind, in which w<
make so nmny aKBiiinptionii, and introduce «o niuny o^uMtable
constants, cannot be t^tiniated merely by iIk numerical agreement
with certain sets of expmmonts. If it liax any ralue it ix because
it enables us to form a mental image of what talces place in a
piece of iron during magnetization. To te«it the theory, we shall
apply it to the c»»f in which a piece of iron, after being mibjectcd
to B magnetizing force A'q, is again subjected to a magnetizing
force A*,.
If tJiv Dew force X, acts in the same direction as that in which Xg
aoted, which we shall call the poeitire direction, then X,, if leaa
than X„, will produce no permanent set. of the molecules, and when
X^ is removed the rcsidtial ma^pictization will he the same as
that prodnced by X„. If A', is greater than A'^, then it will produoe
exactly the §amc efftvl as if .V„ had not acted.
But let U£ suppose X, to act in the motive direction, and let us
Xo = Zeosecilj, and X, = — ituwoetf,.
suppose
A9 A', increase numerically, i?, dimiuishiv. The Rrst molecniefi
OQ which A'l will produce a permanent deflexion aru tliose which
form the fringv of the oo»o round A, and these have an iuclinatioB
when undcflected of 6^ 4 ^o
As soon aa ff,— ^^ becomes hws than ffo + ft ^^^ process of de
magnetization will oommence. Since, at this instant, 0t m 9„{ 2^,
.T,, the force required to Wtjin the demagncl ization, is )et» tJum
K^, the force which pri>ductd thi majjnoliitalion.
If the value of J) and of L were lie same for all the inolecmie
the Bliglieat incre.i« of X, wouM wr(neh the whole of the frin^
of muleculis whose axea have the inclination i^+^o inln a poai.
;4
446]
UAONRTISU ASD TORSION*.
I
*
I
ft
tioD in wlLi<;fa fheir axes are inclined ^i + ^ to the native
axis OB.
Tlioujjh the doronfrnettzatioD does not late place in a uinnner so
Kiul(]«ii ii« t)ii)i, \l fjikcj^ place m rapidly as to aiTord some conlirinu
tioQ oflbin modo ofcxplitinm^f the process,
Let U8 naw ««[ii)oec tlmt t>v ^iviiijif ii proper value to the reverse
tone Xi we havt; exiiotly dtmagnctizcd the piece of iron.
Th« axoK of the molvculcs will not now he arranged indilFer
enllv in ull diTvctioD!<> a« in a piece of iron which has never been
magnetiKed, hut will form three jfroiips.
(1) Within a cone of scminnglu 9i—0o surronndins the positive
pole, th« uses of the molcciito; remain in their primitive positions.
(2) Tlio tuime is the ease within a cone of semian£;lc &^—Pq
sorroundin}; the negatix'e pole.
(3) The directions of the axes of ull the other molecules form
■ conical sheet Eurrounding the ncg^ative iKile, and are at an
inclination O^ + fi^.
When .1^ is {>rcat«'r than J) the secoud group is absent. ^Vben
X is f>roat«r tlian D the first group is also absent.
Tlie stxlc of the iron, therefore, though apparently demognctizml,
is in a different stut« from that of a pieoc of iron nliich lias never
bevn Di»{rn(ti£ed.
To shew this, let lis consider the effect of a magnetising force
X.^ acting in cither the ponitivc or the negative direction. The
fir«t pcmiBiient ifft^ct of such a force n'ill hu on the third gronp
of molecules, whose axes make angles = O^h^a with tlie negative
axis.
If tlie force X, nets in the negative direction it wilt begin to
produce a piTniancnt effect a« soon as (J^ + A) l>econ)es Icsh than
tf]fA). Uxt 18, as Koon a* .V^ becomes greater than X,. But if
X, acts in the positive direction it will begin to remagnetize llie
iron as soon as 0^— ;3 bi^eoraes less than 6^ + ^,,, that i^ when
0, K flffZiS,, or while X.^ is still much less than X^.
It iippcarn therefore from our bypothertis that—
When a piece of iron is magneliiied by means of a force Xg, its
ningnetiBm cannot be increased without the application of a force
greater than X^. A rtverBe force, less than X^, is snfBcient to
diminish ita magnetixation.
If tbo iron is exactly dema^etizcd by the revereed force X, , tin n
it cannot be mn}fncttiKd in the reversed direction without the
application oi' a force greater than .V , but a positive force less than
^tt
86
WEBEBS THEORY OP INDUCED MAfiSETISU.
X, is sufficient to hegin to rcmagneUze lite iron in ite ongioAl
direction.
Thccc rotiiiltK arc onneiiHtiHit with what hu been a«laa11y obMrve
hy Riteliie*, Jncobif. Marianinit, and JooleJ.
A very complete account of the relations of the magnet izatios
of iron and stcrl to inagnctto forces and to mechanical strains ii
givcn by Wicilcinarin in liis Galvanitnus. lly a detailed com
pariKon of the effects of ma^etization with those of torsion, bfl
shews that the ideas of elasticity and plasticity which we derivsl
from experiments on the temporary and permanent torsion of wireaj
can he applied nnth equal propriety' to the temporary and permanent
magnetization of iron and steel.
W?.] Matteuecill found that the extension of a hard iron hax
during the action of the magnetizing force increases it« temporary
magnetism. This has been confirmed by Wertheim. In the ease
of soft hare the magnetism is diminished by extension.
The permanent magnetism of a bar increases when it is extended,^
and diminishes when it is compressed.
Hence, if a piece of iron is first magnetized in ono direction,'
and then extended in another direction, the direction of magnet ,
izatioD will tend to approach the direction of extension. If it h4^
compressed, the direction of magnutizatJoa will tcrnd to bccomc^^
normal to the direction of comprvsstun.
This explains the tmiiU of aa experiment of Wiedemann's,
enirent was i>a»f«d dowimani through a vertical wire. If, eithu
during tlic )ait»agc of tlio current or after it has ceased, the wir
bo twistod in the direction of a righthanded screw, the lower eai
becomes n north poU).
sa
I
F^.II.
Fig. 12.
• pail. .Wo.?.. IS83.
% Ami. Or VUmie H 4i Ptf*rf«, lUS.
f pig,. akh., isai.
CTTASOE OF POBM,
87
I
^
^
Here the aownirard current majrnetizes every part of the wire
in a tan^ntial direction, as indicated by the lettcra .V.S'.
llic twisting of the wire in the directioo of a righthanded screiv
the portion ASCH to be extended alonij the diagonal AC
and coinpTcswd ulong tho diaj^onat BD. The direction of magnet
iuition thvrcforc tends to iippruach AC and to recede from BD,
Rod thus tJic lower end becomes u north pole and the upper eail
a vouUi pole.
Effect ^ ilagnHicaiion on tie SHineniion* of the Magnvi.
448.] Joule*, in 1812, found that an iron Imr hccomes length*
ened when it is rendeied magnetic by an electric current in a
coil which stirrouuds it. He al'terwardsf shewed, by pladng tiie
b*i in water within a glaan tube, that the volume of the iron is
not augmented by th» magnetization, and concluded that it«
tisasverse dimensions were contracted.
Finally, he paseed an electric current through the axis of an iron
tnbe, and back outside the tube, so as to make the tube into a
closed magnetic solenoid, the mugnetization being nt right angloi
U> the axis of the tubu. Tho length of the axis of the tube was
found in this auc to be xhortoned.
lie found that an iron rod under longitudinal pressure is also
elongstrd when it is magnetized. Wlien, liowever, the rod is
under considirable longitudinal tension, the effect of niagnetizntion
in to rfiorlen it.
TliiM was the case with a wire of a quarter of an inch diameter
irben the tension exceeded 600 poundj* weight.
In the case of a hard steel wire the effect of the magnetizing
force was in every case to shorten the wire, whether the wire was
tinder tension or pressure. The change of length lasted only as
long as tho magnetizing force was in action, no Hlti^rution of length
was obecrved due to the {)crmancnt magnetization of the steel.
Jonle found the vlong:itioQ of iron wireu to be nearly prciportional
to the (quare of the uotual mngnetixut ion, ko that the tiriit etfect
of a demagnetizing current wbk to Hluirten the wire.
On ihe otiicr hand, he found that the ahortening effect on wirea
ondcT tension, and on stoet, varied as the product of the magoot
iiation an<l tlie magnetizing current.
Wiedeinann found that if a vertical wire ia magnetized with its
" Siiuwoi'i ^niHil/ o/ Bi«lH<Uv. vol, riii. n. S19.
^^_ t fAiC Jf(V.. \Ut.
88 webgb's teeobt of induced maonetism. [448.
Boutb end uppermost, and if a carrent is then passed downwards
through the wire, the lower end of the wire, if free, twists in the
direction of the hands of a watch as seen from above, or, in other
words, the wire becomeB twisted like a righthanded screw if the
relation between the longitudinal current and the magnetizing
current is righthanded.
In this case the magnetization due to the action of the current
on the previously existing magnetization is in the direction of
a righthanded screw round the wire. Hence the twisting would
indicate that when the iron is magnetized it expands in the
direction of magnetization and contracts in directions at right
angles to the magnetization. This agrees with Joule's results.
For further developments of the theory of magnetization, see
Arte. S32845.
CHAPTER VII.
UAQKETIC UEASURGUeSTS.
»
I
I
44D.] TuK principal magnetic mcaeunemoDts arc the deterniina
tioD or Uie nutgitctic axi« mid maf^netic moment or a inagnet, and
that of tJie direction ftnd intinsity of tbo magovtic furco at a given
pbce.
Since theae meaaurcmente are made ncnr the Eiirfacu of Iho earth,
flie magneti are alwajs acted on by gravity as well us by terrestrial
mapnetisni, and iiiiiue the mugtiets are made of iAimiX tlieir mng
netisni ia partly pGrmanont and partly induced. Tlie permanent
maf^etism is altenxl t>y cbaiiifes of t'em[iL>niturc, hy at.rong in
duction, and by violent blows ; tlic tniliin^d nia^iietisin varies with
every variation of the external magnetic force.
The mottt convciiient way of (>bscr\'ing the force acting on a
magnet is l>y making the magnet free to turn about a vertical
axis. In ordinary compaMcs this is done hy balancing the magnet
on a vertical pivot. Tlie finer the point of the pivot the smaller
is the moment of the friction which interferes with the action of
tlie magnetic force. For more refined observations the magnet
is soapended by a thread composed of a silk fibre without twist,
either single, or doubled on itself ii xullicient number of times, and
eo formed into a thread of parallel lilires, each of which supports
as nearly as possible an equal part of the weight. The force of
torsion of mich a thread is much leas than that of a metal wire
of equal strength, and it may be calculated in terms of the ob
served azimuth of the magnet, which is not the case with the force
arising from the friction of a pivot.
The suspension 6hre can be raised or lowered by turning a
horizontal screw which works in a fixed nut. The (ibro is wound
round the thread of the screw, so that when the screw is turned
the suspcosion fibre always hangs in the same vortical line.
90
MAGNETIC MBASUREMENTa.
[45<
r thfl
on of^
fl
;bt
est
«s
ireo
The snspensioii fibre cnrrite n small horizontal dirided iiircle
called tbe Toreioncirclf, «nd » slirimp with an index, whtoh eaa
he plaoi'd so that tlic iiidox cuincuke with any ^ven division of
Uio toreion circle. The stirrup is so shaped that the nia^^t Ijsr
can bo tittiM) into it with it* uxie horizontal, and with anj one
of its four sides uppermost.
To osLertuin tlic zero of torsion » nonmai^etic body of
snme w^iffht as tie magnet ia pL
in the stirrup, and the poi.ition
the tnrvion circle when in e<}uilihriui
asciTtaiucd.
Till" miig^net itself is a piece
har(l.t4;miicrtil steel. According
(lauss and Weber its length ought
to be at Iwist eight times its greatest
transverse dimension. This is neees
mry whin permanence of the direo
tion of the magnetic axis within
magnet is the most important eoa
sideriition. Where promptness of i
tion is required the magnet ahonld
be shorter, and it may even be ad
viauble in o1:)E«rviiig sndden altera
tiona in magiMttic force to ase a bar
magnet ize<l transversely and sus
pended Htth its longest dimeUfcioQ
Tertical *. ^M
450.] The magnet i« provided wit^^
an arrangement for ascertaining ita
angular position. For ordinary pur
pn^i^s its ends are pointed, and a
^' *' divided circle is placed below the
ende, by which tlieir positions arc rend oiT by an eye placed in «
plane through the nutipension thread ami the point of the needle.
For more oceiimtc observations o plane mirror is fixed to the
magnet, so tltal the normal to the mirror coincides as nearly as
possible with the axis of magnctitation. This is the method
adopted by Gsnsa and Weber.
Another method is to attach to one end of the miignet a l«ns and
to the otlier end a scale engraved on glaM, the diHtunce of tbe lens
5m, JrimriMttr, Vor.SS. ISU.
450]
TUB MIRROR JrETOOD.
91
»
from tie scale being equal to the principal focal length of the lens.
The straight line joinings the zero of lh<? eealo with th« optical
centre of the leas ought to coincide as nearly ti* possible with
the msffiHitic axis.
As these 0]>tical methods of asoeriainin^ the angular position
of mspended apparatus are of great importance in many physical
researches, wc shall here consider once for all their mathematical
ticoiy.
Theory of lie Mirror Metkoi,
VTe shall euppoee that the apparatus whose aagul^ position is
to bo detormincd is capable of revolving about a vertical axis.
Tilts Kxis is in general n tibre or wire by which it is saspendid.
Tltc mirror should be truly plane, so that a scale of milliractirs
may Ix «e<tn distinctly by reflexion at a distance of several metres
from the mirror.
Tlic normal through the middle of the mirror should pass through
tbt axis of sofipension, and should be accurately horiKontal. Wc
shall refer to this normal as the line of collimation of the ap
paratus,
Having rotighly ascertained the mean direction of th« line of
collimation during t)ie experiments which are to be made, a tele
scope is erected at a convenient distance in front of the mirror, and
a little above the level of the mirror.
The t«letM»j>c is capable of motion in a vertical plane, it is
direete<l towards tlic suitjiensitm fibre just above the mirror, and
a fixed mark is erected in the line of vitiiDn, at a horizontal distance
from Uio object gli>s9 equal to twice the dintance of the mirror
from the object glass. The apjiaratus iihould, if possible, be so
arranged lliat this mark is on a wall or other fixed object. In
order to ^eu the mark and the t>uspeni<ion fibre at the same time
tlirongh the telewope, a cap may be placed over the object glass
having a slit along a vertical diameter. This should be removed
for the olh^r oboervatjons. The telescope is tlien adjusted so that
the mark iw seen distinctly to coincide witJi the vertical wire at the
focuH of the telcHcope. A plumbline is then adjusted so as to
pass close in front of the optical centre of the object glass and
to hang below the teleiwopc. Below the teIe»cope and just behind
the plumbline a scale of equal parts is placed bo as to be bisvct«d
st right angles by the plane through the mark, the suspension fibre,
and the plumbline. The sum of the heights of tlic scale and the
93
MAGNETIC MEASURKMESTa
[450
objed; glass should bo equal to twice the height of the mirror from
Uio Q<ior. The telescope being now directed towurds tho mirroFj
will see in it the reflexion of the ecale. If the pmrt of the «cal
where the plumbline crossee it appears to coin<;ide with the verliaii
vire< nf the teleecope, thea th« line of collimation of the tnin
coincides with tiie plane throiipfh tho mark nnd tho optical centre
of the object ^lass. If the verticul wire coioeides wilil anv other
division of the xenle, the ungtiUr po>ition of the line of oolliination
it! to be found an followa :—
Iji't the plittie of the paper be horizontal, and tet the various
points be projected on this plane. Let be tJie centre of th<
object glaa^ of the telescope, P the fixed mark : P and the vertical
wire of the telescope are conjugate foci with re»j>eot to tlie object
glass. Let M be the point where 01" cuts the plane of the mirror.
Let M.\' be the normal to the mirror ; then OMS'=. is the ang\
which the line of collimation makes with the fixed plane. Let J/<
be a line in the plane of OM and UN, euch that MfS = OJCVf
then S will be the part of the scale which will be seen by reflexion
to coincide with tho vertical wire of the telescope. JS'ow, since
r.
I
— ¥^ ^
jl/.V IS horizontal, tho projeefed angles 03fy and .A'JI/i$ in
figure are equal, and OJUS=20. Hence 0S= OMiaQ20.
We hare therefore to measure OH in terms of the divieionfi of
the scale ; then, if i^ is the division of the scale whidi coincides with
the phmibline, and » the obMrred division,
«»,,= OMtm29,
whence 9 may be fonod. In measuring 03f we must remember
that if tho mirror is of gliiBS, silvered at the bdclc, the viKgal in:
of tho reflecting eur£Ke is at a distance behind Uic front sarfni
4
URTH0D9 OP OBSERVATION.
*
of the g^ass =  , vhere t ia the tJticksess of the glass, and n is
tb« index of n>rmction.
We must also rcmcmlwr tlint if the Hue of suepension does not
pa^ tltrougli the point of Mlt^iioii, the position of 31 will alter
with &. Kciice, when it is pufutihle, it i» adviwildc to make the
centre of the mirror votncide with the line of itiuiM>n!iii>n.
It it alao itilvisahle, espeoiall}' wlitm larf>;fi angular motions haT<!
to be obiterved, to make the scale in the form of n concave ojliodrio
Rtufair, whoee axis is the line of suspension. The angles are then
ob«erved at once is circular measure without reference to a table
of tang«nl». The scale should he carefully adjusted, an that the
axis of tiie cylinder coincides with the suspension fibre. The
Dumbeni on the scale should always run from the one end to the
other in the same direction so as to avoid negative readings. Fig. 1 G
Fig. 16.
represents the middle portion of a scale to he luied with a mirror
and an inverting teleeooije.
This method of observation is the best when the motions are
slow. The observer sits at tlie telescope and sees the image of
the scale moving to right or to left past the vertical wire of the
telescope. With a clock beside him he can note the instant at
which a given .division of the scale passes the wire, or the division
of the scale which is passing at a given tick of the clock, and he
oan also record the extreme limits of each oscillation.
When the motion is more rapid it becomes impossible to read
the divisions of the scale except at the instants of rest at the
extremitjea of an oscillation. A conspicuous mark may be placed
at a known division of the scale, and the instant of transit of this
! mark maj be noted.
When the apparatus is very light, and the forces variable, the
.motion is so prompt and swift that observation through a telescope
MAGSOTIC MBASmiMIEirfS.
[45 q
would be useless. Id this cum tlio observer looks at tlie scald
(linclly, And obwervva the motiuus of thu iinag« of tlie vortical win
throwD on the scale by n Ump.
Tt is munifest thut iiincc thi> im&g^e of the ecale reflected by th<
mirror anil refratU'il by the object glass coiocidee with the ver
wire, the imaifc of thu vertionl wire, if suf&oientfy illuminated, will''
coincide with the «cnlo. To observe this the room ts darkened, and
the ooDCGQtrahd rays of a lamp are thrown on the vertieal wire
towards the object glass. A brii^ht patch of li^ht crossed by the
shadow of the wire is seen on the scale. Its mottoos can bo
followed by the eye, and the division of the scale at which it come
to rest can be Bxcd on by the eye and read off at leisure. If it he'
desired to note the instant of the passage of the bright spot past a
given point on the scute, a pin or ■ bright metal wire tnay
placed there so ns to Ihwh out at the time of posaage.
By substituting a small hole in a dia]>hragiii for the cross wire
the image becomes a small illuniiuutcd dot moving to right or loft
on the scale, and by btuhxtitiiting for the scale a cylinder revolving
by clock work about a horizontal axis and covered with photo^
graphic pajwr, the spot of light tracK« out n curve which can
afterwards rendered vinble. Each abscissa of this curve corre«poi
to a particular time, and the ordinate indicates the an^
jKisiiion of the mirror at that time. lu this way an automatio
system of contiimouH registration of all the elements of terrestrial
magnetism hax been established at Kew and other observatories.
In tome «we* the teIe»cope in dixpt^nsed with, a vortical wire
18 illuminated by a lamp placed behind it, and th« mirror is a
concave one, which forms the imago of tlio wire on the scale a4M
a dark lino across a patch of light, ^H
461.] In the Kcw portable api«nitiis, the magnet is made in
the form of a tube, having at one end a leas, and at the other
a glass ecale, so adjusted as to be at the principal focus of the leu
Light is admitted from behind the scale, and after passing through
the lens it is Tiewed by means of » telesctipc.
Since the scale is at the principal focus of the lens, rays Ironi
any division of the scale emeigo from the lens parallel, and if
the t«!e«e4)i>e is adjusted for celestial objects, it will shew the scale
in optical roincidenco with the crow wire* of tlie telescope. If ;
given division of the ecale coincides with the intersectjon of tl
orosa wires, then the line joininer that division with the optic
centre of the lens must be parallel to the line of oollimation oC.
be I
he^
it a^
rire
loft
photo^y
!«nb«fl
>ponda^
nguhu^B
'41
ical^
■ 45271 DIBBCTIOS OP MAOKBTrc TOBOB. »»
the tetascope. By fixing tht> magnet and moving the t«1oRCoptf, we
may ascertaiti tbe angular value of tlie tlivisioiw of tin kchIc, und
(hen, when (he ni3<;net is suspended and the poaition «!' the tt'lu
Kope known, wv may determine the poeitioa of the mugnot at
ony inetant by reading off the division of the scale which coiuuidee
with the CTOfS wires.
Tbe telescope ia aupporbed on nn ana which ia centred in th«
line of tho suspension fibre, and the position of the tele!ico[>c is
read off by vornicrs on tho azimuth circle of the iuatrament.
This arrangement is snttabk for a small portable magMctomelcr
in which tho whole apparatus is supported on one tripod, and in
which the oedlUtioos due to accidental disturbaQcen rapidly
subside.
IDeUrmiHotion of tie Direction of the Axis of the Magnet, and of
tie Direction of Terretlriat Magitelitm.
4fi2.] Let a syntein of axes be drawn in a magnet^ of which tiie
axiK of : is in the direction of the lengtli of the bar, and x and y
perpendicubir ti> the itideH of the liar supposed a paratli'topipod.
Let /, in, << and A, fj, v be the angles which the magnetic axis
and the line of eollimution make with these axes respectively.
Lot J/ he the magnetic moment of the magnet, let IT be the
horizontal component of terrestrial magnetism, let Z be the vertical
oompouent, and let fi he the azimuth in which // actM, rockoncd
from the north townnJs the west.
u Let f be the obscrvni axinmlh of the line of collimation, let
^k be tho azimuth of the stirnip, and ^ the reading of the index
"of the torsion circle, then a—(i is the azimuth of the lower end
»of the suepenaon fibre.
Let y he the value of «— ^ when there is no torsion, then the
moment of the force of torsion tending to diminisli a will be
where r ig a cocfiidcnt of torsion depending on the nature of the
fibre.
To determine \, fix the stimip m that y is vertical and up
wards, a to tho north und r to the weat, and observe the aKimuth
f of the lino of collimation. Then nmove the magnet, turn it
through an angle v about the axis of t and Teplac« it in this
inverted portion, and observe (lie azimuth f of the line of coU
^^imatiou when g la downwards and x to the east,
MAOSETIO MEASCHEMEST9.
Hence
Next, bftiig the stirrup to the suspension fibre, and place
magnet in it, adjusting it carefully so that jt may be vertical an'
upwards, tlieu the moment of the force tending to increase a is
MHsia m sin (5— a—  + ;)— r(«j8— y). (4JJ
But if C is the observed azimuth of the line of coUimatioa
f=a+A.
C^J
BO that the force may be written
J///«ni«sin{«f+iA)r(C+X^j9y). (i
When the apparatus ia in equilibrium this quantity is xero for
a particular value of f. ^H
MTicn the apparatus never eomes to rert, but murt be observeJ^^
in a i>t:itc of vibration, the viiluo of ( corresponding to the positios
of equilibrium may be calculated by a method which will
described in Art. 735.
When the force of torsion is small compared with tJie moment
of the magnetic force, we may put i—(+l—K for the cine of that
angle. m
If we give to ^, the reading of the torsion circle, two diflcren^^
valuee, ^ and ^,, and if d and Ci oro the corresponding values of C
ioa^^
en^B
or, if we put
J/// sin « (CC) = r (C,C.ft +^).
(iU
— — = /, then T = Mllma mV,
and equation (6) becomec, dividing by J/ZTsin m.
If we now reTcme the magnet so tlint y is downwarfa,
Okdjust the appatKUiK till f is exactly vertieal, and if C is the ne^,
valoeof Uic aEiraulh, and ff the corresponding declination, ^M
yf '_/+At'(rX + /8y) = 0.
whence
i*g
= i(f+n + *''tC+r308+y)).
(10)
^
OBSKBVATrOS Of DEFLEXION.
i»7
I
I
Th« reading of tlie torsJon circle should now be iidjuHtoil, ao that
Uie coefficient of r' may l>e an nearly aa poa»ible aero. For ihie
parposG we must clet.i.'rmiDe y, the value of a—fl when there ia no
torsion. This may be (lone by placing a uoq magnetic bar of the
Eame wetfjht ae the magnet in the stirrup, and deteriuining: a— ^
when thcr« is ctiuilibHum. Since t is small, great accuracy is not
requirnl. Another method is to use a torsion bar of the same
wei);ht us the mmgaai, containing within it a very small magnet
whcee n)af>nctic moment is  of that of tho principal ma^et.
Since r remains tlu: itame, r' will become nr, and if ^j and (,' arc
tJie value* of <"«« found by the torsion bur,
»= i(C. + Q+i»T'(C, + C,'2(5+y)). (12)
Subtracting this equation from (II),
2{— l)(/3 + y)=(« + ^)(f. + fn(l+}0(^+O. (13)
Having' found the value of j3y in this way, ^, the reading of
the torsion circle, should be altered till
C+r2O+y) = 0, (14)
as nearly as possible in the ordinary position of the apparatus.
Then, sinoc t is a very small numerical quantity, and since its
coeffleient is very small, the value of the second terra in the ex
pnasioa for 2 will not vary much for small errors in the values
of f and y, which arc the iiuautitiea whose vaJues are lea^t ac
curately known.
The value of h, the magnetic declination, may be found in this
way trith conaiderablo accuracy, provided it remains constant during
the cx>crimcntB, ko that ne may a^ume E'= E.
When gnat accuracy is required it is necessary to take account
of the variations of S during the experiment. For this purjHiBC
ebaervations of another suspended magnet should be ma<le at th«
■una instants that the diflerent values of ( are observed, and it
If, ij' ate the observed azimuths of the second magnet corresponding
to {and (", and if 2 and 1/ are the corresponding values of &, tlieu
Ueooc. to find the value of ft tve must add to (1 1) a correction
The declination at the time of the first obBvrvation ii therefore
« = i(c+c+it')+^'(c+r2^M
(16)
VOL. U.
98
MAOSETIC MEASmtBSrESTS,
[453.
To fiud the direction of tiie magnetic uxis nitliiD the miignct
fubtraet (10) from (9) and add (15), i
/ = A+HCni{7l')^4''(Cr+2X»). (17)1
By repeating thu tfxpcnraents nitb tUe bar on iU two «dgea, bo
that tlie uxix of ■s in Tcrticallj' upwards and dnwntrards, we can
find tlie value of m. If Ike axis of collicnation is capable of ad
jnstmvnt it ought t<i be made to coincide with the niB^etiti axis
IIS nearly as po^naiblc, so that the error arising from the magnet not
being exactly inverted may be as small aa poiwible*.
On the Measurement of Ma^etic Foreei.
453,] The most important mcasurementn of magnetic force are^
thoae which dttermine M, the magnetic moment of a magnet,!
and //, the intensity of the horizontal component of tcRCStnalj
magnetism. This is generally done by combining the resnlts of j
tno ex))eriments, one of which determines the ratio and the oth«r{
the product of these two quantiticB.
Tha intensity of the magnetic force due to an infinitely smull
magnet whose magiictio moment ie ^f, at a point di^tAut r from
the centre of tlie magnet in the positive direction of tlie uxis of
Uie magnet, ia
=?
(Ill
and is in the direction of r. If the magnet is of finite siz« bub 
spherical, and mngnetized aniformly in the direelion of itN axiiiij
this value of the force will still bo exact. If the magnet ia m
BolcDoidal bar magnet of length 2L,
If the magnet be of any kind, provided its dirocniuonfi are alll
small, ctnnpared with r.
It = 2~(l+J,l+J^^ + &c.),
(3)
where A^, A^, &c. are eoeHiHcnts depending on the di»lributioa of
the magnetization of the iMr.
Let If l>o the intenftity of the horizontal part of tArmtrial
mtignettim at any place, il it directed towards magnetic north.
Let r l>e meastired towards m^nclic west, then the magnilic force
at. the extremity of r will be // towards the north and R t')warda
I
• Sm ■ rapor oD ' ImjicrfsM IttTCndnii,' b* W. S»»».
rA. Mi (1«6[^), p. }li>.
TrttM*. a. S. Bilm
I
4531
DEFLEXION 0B6KRVATI0NS.
m
tbe west. The resultant force will make an angle with tbe
magnetic meritliaD, mcmured towards the weet, and such that
B = lltao0. (i)
n
Hence, to cIet«rmino jj we proceed as follows : —
The dir««tion of the mugnetic north having hven asecrUtned, a
magnet, whoKC dimensionit nhould not be too givatj \a, anspendud
aa in the former experimentit, and the dellecting magnet M ia
placed 80 tliut \is centre is ut a disttuico r from that of the gus
peuded magnet, in the same horizontal plane, and due magnetic
The axis of M is earcfiillv R<)juste<l so as to be horizontal and
in the direction off,
Tbe 3u»[>inili'd magnet \* observed before M is brought near
and also sAer it is placed in position. If is tlie observed deilcxioo,
we have, if we use the approximate forniuin (1),
M T
H=Y^°^'
or, if we use the fornnU (3),
l+Ji~+J^^+&c.
(6)
(«)
I
Here we muHt bciir in mind that though the dcHexifio can
be ohwrved with great acciirncy. the distance r between the centres
of the magnetH is a quantity which cannot be precisely deter
mined, unlv«a both magnets are fixed and their centres dellned
by murks.
Thi* difficulty in overcome thus :
Tlie magnet .1/ Im plucird on a divided BMie which extends eOBt
and west on both sides of the Eut>[>eDded magnet. Tlie middle
point between the ends of M is re<'koned trlio centTo of the magnet.
Thin point may be marked on the magnet and its jioifition observed
on the scale, or the ponitiunM of the ends may lie obaierved and
Ihe aritlimetic mean taken. Call thiit <>,, and let the line of the
suapemiioD fibre of tJic suspended tuugnet when produced cut the
scale at fg, then ri=*j— *„, where «, is known acouiately and a^ ap
pr^xiniately. Let 0, be tlie deflexion observed in this position of M.
Now reverse M, tliat U, place it on the scale with its ends
rrversed, then f) will be the same, but M and vf,, A^, &o. will
have their signs changed, so tliat if dj u tJte dcllexion.
I ff
1
!
  T. r,Manl»„= i_J, _+4 — _&
2 J/
('>
B 1
100
HAGKCTIC UEASUBBUfiXTS.
[45^
'liking the aritlimi.'tica! mean of (6) and (T),
1 U
1
(0)1
Now lemove M to the vest sido of tb« Euepoodwl ma^Dot,
place it with its centre at tlie point marked 2«o— « on the scaler
Let the deflexion when the axis is in tho first poeition be (?,, and
when it is in the eecond 0^, then, as before,
Let us suppose that the true position of the centre of the ei
peoded magnet is not «„ but Ja+<''> t'"^
fi = r— ff, rjsr+o, (10)
and since ^ may he neglected if the measnremeats are car
made, we are sure Hint we may take the arittimetical me*n of f)'
and r.j" for r'.
Hence, taking the arithmetical mean of (8) and (9),
1 /A I
 j^f*(tantfi— tan^j + tan^,— tantf^) = l + ^i+&c, (12)
ot, making
j (tan(Ji~tan^+tAn^jtaD0J = D, (U\
464.] We may now regard J) and r as capable of exact detei
minatioD.
The quantity ^ can in no case exceed 2X*, where L is half th«
length of the ma^fuet, so that when r is considerable compared
with Z we may ncglo«t the term in J^ and determine the nUio
of // to M nt once. Wa cannot^ however, aasame that A^ is oqaftl
to 2L*, for it may be \tm, and may even b« negative for a mngn«t
whose largest dimensions arc trnnsTerae to the axis. Hie
in vJ,, and all higher terms, may t«fely be ncglectod.
To eliminate A.^, repeat tlw exprriment, using distances r^, fjT^
ftc, and let the v^ues of Ji
m
2M , 1
'ii "»• ^i
A
&c.
&o.
DEFLEXION OBSERVATIONS.
101
If we suppose tli»t. the probable errors of these equatiofui are
e4]UAl, iiM th«y will be if thcj dopen<l on the determination of J)
only, nod if tlierc is no unccrtuinty tiboiit r, then, by muHipIyiog
each eqnatioD by r~^ aod ndtlJit^ the results, we obtain one e<]uation,
and by multiplying^ eiu.h cquntion by r''' and adding we obtain
another, accordinj; to the general rule in the theorj of the com
bination of fallible m(.'H«urc8 when the probable error of each
equatioD is supposf^ the same.
Let UN write
2(01') for /),r,» + i>,r.» + Z)jr.» + 8tc.,
and uac similar oxpresxions for the sums of othi'r groups of symbols,
Ifaea the two resultant e<iuations may be written
2 M
I
»
and J,{S(7>r»)£(r~'<')£(2>/'')S(r')}
= 2 (ft") 2 (r«)S (Dr^) 2 (r*).
The value of A, derived from these equations ought to be less
than half the square of tlie length of the magnet M. If it is not
we may suspect some error in the observations. This method of
obaer\'ation and reduction was given by Gauss in the ' First RcjKirt
of the Magnetic Association.'
When tbo observer can make only two series of experiments at
2M
distances r, and r,, the values of  . and A^ derived from these
experiments are
r,"— r.
If iDy and hD^ are the actual errors of the observed deflexions
Z>i and b^, the actiuil error of the calculatetl result Q will be
r * r *
If we snppoeo the errors ?7>, and ft/)j to be independent, and
that the probable raluo of either it tl), then the probable value
of the error in tho calculated value of Q will be S Q, wher«
 1" A. r "•
MIOSETTC MEASntRMBSra.
If wo sapposc tbat one of thceo dUlnnccw, tay tli« smalW, ii
<;ivin, the vnlue of the greater <Ii<itiiiioV may be determined to M
t(> make bQ a minimum. TbJs couiiitJoa leads to an crjuutioii of
the fiHh do^rcM) in r,", nhich hu only one nsal root greaiat than
fj*. From this the Lwgt value of'/, Ik Tound t« be /■, = 1.318flr,'
If ODe obsen'ation only is taken Ibe IxMt distance ift wlien
where i/> is the probable error of a measurement of defiexion, and
br IK the probable error of a measurement of distance.
MelimI of Sina.
455.] Tlie method whieh we have juet deeoribed may be called
the Method of Taoffentt!, bt'cutiec the tangent of the deflexion is
a measure of the magnetic force.
If the line r, . iustend of liein^^ measured cant or west, ih adjujrted
till it is at right niiglefl with ihu uxih of the deflected ma^et,
then S is the Kume as bi^fore, but in order that the suspended
magnet may remain pcqiendteular to r, the revolved (lart of th«
force If in the direction of r miiHt be niual nod opposite to S,
Ilenee, if is the deflexion, H = II sin 6.
This method in ealle<l the Method of Sine*. It can be applied,
only when /f is lf» than //.
In the Ken jiortable a]>paratit8 thiit method ia employed. The
eutipeuded magnet hang§ from a part of the apparatus which
revolves along with the telescope and the arm for the defle^ting
magnet, and the rotation of the whole is measured on tlio azimuth
circle.
The apparatus is first adjusted so that tlie axis of the telescope
coincides with the mvaii position of the Hov of coUimation of the
magnet in iU undiirturbed nlul«. If the magnet Is vibrating, the
true azimutJi of magnetic nortlt \» fotmd by obecrring the ex
tremities of the oscillation of tlie transparent acole and nuiking tb«
proper correction of the reading of the azimuth cirelo.
The deflecting magnet is then placed u>on a Mtraigbt rod which
posMS through the axis of tlio revolving apparatus at right angle«:
to the oxia of the telescope, and is ndjuFled 6o that the axis of the
deflti'ting magnet is in a line poHHing tlirougb the eent(» of the
snspi'iidtd magnet.
The whole of tlie revolving apparatus is then moTod till the Uu^
• See Mtyt Mayn^Unt.
I
d
456.]
TIME OP VIBRATIOS.
103
I
I
I
of ooUimAtion of the saspondwl mag^net afrain coincides with the
axis of the telescope, and th« new azimnth rtadinf is contctwl,
if neeesaary, l>y the mean of the ecalo readings at the extromitic«
c^an osnllation.
The difTereRce of the corrected azimuths gives the deflexion, alUr
which we proceed as in the method of tangents, «xocpt tluit in the
expression for 1) ne put sin Q instead of tan 9.
Id thia method there is no correctioo for the torsion of the eue
pending fibre, since the reUtive position of tho fibre, telescope,
a&d majcnet is the same at every observation.
Tlie axea of Uie two ma^ets remain alnays at right nn{;1ca
in this method, so that the correction for length can he mora
acoaratcly made.
456,] Having thus measured the ratio of the moment of the
deflecting m:^rnet to the horizontal component of terrestrial ma^
netixm, we have nest to find the product of these qiiantilies, by
determining the moment of the couple with which terrestrial mng
netism tends to turn Uie same magnet when its axis is deflected
from the magnetic meridian.
There are two methods of making this measurement, the dy
samicalj in which tho time of vibration of the mngBct under the
action of t^^ricstrial mnguetism is observed, and the statical, in
which the magnet is kept in equilibrium between a measurable
statical couple and the magnetic force.
Tlic dyniimical method requires simpler apparatus and is more
■CGnrat« for absolute menaurement^, but takes up a considciablo
time, the statical method admits of almost instantaneous mcasure
mml, and is therefore useful in triicing the changes of the inten^itj
of tho magnetic force, but it requires more delicate apparatus, and
is not m accurate fur absolute measurement.
jWrAiorf «/■ Vlhratiom.
The magnet is suspendetl with iU( magnetic axis horizontal, and
is eel in vibration in ainall :irc». Tiie vibrations are obwrved by
mans of any of the methmU already dtwcribed.
A point on the scale ia chosen corresponding to the middle of
the arc of vibration. The instant of paa.inge tlirough this point
of tho scale in the positive direction is observed. If there is safii
cient time berorc the return of the magnet to the same point, tlie
instant of juusage through the point in the negative direction is
iihu> observed, and the process ia continued till a + 1 positive and
104
MAOSETIC HBASCREMBKTS.
[456.
» no^tive pasmf^es have been obserri'd. If th« vibrations an
too rapid to allow of every consecutivo paamge boin^ obcerved,
everr third or every 6Ftb passage in obwrved, oare bein; taken tliati
the observed paessges are alternately po.iittv« and iwgiUve.
Let the observed tiroes of passage be
we put 1
r,. r,, r»^„ then
then TV,.! is the mean time of the potitive pusa^a, and oagh
to agree with 7",,, , the mean time of the negative passagee, if tha
point baa I>een properly chosen. The mean of theao resulta ia
to be taken an ti\a mean time of the middle pasmgc
After a large number of vibrations have taken jilaoe, bnt befora
the vibrations have ceased to be distinct and regular, tho observer
makeK another series of observations, from tvliich lie dcdnces th
mean time of the middip pas^afr* of the secom) series.
By calculating the period of vibration either from the fi
HCries of observations or from the E«cond, he ought, to be able
be certain of the number of whole vibrations which have tak
>hice in the interval between the time of middle pastxage in (Jii> two
Mtrie*. Dividing the interval between the mean timen of middle
[WK^age in the two series by this number of vibrations, the meua''
time of vibration in obtainivl.
The observed time of vibration is then to be rcdnc«d to lJi<
time of vibration in infinitely small arcs by a formula of the sama
kind as that used in pendulum obttervations, and if the vibrations
are found to diminiali mpidly in amplit«idc, there is aoolher cor
rection for resistance, ttee Art 740. These corrections, however, ara
very nnall when tlie magnet bangs by a Hbre, and when the arc of
vibration vs only a few desreois.
Tito equation of motion of the magnet ts
Jiff
where 6 is the angle brtwi>en the magnetic axis and the direction
of the foTOO //, A is the moment of inirtiu of the magnet an
iHi*i>ended apparatus, .1/ is the magnetic moment of the magnvi
// the intensity of Uie hori«ont«I magnetio force, mnd J/Z/r' tJi
ooeffioient of torsion : t' ts dvlenniniHl as in Art. 452. and i«
vciv smaH quantity. The value of tf for equilibrium is
% = {^j' • n ^^ «iia!l angie,
1
re 01
i
457]
BLIMISATIOS OF IITDOCTIOK.
105
MU =
Mtd the eolutioB of th« equation for small values of Uie amplitude, is
where T in tlie periodic time, and C tbe amplitude, and
3/7/(1, t')'
whetioe we lind tJie value of MU,
Here T is the time of a poinjiletc vibration determined from
obdervatioD. J, the ntoment ol' intrtio, in founil once for all fur
tbe ma^ety eiUier by wt^ghing ttnii meuKunnfr it if it iit nf »
regular figure, or by a dynaaiieal procuss of coin[iiirisoD with a body
wbwe moment of inertia is kuowu.
^ Combining tlita value of M/f with that of ^ formerly obUinud,
weget J/« = (J///){j = y,^/*^.
457.] We have suppoiied that // and ,1/ continue constant during
tbe two series of experiments. Tlio flugtuatiuus of // may be
aecertained by simiiltuiKmis observatiouH of the biRlar magnet
ometer to be preacntly d<.WTib^(l, and if the magnet has been in
use for some time, and i* nut exposed during tiie experiments to
changes of temperature oi' to conoussion, the part of M which de
pends on permanent ma^ietism may be assumed to be conrtant.
All steel magnets, however, arc capable of induced magnetism
depending on tlie action of exti^rnal mugnetic force.
Now the magnet when employed in the deflexion cxpcrinaent*
i» placed with ita axU cant and we^t, to that the action of ter
restrial nmgnetiam is transverse to the magnet, and doc^ not tend
to increase or diminish 31. When tbe magnet is made to vibntlv,
its axis is nottb and south, so that tbe action of terrestrial mag
netism tends to magnetize it in the direction of the nxin, and
therefore to iDcrcaee its magnetic moment by a quantity i fl, whore
it is a coefficient to be found by experiments on the m»}^ii!t
There are two ways in which this souroe of error may be avoided
without calculating i. the experiments being arranged so that the
magnet shall bo in the came condition when employed in deflecting
another magnet and when iteelf swinging.
I
I
106
MAOSBTIC MEASntEMKKTS.
[451
W(! may plitcc tho d<;lloctin^ ma^ct with its »xi* potntuif;
north, aL a. (lisUitcu r from tho centre ot tlifl etipi#nd«d inagn«t,
Ihv line r miiking iin iingk wliow cosido in V j willi the magnetic
meridian. Tlie iietion of the tlf(i«clin{r miM>Dct on tlie sii^ix^ailed
one i» then At right unifies to its pnrn direction, tttid ia equal to
Hera Jf is the magnetic moment when the axis points tiorth,
RK in thir experiment of vibration, so tliKt no corfts;tion has to li^^l
muilc for induction. ^^
Thi* method, however, is extremely difficult, owing to the largo
errors wbieli would be introduced by a Blifflit dioplaeement of the
deflecting magnet, and as the correction by levorsiiig the deflecting
mugnct iff not uppliLablc hero, this method i» not to be followed
except when the object is to determine tlie eocffieicnt of induction,
'llie following method, in which the magnet while vibrating
fnod from the inductive action of terrestrial magnetism, is due
Dr. J. P. Joule*.
Two magnets are prepared whose imigiietie moments are
nearly equal as possible. In the deflexion experiments these mag^J
net* are used separately, or they may be placed simultanoousl]
on opposite sides of the suspendnl magnet to produce a greater
deflexion. In themt ex[Hriment« the inductive force of terrestrial^'
magnetinm is trantiverM to the axis. ^M
Let one of these msgiietw be suxpended, and let the other be
placed )iaral)el to it with itit centre exactly below that of the Bits
ponded magnet, and witli iU »x\s in the same direction. The force
which the fixed magnet exerts on the sospended one is in the
opjiosite direction from that of terreatrial magnetism. If the 6xcd
magnet 1>e gradually brought nearer to the snapended one the time
of vibration will ineT«i.>e, till at a certain point the equilibrium will
cease to be stable, and hi^yond this point the suspended mugne^^
will make oscillations in the reverse pontion. By experinicntinj^
in this way a position of the fixed magnet, is found at which it '
exactly neutralizes Uiv effect of terrestrial magnetism on tlie iras
peiMled one. Tho two magnets are fastened together so as to ^__i,
parallel, with their axcH turned Uie snme way, and at the dintanO^H
jiiet fotind by experiment. Tliey are then suspended in tlio ueua^V
way and made to vibrate together through small aiva.
• Prec. PItit. S; KauAnUi, Mkrdi li>. ISOi.
I
IKTEKSlTr OP HAOKXTIC FOBOB.
t07
The lower maj^Det exactly neutralizes tUe effect of terrestrial
niagnetiiira on the upper one, and since the maijTiets are of equal
niuinent, the npper one oeutralizes the inductive action of tlie earth
on tliG lower one.
H The value of 3/ is therefore the same in the cxperinii'Dt of
^^vihration a^ in the experiment of detlesion, and no corri^ction for
induction ii* required.
468.] The most accumtc method of asoertaininj^ the intensity of
the horizontal magnetic force is that wliieh we have just described.
The whole series of experiments, however, cannot lie performed with
sufficient accnraey in much tesH than an hour, so that any changes
in Uie intensity which take pJaoc in periods of a few minutes would
escape oIiMTvation. lleiiw a didVrent method is required for ob
M.rving' the inten«i(y of the majs^ietic force at any instant.
The statical method eon^ists in defieeting the magnet hy means
(if a statical couple acting in a horizontal plane. If £ he the
moment of this couple, M tlie magnetic moment of the m^not,
Jf the horizontal component of terrestrial magnetism, and 9 the
deflexion, MIIsin$z= L.
llenoe, if L is known in terms of 9, Mil can he found.
The oouplp L muy be generated in two waj», hy the torsional
ity of It wire, iix in the ordinary torsion halancv, or by the
ight of tlie sui^pindcd appnnitui;, us in the bifilar euspenition.
In the torsion bnlancu the magnet is fastened to the enil of a
vertical wire, the upper end of which can be turned round, and ita
rotation mcosurvd by means of a torsion circle.
We have then
L = T{a—a„—0) = MllsmB,
Here a^ is the vnhic of the reading of the torsion circle when the
oxia of the magnet coincides with the magnetic meridian, and a is
the actual reading. If the torsion circle is turned so as to bring
the magnet nearly perpendicular to the magnetic meridian, no thut
then
I
.=1^.
or
r^,a~<,,^"^.(r)^Mn{\\ff%
MH=T{\+\ff*)(aa^~~ + 6').
By observing!}', the deflexion of the magnet when in equilibrium,
wo can calculate Mil provided we know r.
If we only wish to know the relative viiiue of // at diflerent
times it is not neoessary to know either M or r.
Wo may easily determine r in absolute meoiure by suspending
108
MAOKKTIC lIBASCREKCrrS.
a nonmagiietic body trom tlie mme wire and observing its time
ortMciUiition, thcn if A in Ibe momont of ioertia of tbts body, am
T th« time of a complete vibration,
4 It' .4
Tbe tiut! objection 1o tbe luo of the torsion balance i» that tb«
zeroreading a„ in liable to change. Under the constant twiEtinp
fbrcv, iirii^![i(; iwm the tendency of the magnet to torn to the north,
tl>e wire graduMly aoquiFeti a permanent twist, eo that it becomes
noccsMry lu ili^U'rinine the zeroreading of the torsiou circle afrc»h
at ithort iiitervatn of time.
BiJUar Siupention.
459.] The method of sn^tpending the magnet by two wires
Dbrw was introduced by Gausa and Weber. As the bifilar s
pennon is naml in many electrical iostruroents, we shall investigattt
it mora in detail. The general appearance of the suspension ia
shewn in Fig. 16, and 1%. 17 represents the projection of the wires
OD a horizontal plane.
AB and A'B' are the projections of the two wires.
JA' and BB" are the lines joining the upper and the lower
of the wiree^
a and 6 are the lengths of the lines AJ" and SR.
a and ^ their aziniiiths.
ff' and H'' the verti^nl comjicnenUi of the tennons of the wires.
Q and Q" their horicontal component*.
i llifl rerticml distance between A.f and BB'.
Tl)e forres which act on the magnet ar^— its weight, tbe conpl
arising l'Tt<m tcrreetrial magnetism, the UmJon of the wires (if anyj
and their tensions. Of these the etTects of magnetism and of
ti»«ion are of the nature of couples, lleoee the resultant of the
tcUN<>ns nniit eonaist uf a vertical force, equal to the weight of the
nagnet> together with m coaple. The reenlunt of the vertical
components of the leosions ts thervfore alone the line wboM pn^
tioo is 0, the intersection of Jj" and BB", and either of
lines is dividej in in the ratio of tt' to M'.
Thi> boritoiital euinpoaents of the ttnaiou fonn a coaplo,
are tberefont e>)iial in magnitodv and pMiQal in duMtioo. Colli:
eitber nf them Q, the moment of the couple which th«y form ia
when PP" is the diateaea betvaan the pwalU Uaca JM ud JTJF,
in^^
459] BiriLAB SUSI'BNSIOS. 109
Tu find Lhe value ot Z vrts bave Uic cqautions of momenla
QA=Jf'.AS = tF'.A'B',
and the gcomctriml cc^uation
{AB + J'li') py= ah sin (ofl).
wbeoc« wc obtaiD,
(2)
Rjl*.
Fig. ir.
ir n ii tfao mass of the ButtpondiHl appnmtiiH, and ^ tho intciifity
of gravity, ;r+ »"= mg. (5)
If we also writ* r W'=»mg, (6)
I wo find i= i(lii»)«i^ ^am(a^). (7)
llw vsliw of X is therefore a maximum with req>oct to s wfa«n s
no
MAOSETIC UEASUREMESTS.
[459
is zero, that is, when the weight of the 8usi)«D(l«d muss is equally
borne by thv two wim. ^M
Wo may mljnst the U'Dsionx of Ihc wires lo i>quaUty l»y observingf ^^
the time of vibnition, mid milking it u minimum, or we may obtain ^i
a Kclfucting uiljustmc^nt by uttaching the ends of Ihv wires, u^f
in Fig. IG, to » pulley, which turns on its axis till the t«iisioiu^^
arc equal.
The distance of the upper ends of lUu nuiipcniiion wires is rej
gulated by mCAiiti of two other pullies. Tlic dUtanee bctwoon the>j
lower endK of Uie wires is also capable of adjuatnient.
By this adjustment of the tension, the couple artslD^ from the!
tensions of the wires becomes
The moment of the coaple arising from the toraion of the wirei i
is of the form j (y—^),
where T is the sum of the coefficients of torsion of tlw wires.
The wires ought to be without torsion when a = ^, we may I
then make y = a.
The mumi^ut of the couple arising from the horizontal mo^etio 
force iH of the form
where i is the magnetic declinafion, and $ is the azimuth of the
tis of the magnet, Wc shall avoid the introduction of unncceeeary
eymbols without siicriGciDg generality if we sssume that the axil of
the magnet ia parallel lo SB", or that ji = $.
The equation of motion then Wcomes
A ^J^ = J/// sin (8ff) + 1 ^ «y sin (a(Jj+r(o»). (SjJ
There are three principal positions of this apparatns.
(1) When a is nearly equal to A. If 7*, ts tbe time of a complete j
ovcillation in t^is position, then
iv'A I ai „„ , ,
mg^T+itl/. (9)
r.»
4 i
(2) Wlien a ia nearly equal to 8+ir. If 7, is the time of
complete oscillation in tliis position, the nortb end of the magar'
being now turned towards the south,
iii*J lab
^  i T "'■*■'•— *'"• *'*•)
The quantity on the cigfathaod of (his equation may be m:
I
459]
BIFILAH SFSTENSTOX.
Itl
^f M small as we plense by diminisliing a or i, bat it mwit not ht
made ac^stive, or the equilibrium of the magnet will bevome un
Btxhle. The magnet in thia ]>oaition forms an inatrument by which
small vanatiouB ia the dirixtion of the maj^netii; force Piay be
rendered sensible.
I For wbeo i~$ is nearly equal to ir, na (8—^) is nearly equal to
0—h, and we tind
J.
1 l** .r»r
By diroinishin^ the denominator of the fraction in the InH t«rin
we may make the variation of S very large eomiiared with thitt of f>.
We should notice that the coetHi'ieut of 8 in this expri^itsiriii is
negative, so that when the direction of the magnetic force turns
in one direction the magnet turnn in the oppo&itii direction.
(3) In the third jxHjit.ion the \x\>\k:t part of the suspension
BplKirattn is turned round till the axis of the magnet is nearly
pcrpendicuhir to the ma<<;ne)tic meridiao.
If wc make
$h = +(f, anil otf = /3ff',
tlie equation of motion may be written
(12)
If tb«re is equUibriuin when II — If^ and f = 0,
(13)
(H)
and if ff is the value of the horizontal force corretponding to a
•mall angle S",
Vand
H «na
V V ^■^nysin/S+r^
Id order that the magnet may be in stable equihbrium it ia
nece«MUy that the numerator of tlie fraction in the s^ond member
■hoiUd W poaitirc, but the more nearly it approaches sero, the
tnon: «en«itive will be tlic inittrument in indicating changes in the
value of the intensity of tbe horizontal component of t«'rre«trial
magDetiiim.
The statical method of estimating the intensity of the force
depend* upon the action of an instrument which of it«elf assume*
(^myc08j3 + r \
(15)
iia
MAOKETIC MEASUREMENTS.
[46a
diflbrent positions ni* equilibrium for dilTeTent Tallies of the Toret.
Heuoe, bj nieaaii of a mirror attacbed to tbe magnet and throning
* 3{>ot of light upon a photographic surface moved by clockwork,
• cun'e may be traced, from which the inteaeitj* of tbe force at uijr
inirtimt m»y bt dctormin(Hl according to a scale, which we may for
tlie pr^'sent consider uu arbitrary one.
460.] In an ub8ervat'>ry, where a continuous system of reffie*
tratioa of decliaatioa and iutenttity is kept up either by eye ofa
torvatioD or by tlm automatic photographic method, tJte abttolut
valuex of the ilfcltnutioii nud of the intensity, as well as tbe positioa^
and moment of the magnetic axis of a magnet, may be detemuned
to a greater degree of accuracy. ^M
For the di>cli no meter g'wcs tbe declination at eveiy instant afTecte^H
by a constant error, and tlieblRlar magnetometer gives tbe intensity
at every instant multiplied liy a constant coefficient. In the ei
pcriments we sab«titute for $, S' + Sg where 2' is the reading of tb^.
declinometer at the given instant, and 8^ is tbe unknown bat eon^
Btant error, so that i'*f ^ is tbe true declination at that instant.
lu like manner for //, we substitute 67/' where //' is tbe reading
of the magnetometer on it^ arbitrary scale, and C is an unknown
but cooHtant multiplier which converts these readings into absolut«
measure, so that CJI' ia the horizontal force at a given instant.
Tlie experiments bo determine tbe absolute values of the quan
tities must be conducted at a sufficient distance from the declino
meter and magnetometer, so that the dilferent magnet* may not
sensibly disturb each other. The time of every observation most
be noted and the corresponding values of h' and //' inaettcd. Tbe
equations are tlien to be treated so as to 6nd 6^, the constant error
of the declinometer, and C the cotfficient to l>e applied to the
reading of the magnetometer. Wlien these are found the readings
of both iustriunents may be expressed in absolute measure. The
absolute measuremeots, however, must be freij^uently rep«Ated in
order to take aocotutt of changes which may occur in the magoctic
axis nod nuiiniettc momeut of the magnets.
461.] The moihods of determining the vertical component of the
terrestrial magnetic force have not been brongbt to the anm^ degree
of precision. The vertical force murt act on a magnet which tnms
about a horizontal axis. Now a body which turns about a hori
Kontal axis cannot be made so seasitive to the action of small forcn
as a body which is snspeaded by a fibre and turns abont a vertical
axis. Uesides this, the weight of a magnet is so largo compara^^
461.]
DIP.
»
»
»
I
with tbe magneiic force exerted upon it that a emnU displocv
ment of the centre of iDdrtia bv unequal dilutittioa, &c. produces
a gKAtjet cflect oa tho poeitioa of the ma^et than a confii<Ier»lilc
ebao^ of the magnetic force.
Uenoe the measurement of the vertical force, or the comparison
of the vertical and the horizontal forces, ia the least periVct purt.
of the system of ma^etio measurements.
TTie Tertical part of the magnetic force is ^nerally deduced from
the horizontal force by determining the direction of the total force.
If i be the angle which the total force makes with its horizontal
component, i is called tbe magnetic Dip or Inclination, and if ff
is the horizontal force already found, then the vertical force is
77 tan i, and the total force is //sec /.
Tbe magnetic dip is found by means of the Dip Needle.
The theoretical dipneedle is a magnet with an axis which passes
through ita centre of inertia perjjendicular to the Tiiagnetic axis
of the needle. Tbe ends of this axis are mode in the form of
cylinders of small radius, the axes of which are coincident with the
line passing through the centre of inertia. Ihese cylindrical enda
rest on two horizontal planes and are free to roll on them.
When the axis is placed magnetic cast and west, the needle
is free to rotate in tbe plane of tbe magnetic meridian, and if the
instnimeat is in perfect adjustment, tbe magnetic axis will set itself
in the direction of the total magnetic force.
It is, however, practically impossible to adjust a dipneedle ¥0
tliat ita weight does not influence ita position of eijuilibrium,
because its centre of inertia, even if originally in tbe line joining
the centres of the rolling sections of the cjlindrical ends, will ccaec
to be in this line when tbe needle is imperceptibly bent or un
eqaally expanded. Besides, the determination of the troe centre
of inertia of a ma<>net is a very difBcutt operation, owing to tbe
interference of the magnetic force with tliat of gravity.
Let us suppose one end of the needle and one end of the
pivot to be marked. Let a line, real or imaginary, be drawn on
the needle, which we shall call tbe Line of CoUimation. Tbe
position of this tine is read off on a vertical circle. Let $ be the
angle which this line makes with the radius to zero, which we shall
suppose to he boriiontal. Let A be the angle which tbe magnetic
axis make« witlt the line of coUimation, so tliat when the needle
is in this position the line of colliinatton is inclined $+K to the
horizontal.
VOL. XI,
114
MAGNETIC MEASUREME5T8.
Lctp 1)0 the perpendicular from the cmtn; of inertia on the plaoe
on whioh tlie axis roilii. then p will bo n function of d, nhul«vi
bo the Khajie of the rolling »urf<iccs. If Iwth tho rolling siAtio
of the endH of th« axis urc circular,
j> = e—a ein (fl + o) (:
where a ie the distance of the centre of iaertiii from the line joininff
the centres of the rolling eeotions, and a is the an^le whidi
Uoe makes with the Uoe of ooUimation.
If M is the inapietic mometit, m the maae of the ma^ot,
y the force of gravity, / the total magnetic force, and i tho dip, th<
by tlic conservation of onergy, when there is stable equilibrium,
itlooB[0iK—i)—mgp (!
must be a maximum with reapect to $, or
oe
]
.'IP
(I
JW/sin (ff + X— i) s— Bij^,
= IW^flC0«(^ + o),
if th« ends of the axia are cy)in<lnc«l.
Also, if 7* be the time of vibration about the positioD of eqn
librium, ■ /* V ■<»''^ /J
MI+Mpamn{9+a)= yj {4
where A i» the moment of inertia of the needle about its uda of
rotation, and is determined by (3).
Id det«TmiDing the dip a readiDg u taken with the dip circle in
tbe DUgnetic meridian and with the graduation towards the weet.^
Let 0, be this reading, then we have
jl//sin(^jf X— i)= ni^<ico8(d, + a). {i
The instrumeDl U now turned about a vertical axis through 180*,
so that the gnuluatton is to the east, aod if tf, is the new reading',
J//»in (flj+A— x+i) = wyaeoa (ff(+a). (6)
Taking (0) from (5), and remomboring that dj ia nearly equal to
I, and 9t nearly eqtud to %—i, and that X ia a Rmall angle, such
that rngaX may be neglected in compariaon with 311, ^M
3fl(0i~0. + v~2i)= 2mffacmict»a. {^^
Now take tlic nuignot from its bearings and place it in
deflcaioo appantua, Art. 453, so as to indicate iu own magnet
moment by the dellexion of a suq>cndcd magnet, then
3f=kf*ir2)
where J) is the tangent of the deflexion.
IB
1
DIP CIBCI.!!.
Nexi, reverse tbe msgnetisai of tbe needle And dctcnnine its
Bew magnetic moment J/', by observing a new deHcxion, the tan
gent of which is 1/ j/'= J^V/i/, (9)
wUenoe MI/= M'D. (lOj
Then place it on iU bearings nnd take two readings, 0, and $^,
in which d^ is nearly ir + t, and S^ nvarly — t,
J/'7sin(flj + A'— IT— i) = mgaaM{e^^a), (ll)
i/'i8in(dj + X'+i) = m^flco9(a« + a), (12)
whence, as before,
if'/(tfj— d,— IT— 2i) = 2in^(»co8icoBa, (13)
and OB adding (7),
Jf/(fl,tf^+if2i)+jl/7(tfjfl«ff2i) = 0, (14)
or J>(fl,fl, + »20+ jy(^,fl4ir2i) = 0, (15)
wheDco w« find the dip
1
■ i?(gi<?34^) + P'(tf,<i«ir) ^
(ICJ
lere i) and i/ are the tangents of tbe dodcxioDB produced by the
needle in it* first and second magnet ixations respectively.
In taking obiwrvatione with the dip circle tbe vertical axis is
carefully silju.tti<d no that the plane V'aringw upon which the axis of
Uie magnet re^ are borixontsl in oveiy azimuth. The magnet being
magnetized so that tlie end A dipG, is placed with ite axis on the
plane bearings, and obdvrvati'iOB are taken with tbe plane of the circle
in the magnetic meridian, and with the graduat«;d side of the circle
east. Each end of the magnet is obeervid by means of rending
inicroeoopes carried on an arm which moves concentric with the
dip circle. ITie cross wires nf the microscope are made to coincide
with the image of a mark on Mw niiignct, and the position of tbe
arm » then read olT on tbe dip circle by means of a vernier.
AVe thus ohlaio an observation of Uie end A and another of the
end B when the gradnations aie east. U is necessary to ohsorvo
botli ends in order to eliminate any error arising from the axle
of tbe magnet not being concentric with the dip circle,
The graduated side is then turned west, and two more obscrva*
ttona aie made.
The magnet is then turned round so that the ends of the axle
are reversed, and fonr more observations arc made looking at the
other side of the magnet.
1 3
110
MAQKBTIC WRASCBEUBST8.
i
The mafrnetizsHoD of Ibe ma^et is then rereraed bo that the
end B dips, the magDetio tnoniont is ascertained, and eij^ht observa
tions ar« takoQ \q this state, and tlio sixteea obserratioos combined.,
to determine the true dip.
462.] It is found that in spiU) of the utmost care the dip, ns tbiisi
dcductd from obsiTvutions muile with one dip circle, differs pur
eeptibly from thut diduocd from oWrrutioiis with another dip
eirele at tlic Mmc pliicr. Mr. Brotiii hits pointed oat the eflV
diie to ellipticity of the benrini^ of the nxlc, and how to w>rrec
it I>y taking observations witli the magnet mngnetiMd to differen
strengths.
The principle of this method may he stated tku. We »h*ll
suppose that the error of any one observation is a amiiU quantity
not exceeding a degree. We shall uUo Hiipposc that nome unknown
but re^^ular force aets upon the ina^et, disturbing it fhim ita
true position.
If X ts the moment of tliia toroe, 0^ the true dip, and 6 th«
observed dip, then
L = Mlsia{d~$a)> ('^
= MI{ee^), (18
sinee 0—6^ is small.
It is evident tEiat the greater M hMomea the nearer does the
needle approach its proper position. Now let the operation oE
taking the dip be perforuiod twice, first with the nu^netixationl
fs^xnA to Ml, the (ficate&t that the needle it capable of, and next*
with the magnetization eiual to ,1/j, a much smaller value but
Kufliricnt to make the readiu^ distinct and the error still moderateu^M
Let 9, and 0^ be the dips deduced from these two sets of obattrva^^
tions, and let L ho the mean vsloe of the unknown disturbinff
force for the eight positions of each determination, which we shall
auppoae the same for both determination*. Then
If we find that several experiment* give nearly njnal values for
L. then we may consider that B^ muitt be vety nearly the true v«lae
of the dip.
46S.] Dr. JohIc has recently construeted a new dipcircle,
which the axis of the neodle, instead nf rolling on horixontnl agat
pUnes, is slung on two (iUments of silk or spiilKr'it lUtijid, the
Henoe tf„s3'
(20)"
463]
JOULES 8t;8FBM.SIOM.
117
I
I
I
of tlie flUmcnt* bcin^ nttnch^d to the arms of a delicate balance.
Tb« niit of Uic ncvdle thu« rolls on two loops of silk fibre, and
Dr. Joule ftt\t\» tliat its rrccdom of motion is much greater than
whim it roll* on agtile plance.
In Kip, 18, NS is the n4MdIe, CC is its axis, conn&Ung of a
■tnight cylindrical wiru, and PCQ, P'C'Q' arc the filaments on whicli
the uts rolls. POQ i» the
balanoo, consistinfp of a doubl«
bent luvor siipjiortcd by a
wire, 00, stn'tcl>i'd horizont
alljr between the prongii of
a forked piece, and liaviiij;
a counterpoise Ji which can
be screwed up or down, so
that the iMtlance is in neutral
equilibnuD) about 0.
Id order that the needle
may be in neutrsl equilibrium
as the needle rolls on the
filamcDtfl the centre of gra
vity must neither rise nor fall.
Ht^nce the distance OC must
remain ccniKtunt an the iiectllo
rolls. This condition will be
fulfilled if the arms of the
balance OP and OQ are equal,
snd if the filaments are at
rif^ht angles to the arms.
Dr. Joule finds that the
needle should not bo more than
five inchirK long. When it is ci»ht inch«i hmg, the bcmling of th<!
ntnlle tind^ tu dimiiii«h thi; apjiurcnt dip by a fraction of a minute.
The axis of the nocdle was origiiinlly of Bt«ol wire, stniighteuetl by
Iwing liMught to a re<I lii'at while ^itntcht'd by a weight, but
l)r. Joule found that with the new suspLntiion it is not nocwssary
to use steel wire, for platinum and even standard gold are hard
enough.
The batanoe is attached to a wire 00 about a foot long stretched
horiz^Hitally between the prongs of a fork. This fork is turaod
round in azimuth by means of a circle at the top of a trii>od whteh
supports the u hole. Six complete observations of the dip can be
PtKl8.
118
U&OITBTIC UEASTBEMCSTS.
obtained in one hour, and the aven^ vrror of n nogle obwrvattun
is a fractioB of ft minut« of arc.
It is proposed that the dip ncedl« in the Cambridge Pti;si«^^
Labontory shall be obscTvod by meanit of a double image iMlrq^l
meat, eonsistinfr of two totally rullMting prisms pbocd an in
Fig. 19 nod mounted on a vertical graduated circle, so that the
pluDc of reflexion may be turned round a horizootal axis nearlj
coinciding witJi the prolongation of the axis of the suspended di{
needle. The needle is viewed by means of a t«Icscope pla
behind the prisms, and the two ends of the needle are seen togethi
as in Fig. 20. By turning the prisms about the axis of tlie vertit
circle, the images of two lines drawn on the nuv<lle may be ma
to coincide. Tlie inclination of the needle is thus detennined
the reading of the vortical circle.
""~~...
/ =
The tots) intensity /of the toagnvtic force in the Une of dip n
be deduced as follows from the times of vibration 7„ 7,, Tg,
ia the four positions alrmdy deKcribed,
The values of J/* and J/' must be found by the method of dcflcxi
and vibmtion formerly described, and A u the moment of inertia
the mu(^ut about its axle.
The observations with a magnet anspended by a fibre an
much more acearate that it is usual to deduce the total force from'
the horizontftl force from the equation
where / h tlio total force, Jl the horisontal force, and $ the dtp.
404.] The procc« of detonniniug thi^ dip being a tedious one. w
not suitable fur dct«rminiiig the continuwut variation of the mogtMtti
I
TERnCAL FORCE.
1
r
force, .tlie most convenient instrument for continuous observa
tions is the vertioil force maf^Dctomitcr, wbich is simply k mag^«t
balanced on knife ed^^s so as to be in stable equilibrium with it«
magD<>tic axis nearly liorizoiital.
K Z is the vertii'fll component of the magnetic force, M the
mof^etic moment, antl the small an^le which the magnetic axis
makes with the horizon
JHZ = wya cos (a—$),
where » is tlic man of thi; magnet, ^ the force of ^mrityi a the
distance of the centre of gmvily from the axis of suspension, and
a the angle which the pinoe through the axis aud tlie centre of
gravity makes with the ma^uetic axis.
Hence, for the small variation of vertical force t2, there will be
a variation of the angular position of the mngnot 16 such that
MlZ= n>yaBin(a~0)80.
Tn praetiee this instrument is not used to determine the abtiolute
value of the vertical force, but only to register its email variatirns.
For this purpose it is sufiieient to know the absoluM valuL ol Z
ilZ
when = 0, and the value of
d9
B The valne of Z, when the horisontal fojcc and the dip are known,
is found from the equation Z = //tan 9„, where 6„ is the dip and
(// the horizontal force.
To find the deflexion due to a given vnrintton of Z, take a magnet
and place it with its axis east mid west, und with its centre at a
known distance r, east or west from tlie declinometer, as in ex
periments oil deflexion, and let the tangent of deflexion be Dy.
H Then place it with il« axis vertical and with it« centre at a
^ distance r^ above or below the ociiLre of the vertical force mag
netometer, and lot the tungcnt of the deflexion pro<Iuccd in the
magnetometer be />;. Then, if the moment of the deflecting
, magnet is M,
Henoe
2.v = irvi>, = 2r,»/>s
i^^Jl^^^.
do
A
The actual value of the vertical force at any instant is
JZ
Z==Z^ +
49
where Z„ is the value of Z when 6= Q,
For continuous observations of the variations of magnetic force
120 HAQITETIC HEASUBEHENTS. [464
at a fixed observatory the Unifilar Declinometer, the Bifilar Hori
zontal Force M^oetometer, and the Balance Vertical Force Mag
netometer are the most convenient inBtruments,
At several observatories photographic traces are now produced on
prepared paper moved by clock work, so that a continnous record of
the indications of the three instruments at every instant is formed.
These traces indicate the variation of the three rectangular com
ponents of the force from their standard values. The declinometer
gives the force towards mean magnetic west, the bifilar magnet
ometer gives the variation of the force towards magnetic north, and
the balance magnetometer gives the variation of the vertical force.
The standard values of these forces, or their values when these
instruments indicate their several zeros, are deduced by frequent
observations of the absolute declination, horizontal force, and dip.
CUAPTER VIII.
ON TEBRESTBUL MAGNETISM.
465.] Ora knnwkdcc of Terrestrial MagTietiBni is derived from
U»e fUidy of llic dilstrilnition of miifpictk' force on the earlb's sur
fac« »t nnj one time, nod of the changes in that dietributioa at
different tinu<».
The rDi4pii>tic force nt any one place and time is known when
itK three courdinalog urc known. These coordia»ti.8 may he given
in Uie form of the dLolinatioo or azimuth of t)ie foree, the dip
01 inclination to the horizon, and the total iuteoeity.
^e mcwt convenient method, however, for iDveatiguting the
g«nend diittribtition of magnetic furce on the earth's surface i» to
con»Jd«r the maj;niliidea of the tliree compoueuta of the force,
A' = IIvo» h, dir.ct«d due north, I
y= Ntin 4, directed due west, ( (1)
Z = Htan 0, directed rerticsllf npwards, I
where JI denotes the horizontal force, A the declination, and
the dip.
If r is the maj^nctie potential nt the earth's surface, and if we
consider the citrth a sphere of radius a, then
a eoB I dk ir
where I is the latitude, and X the lon^^ittidc, and r the distance
bom the centre of the earth.
A knowledge of Tovvr the surface of the iitrth may be obtained
from Uie observations of horizunlul force alone as followi*.
Let Vg be the value of F ut the true north pole, then, taking
the lineintegral alonf; any meridian, we find,
I
for the value of the pot«DlJal on that meridian at latitude /.
KAOSmSV.
[466.
ThiiE the potential mair be founil for any point on ti>0 Mrtb's
Eiirfacv provided we know thu valuo of A', the northerly oomponctit
at cxf.ry point, nnd ^, the value of Tat thv polu.
Sinoo tlie (onxa depend not on the absolate value of F
on its derivatives, it ie not necesssry to 6x any particular val'
for f„.
Tlic value of V at any point nay be ascertained if wo know
the value of X along any given meridian, and also that of Y
tlie whole snrfiice.
ilnS
Let
■'1
where the integration is performed along the given meridian from
the pole to tlie pAiallel /, then
r = r,ajYcoBUk, (sj
where the integration is performed alon^ the parallel t from tb
(riven meridian to the required point.
Thcso methods imply that a complete magnetic survey of th«
earth's surface has been made, so that the ralue^c of .V or of F
or of both are knonni for every point of tbo earth'« tur&ec at n
given epoch. liVhal wo actually know are the magnetic com*
poDcnts at n certain number of vtationx. In the civilized parts of
the earth these ttatiooa arc comparatively numerous ; in other places
there are laT;ge tracts of the earth'* siirfacc about which we have
no dats.
Sfaffjulic Surrrjf.
4C6.] Let us sup{>oce that in a country of moderate size, who:
greatest dimensions are a few hundred nJIes, obeervatioiu of tl:
declination and the horizontal force have been token at a ood
siderable number of stations distributed &irly over ibv country.
AVithin this district we may suppose the value of /' ta be
pMsented with sufficient aocutscy by the formula
whence X=><i + *,/+fl,A, (T)
Let there be m statioas whose latitodes are t^, t,, ... &c and
longitudes A,, X,, &e., and let T and F he found for each sUtioa.
Let
i,mi^{ty
and \, = jX{X),
di
MAGNETIC SCETET.
123
/g knd Ag maj be called th« latitude and longitude of the central
Btatiott. Lot
Xa«S(.T), and ToCOb/.^  2{rcoB0. (lO)
thea Xg and fg are the raliiea of X and K nt tlif tmagiDary central
HtatioD, then
Xx=X^ + Ml~l,) + B,(\\^), (11)
rcoB;= l'„co»/„ + 5,{//„)+5,(AAo). {17}
W« tiavc » MjoatiDns of the form of (11) and n of the form (12).
If we denote llie j>roI>able error in the det*rmin«l ion of A' by f,
»iul that of y coft I by t], then we may ntlcuiate ( and t] on
the suppontion that they arise from errors of obserTation of //
^ and >.
(Let the probable error of // be ^, and that of h, d, then since
iX = COB d .4 /IH<iin bM,
P (» = .*« cos" 6 + rf« //* wn" fl.
Similarly t)> = ^' »in* 8 + </*/?' co«« 8.
If the ruriatiouj* of .V and K from their vataes as given hy equa
tions of the form (H) and (12) oonsiderably exceed the probable
errors of obeerration, we may conclude that they are due to local
attraction!!, and then we have no rea^D to give the ratio of f to r
any other value than unity.
Aoeording to the method of least squares we multiply the equa
tions of the form (II) by ij, and those of the form (12) by f to
make their probable error the enme. We then multiply each
equation by the coefficient of one of the unknown qiinDtities £,,
IB^, or Sj and a<Id the results, thus obtaining; three e(iuution)) IVom
whiohtofind.fl„£j, 5,.
in which we write for conciseness,
di=S (/»)«/„', &, = S(/A)«/«X„. *, = 2(X*)«V.
P, = 2 llX)nl^X„ Qt = S(;rco« l)^nl,y,i^l„
i»,»2(AJr)i.AoXo, Q,= S(Arco«/)»\,j;co»/„.
By calculating B,, S^, and B^, and substituting in equations
' (II) and (12), we can obtain the values of X nod i'at tiny i>oint
within the limits of the survey free &om the local dt»turbani;e9
^ will
124
VBB&liTBlAL UAGKKTISU.
[467.
;«
which nro fonod to exist where the rock near the etation is msf;netic,
as most igneous rockti are.
Siirvvys of this kiud can bo nuule only ia ooontriee where mag
iictic in8trumeDt« can be carried ohout and set up id s gT^nt: maoy
slations. For other parte of the world wo must be content to find j
the dintribiition of the magnetic elements by interpolation betwoeo I
their vahiee Ht a few stations ut (frcat distances from cacb otbor.
467*] Li*t us now suppose that by processes of this kind, or]
by thr cqiiividcnt gnphical process of constructin^f charts of the
lines orc({<ial values of the ma<;oetic elements, the values of A' and
y, and tliciice of the potential F, are known over tho whole surGic«j
of the globe. The next stt'p is tu expand F in ihe fonn of a siTies
of spherical surface harmonics. ^j
If the earth were m»rnctir.od uii i form ly anil in the tame directiou^l
tliroujifhout ltd interior, I' woiiM lu' it harmonic "f the 6r»t degree, '
the mag'netic mcridinuK woulil be great circleit pa:«ting through two ^^
magnetic polew diametrically opposite, the magnetic equator would ^
he ft great circle, the hori'/ontul forco would be equal at all point* ^^
of the magnetic e<uator, and if Jl^ is this constant value, the value
ut any other point would he If = Iff, cos C, where f ia the magnetie
latitude. The vertical force at any point would be ^= 2J7,BiD/',J
and if 6 is the dip, tan 6 would ho = 3 Ian T.
In the case of the earth, the magnetic equator is defined to bej
the line of no dip. It U not a great circle of the sphere.
The magnetic polos are defined to be the points where there is^
DO borizoiitat force or where the dip is 00*. There are two such
p'jints, one in the northern and one in the southern regions, but
they are not diametneiilly op{ioBitc, aiul the line Joining th«in
not parallel to the maguetic axis of the earth.
468.] The magnetic poles are the jmintc where the value of
on the surfitoo of the earth is a maximum or minimum, ot
etationarj.
At any point where the potential is » minimum the north end
of the dipneedle point* vertically downwards, and if a compass
needle be placed anywhere umt such a jioint, the north end will
point towards that point.
At points where the potential ts a maximum the soutJi end of
tlie dipnwdle points downwards, and the souUi end of the """p— »i
Di«dle point« towards the )>oint. ^M
If there are p minima of J' on tho earth's surface there must he^^
^—1 other points, where the north end of tfao dijiuradle pointy,
4
]
I
I
I
470.] MAQinrnc tfattius of trr EAiiTn. 12s
downwards, l>ut whirn tho oonipiMitnAodle, when carried in a circle
raund the point, instead of revolviiif; i>o that iN north end point*
cnnstantly to Ute centre, rerolves in the oppottite dirt'cttoii, so as to
turn Bometimes ita north end and sometimes it« south end towards
the point.
If we call the points where the potential is a minimum true
north poles, then these other points may be trailed false north polea,
because the compassneedle is not true to them. If there are p
true north poIe», there must he^— 1 fal«e north polea, and in like
manner, if there are q true south poles, there must he f— i false
south poled. The number of poles of the same name roust he odd,
so that the opinion at one tJme prevjilent, that there are two north
poles and two ."iouth poles, is erroneous. According to Gauss there
is in fact only one true north pole and one true south pole on
the earlli's surface, and therefore there are no false poles. The line
joiaini; these poles is not a diameter of the earth, and it is not
parallel to the earth's magnetic axis.
460.] Jlost of the early inveatigitors info the nature of the
nrth's ma{rQ(^tiaiu endeavoured to express it as the result of the
action of one or more har ma^ets, the position of the poles of
which were to he determined. Gauss was the firet to express the
distribution of the earth's magnetism in a perfectly Reneral way by
expanding its potential in a series of solid harmonics, the coefficient*
of which he determined for the first four degrees. These coeffi
cients are 2't in number, 3 for the first degree, 5 for the second,
7 for the third, and 9 for the fourth. All these terms are found
nraessary in order to give a tolerably accurate representation of
the actual state of the earth's magnetism.
Tojind villi Part oftJit Obtervfd Magnetic Force u due to Szfemal
and what to Internal Cautei.
470.] Let us now suppose that we have obtained an expansion
of the magnetic potential of the earth in spherical harmonicti,
ooniiatt^t with tho actual direction and mngnituilc i>f the hori
zontal force at every point on tJie earth's xurfiK'c. then GausK ha*
shiwn how to determine, from the ohsorvcd vertical force, whrthcr
the magnetic forces arc duo to caused, ^iich us magnetization or
etectric ciirn^nli, within the earth's surface, or whether any »rt
is directly due to cause* exterior to the earth's eurfaco.
Let y hit the actflal potential espanded in a double series of
I ii)heri>ca] harmouics, • ;*
126
iESTRTATi MAG17ET19X.
[471
('+»
The first eeries repreeente the part of the potentift! doe to eauttS*
exterior to the euih, and the second serie* repMsftnta the Nu:t
to causea within the earth.
The observntions of horizontal force give wa the mim of the»
series when r = a, the mdius of the earth. The term of the order t is
The obBcr\a(.ionB of Tertical force give us
and the term of the order i tn aZ is
Hence the part due to external causes is
^ — iTTi — '
and the part due to causes within the earth ia
il'taZ,
B,=
ii+l
The expannon of T lia« hitherto been calculated only for
m«in vala« of F at or near certain epochs. No appreciable port]
of thin m«ui value appears to he due to causes external to tbej
earth.
471.] We do not yet know cnoug^b of the form of tha 1
of the solar and lunar parts of the variations of T to
l^ Hit metAod whether any part of these variations arises from
maffnetic force acting from without. It is oortain, however, aa
thfl caleulatioDa of MM. Stonoy and Clianbers have shewn, that 1
the priiK))«] part of these variations cannot ariw from any direct^!
niH^oetic action of the sun or moon, supposing these bodies to be ^^
nuignetjo *.
473.] The principal changes in the magnetic force to which
attention has been directed are as follows.
• Pnibaor IlanulelD of FruiM kM dbronrol ■ pwiaje ehMgs tn tfw
ch la S9JIS iby*, BlwaM ■lacUjr W '
•jnodi« rarolotioa cf llio kid, u deduoad (ruia tki o1i«mitian of >bd lyoli
•lenrntK, ttia p<vlad of which
which
ik<rihs^
S«Jia day*, >I<iu>M oMUjr mmI (o Oak 3t Ita;
loduoad (ruiB tki olwvTation nt > bd ifoli mw U*
MOlor. TUi nwUiodnf diieowriaK «>»* "»■« of reOtfsw of th» iiaiiii mM bdy «f
tb* van 17 lu aflecti » IW nwffnMlo maSbt k tbe Bnt tmHaliiwnl nf Ilia r«pkjiiiMt
bt AUrnMiHii of tU lUbt M AalrooMnr Ahtd., Wiok, Jniw IB, 1971. Hau Prvt.
^ s7Ko». 1», 1871.
U74]
eOBTEBBASBAH OR CBLBSTIALT
I
I
T. Tia more Bfgular Varlationt.
(1) The Solar variittioos, dcpcndio^ on tLo boor of th« day and
the time of the ycsr.
(2) Tlic Lunitr vnrinliont!, tlppendtng on the moon's boor angle
and on hi?r other demouts <>f portion.
(3) Thcec variutions do not repent Ihom^^elves in difTorcnt years,
but Ewem to be subject to a variation of longer period of aWut
eleven years.
(4) BeftideB tbia, tbere is a secular alteration in tbe state of tlie
earth's ma^etism, wbicb has been ^ing on ever since ma^nutic
obmrvationn have been made, and is producing changes of the
magnetic elements of &r greater magnitude than any of the varia
tioQH of dntall x>vriod.
II. The Disluriaacet.
473.] Besides the more regular cbangcif, tbe magiietic eleinenU
are subject to sudden disturbanees of greater or less amount. It
i« found that these disturbaticis are more powerful and frequent
at one tjme than at another, and that at times of great disturbance
the laws of the regular variations are masked, though they are very
distinct at times of amull disturbance. Hence great attention has
hetn paid to these disturbances, and it has been foun<l that dis
turbaaees of a j»iiicu1ar kind are more likely to occur at eertaia
timea of the day, and at certain Ecasnns and intervals of time,
tJtoDgh each individual diHturban<'e appears quite irregular. BcGides
these more ordinary diflturhances, tbere are oecanionally times of
excessive disturWuce, in which the magnetism is strongly disturbed
for a day or tvro. These are called Magnetic Storms. Individual
disturbances have been sometimes observed at the same instant
in stations widely distant.
Mr. Airj' ha^ found that a largo proportion of tbe disturbances
at Greenwich correspond with the electric currents cullovted by
electrodes placed in the earth in the neighbourhood, and are such
a» would be directly produced in the magnet if the earthcurrent,
retaining its actual direction, were conducted through u wire placed
tiwderatath the magnet.
It has been found that there is an e[Mcb of maximum diHturbanoe
every eleven years, and that this appears to coincide with the epoch
of maKimum number of spot^ in tbe »un.
474.] The Beld of investigation into which we are introduced
128
MAQXETISJf.
4ri
by tlie study of terrestrial inagnetitm is «s profound u it u CX'
U'lmivc.
We know that the aun and moon act on the enrth'N magnetimi.l
It has been proved that this notion cannot be exjilaiiied by «U[>1
poniii)* the^e Ijodies magnets. The action is therefore indirect. Is
the cusie of the sun part of it may be tliermal action, but in the
caite of the moon we cannot attribute it to this cause. Is it k»
sible that the attraction of tJiese 1>odies, by causing: strains in tfa«
interior of the earth, produces (Art. 447) changes in the magnetism
already existing in the earib, and so by a kind of tidal action causes
the semidiurnal variations ?
Hut the amount of all these chani^ is Yory smalt compared witJ
the great secular elianges of the earth's msgoertiani.
IVltat cause, whether exterior to the earth or in its inner depth;*
produces such enoroions chanties in the earth's magnetism, that its'
magnetic poles move slowly from one part of the globe to another ?^i
When we consider thst the intensity of the magnotizatioa of th4i^
great globe of the earth ts quite componble with that tvhich we
produce with much difficulty In our atoel ma^ets, these immense^:
changes io so large s body force us to conclude tluit wc are not yd^l
acquainted with one of the most powerful agents in nature, tho
bceue' of whoee activity lies in those inner depths of the earth, to
tlie knowledge of which we have so few means of access.
PART IV.
ELECTHOMAGNETISM.
CHAPTER I.
»LBCTR0MA0NBT1C FOBCB.
47S.] It lutd been noticed by many differeDt obaervers tliitt in
nrtain cnsce m^netisTa is produced or destroyed in needlra by
electric discharjr^ through them or near them, aod coiijectures
of various kinds had been made as to the relation between roag
n«tisra and electricity, but the btwe of these phenomena, and the
form of these rvUtions, remained entirely unknown till liana
Chri»Uui Untted *, st » private lecture to n fctv advaueod students
it Co]><!tihBgeD, obtcrved that a wire connt'ctinpf the cuds of a
voltaic Ixtttcry afleeted u magnet in its vieinity. This discovery
be [MihliKhed in a trmct entitled Experimfitta cirea effectum CouJIicliit
HUttrici in AfHm Majfutiieant, dated July 21, 1820.
ExjX'riinento on the relation of the ma^et to bodies charged
with elei:trieitv h^ been tried without anv result till Oriited
endeavoured to uMertain the etl'eot of a wire ieated by an electric
current. He diHcovered, however, that the current itself, and not
tbt* heat of the wire, was the cause of the action, and that ihe
'eleotrio conflict acts in a revolving manner,' that is, that a magnet
placed near a wire transmitting an electric current tenda to set
itaelf perpendicular to the wire, and with the same end always
pointing forwards as the magnet is moved round the wire.
476.] It appears tlierefore that in the space surrounding a wire
* Sm nAoibaT aeoount of Ontvd'a dfacovvrr in k lottRr trtaa Pmrouar lUnaUoi la
ths Ltfi «f F»ni4ag b; Dr. Bcmoe Jcoico, vol. ii, ji. 30fi.
TOL. U. K
ISO
■LECTROMAGSETIO TOROl.
1 oi
transmittinf; nn electric current a [nuj>Dft i» acted on by forces
tlcpciulciit on tlic iiosition of i\w win? und on the streo^li of ihv
curnnt. Tlic hi>im;i; in wliitili tliOMB foreea act may thcnfwre W
coneitlvred ui a magootic flelil, au<I we amy study it in tlic came
way ft> we hnve already studit d the f\M in the neighbourhood of
ordinary magnets, by tracing the courae of the lines of magnetic
forco, and measuring the intensity of the force nt every point,
4771 ^'' ^^ begin with the case of an indefinitely long etraigb
wire carrying an electric current. If a man were to place bimselj
in imagination in the position of the wire, so that the cmrrent
should flow from his head to his fcot, then a magnet suspended
freely before him wotd<I set iteelf so that the end which gmints ttOr(J^_
would, under the action of the current, point to his right hand. ^
The lines of magnetic force are everywhere at right angleis to
planes drawn through the v/'in, and are there
fore cireW each in a plane perjiendicular to
the wire, which passes through it« centrt^l
The polo of a magn«t which point* north, ii^^
carried round on« of those cireles from left to
right, would eipericncc a force acting always
in the direction of its motion. Ilie other
pole of the t>amo magnet would esjierieng^^
a force in the opi>oeito direction. ^^^H
478.] To comjwm thc«e forces let the wire
be apposed vertical, ood tJic current a de^^
scending one, and let a magnet bo placed M^
an ap))aratus which is fr«e to rotate about a
vortical axis coinciding with the wire. It
ic found that under these cireunistancee the
cumnt ha)i no I'lTcct in causing the rotaliuo
of the apparatus as a whole about itoclf as oa axia. Ht^ncc tlia
action of the vertical current oa the two pobw of the magnet is
such that the etaticid moments of the two forces about the current
as an axis are equal and oppont«. Let M] and ai, he the strengths
of the two poles, r, and r^ their distances from the axis of Uie wir«,
7*, and 7*, the inteniiiticM of the magnetic force doe to the current at
the two ih.Ioi rfspectively, then the force on «, is m, 7",. and
once it is at right angles to the axis its moment is m, 7*, r^.
Similarly that of the force on lh« other pole is m^T^r^, and since
thccu is no motion obMTVed,
IV SI.
48o.]
BLECTROMAONBTtC POTESTTAt.
X = ~2i^,
I
But vu know that in nil magnets
M, + Mf = 0.
Ileoce 7, r, = T^ f,,
or the e1«ct<on)ag:n«H« force due tn a iitmig^ht ciirrent of inliRite
len^i is perpendiouUr to the current, and varies inversi^Iy oa the
distance from it.
479.] Since the product Tr dejiciid* on tlio strength of the
current it may be employwl ait a measure of Ihe current, Thi»
method of measurement is difiercDl from that foonded upon dec
tro«t»IJo phirorimena, and as it depends on the magnetic phenomena
produced l>y olpctric currunts it is called the Ki«;tromagm'tio«y>iteni
of measurement. In the elwtriiniiignetic system if i is the current,
Tr ='2».
480.] If the tvire be t;iken for the axis vf :, then the rectangular
OompODCDt« of T OTO
Ilere Xi£rf Tdjr+ZtU U a complete dlfferentiftl. Wing that of
2tt»n'^+C.
BeDM: the mognetto force in the field can be deduced from a
potential function, ati in aevcral former instancee, but the potential
is in this case a fnnction baviuff an infinite seriea of values whoso
common dificronce is 4iri. Tl)e differential coefficients of the
potential with rettpect to the coordinates have, however, definite and
single values at every point.
The existence of a potential function in the field near an electric
currant is not a selfevident result of the principle of the con
servation of energy, for in all actual currents there is a continual
expenditure of the electric encr;,'y of the battery in overcoming the
reeietanee of the wire, so that unless the amount of this expenditure
were ncc«cnt«iy known, it might be suspected that part of the
taergy of Ihe battery may bo employed in causing work to be
done on a ma^et moving in a cycle. In fact, if a magnetic pule,
St, moves rotmd a closed curve which embraces the wire, work
is actually done to the amount of 4irmi. It is only for closed
paths which do not embrace the wire that the line'intcgraJ of the
force vani«but. We must therefore for the precent consider the
law of force fliid the extKtciiev of a potential as resting on the
evidvDOO of Uifi experiment already described.
K 3
132
ELSCrnOMAOKRTtC FOBCR.
'1
ea
1
it«
it«
cait
I
4S1.] If we consider the Kpatw RiirToiinding: an infinite ittraigfat
liue we sball en tbiit it is n cvclic span^, bocansc it returns into
itself. If we now vonoeive a plau«, or any otber surface, com
mencing at the Ktraiglit lino and extending on one side of
to inlinitf, tliie etirfiico may lie regarded as a diaphra^i wbichl
reduces the cyclic iqnce to an acyclic one. If ^m any fixed point
lines be drawo to any other point witbAUt cutting the diaphragm,
and tbe potcntiul bi; defined as tlie lineinte^al of the force token
along one of Ihesu lines, the jiotential at any point will thon have
K single dvfiuitv value.
Tlic Diabetic field i» now identical in all respects with thai d
to a magnetic ibcll coinciding with this snriace, tbe ntr^gth of
the shell being t. Tliin ^ihell is bounded on one edge by the infinite
struight line. The other part« of its boundary are at an infinite
dii^ljince from the piirt of the field under consideration.
482.] In ull ucttial exjiertmenta the current fonns a closed circuit
of finite diiiifrnsions. We shall therefore compare the magnet
aetion of a Unite circuit with that of a magnetic shell of vhiob
circuit is the bounding edge.
It has been shewn by namerona experiments, of which
earliest are those of Anip&re, and the most accurate those of Weber,'
tliut the magnetic action of a small plane circuit at distances which
are great comp«re<) with the dimemiions of the circuit is tbe same
as that of a magnet whose axis is norma) to the plane of tbe circui
and whoee magnetic moment is equal to tbe area of the cireail
inulti])lied by the strength of the current.
If the circuit be supposed to be tilled up by a surface bound
by the circuit and thus forming a diuphtagm, and if a moguetio
shell of strength i coinciding with this surface be substituted for
tbe electric current, then the mngnetlo action of the shell on all
distant points will he identical witli that of the current.
483.] Hitherto wo have supposed the dimensions of the circoi'
to be smalt compared with the dist»iKe of any part of il fi
the part of Uie field examined. We shall now suppoee the circui
to 1*0 of any form and si&a n hatever, and examine its action at naf
point P not in the conducting wire itself. Tbe following method,
which has im[H>rlnnt goomctrini) applications, was introduced by
Aiu>^ for thin pur)>o«e.
Cuncvive any Kurfare S botindcd by the circuit and not passing
tlirough tJie point P. On this surface draw two sertce of line»
srosring each other so aa tu divide il into vhmcntary poKinns, ih
, Ul^
484.]
WAOKETIC SnELL IN PLACE OF CDRBEKT.
133
I
dimensioaB of which arc small compared with thcir iliHtonou from
P, Aod with the radii of ourvature of the curfAco.
Round eax\i of thaw elemfntK ooDCciv« a current of Bt.i«n{*th i to
flow, Um direction of oiroubtioii hciii^ the samo in all tiie elements
as it is in the orig;inal circuit.
vMoDfT ev«iT line fonniiig; tlii> diviiiinn between two conUguous
elements two equal currents uf Hlixnglh (' flow in opponte direc*
tioite.
Tlic effect of two oqiul and oppoaife cuirenftt in the same place
is ahHilutely zero, in whatpver asjieet we consider the currentii.
Hcncv tlteir ma^etic f.S\xil \s xero. I'he only portions of the
elcnicntttry circuits which are not neutralized in this way are those
which coincide with the original circuit. The ixital eifect of the
elementary oircnits is therefore equivalent to that of the original
circuit.
484.] Now since each of the elementary circuits may be con
sidered as a small plane circuit whone diKtanoe from P is ^reat
compari'd with its dimensions, we may substitute for it an ele
mentaty maffnetic shell of strength 1 whose hotinding edge coincides
with the elemeutaiy circuit. Tlie magnetic effect of the elementary
shell on i* is equivalent to that of the elementaty circuit. The
whole of the elementary shells constitute a maffiictic shell of
strength t, coinciding with the surface S and bounded by the
original circuit., and the magnetic action of the whole shell on P
is eqaivalcnt to that of the circuit.
It is manifest that the action of the oircoit is independent
of the fonn of the surface S. which was drawn in a perfectly
arbitTsry manner bo as to fill it up. We see from thin that the
action of a magnetic shell depends only on the form of its edge
and not on the form of the shell itself. This tcnuU we obtained
before, in Art. 410, but it is instructive to see how it may be
deduced from clectToma^nctic consiilcrntioiix.
'fbe ms^ctic foroo due to the circuit at any point is therefore
identical in magnitude and direction with that due to a magnetic
shell bounded by the circuit and not passing through the pointy
the strength of tla shell being numeriually equal to that of the
eumnt. The direction of the current in the circuit is related to
the direction of miignetixation of tlie shell, so that if a man were
to stand with his ftwt on that side of the shell which wc call the
positive side, and which tends to point to the north, the current in
Iroab of him would be from right to left.
134
ELECTROMAONETIC FORCB.
[485.
485.] The m&t^otic [wt^ntiitl of th« circait, howcrer, difl«n
from that of tho maj^otie shill for thowc ])omt» wliiuli are in the
anl^stftnco of the mngnotic »h«\\. ^^
If at is the solid Angle isiibtoiulotl st th« point P by t1>« nwgoetid^^
ehcil, rockonci] jionitivv wlien tliv jxiHitire or auxtrul «)<lv of the tihell
ie nest to /', thvn the inngnftic )Ot«titiat at any ioint not in tlie
shvll it««lf is loif), wht're <t> a lh« rtmnglli of the ithell. At any ^^
point ill tlie inibttjinco of the shell itwlf vie may nippofte thn «helt^
<livi<!<Hl intn two pnrt« ivlioiie stKiigths are 4k, and 0,, where ^1
^, f ^j = 41, suHi that the )>oint is on the ]M»iitive eide of ^ and
on tJie negative side of 0j. The potential at thin point is
On the negatiye side of the shell the potential becomeH ^(w— I r).j
In this case therefore the potential is eontinunns, and at ever
point hati a Htn^le d<>terminat« value. In the case of the electrit
vireuit, on the other hand, (lie magnetic potential at everr point
not in the conducting nire itself is eqoal to la, where i ie the
strength of the current, and 10 is the solid angle uihlended by the
circuit at the point, and is reckoned positive when the current, as
seen from P, circulat«B in the directionopposite to that of the hands
of a watch.
The quantity Jw is a function having an inlinite aeries of valae
whose common difFereaoc is 4ni. Tlie diftV'reutial cueflBcients
iw with reeprct to the coordinates, ]uve, however, single and de
terminate valued for every point of space.
4B6.} If n long thin flexible solenoidal magnet were pUced in
the neighbonriKxid of an electric clrtnit, the north and south ends
of the solenoid would tend to move in opposite directions round
the wire, and if they were fiee (0 obey the magnetic force the
magnet would fiually become wound round the wire in a close
ooil. If it WfTT' powibh^ to obtain a magnet having only one pole,
or poU'n of untinittl strength, soch a niagtiit would be moved round
and round the win^ continiadly in one direction, but sinoe the poles
of ever_> magmii are iqunl and opposite, this result can never occur.
Faraday, however, hiut Klienn how to produce the continuous rota
tion of one pole <^ a magnet round an eleetrie current by malting
it powiiblc for one pole to go round and round the ourretit whttd
the other pole docs not. That thin process may be repeatMl ttJ
definitely, the body of the magnet mnsi be transferred from nam
sitif of the ciirrciil to tin iitlier ouce in each rcvohitioQ. IV) tM
this nithout interrupting the Aow of electricity, llui current is aplS
HBVOLTIWO MAOSKT.
1S5
I
Nli U»l
into two bnin«hc», » tlwit when one branch is opened to let thft
taagoei poxs tho current continues to flow thron^h the otJier.
Vami»y utcil fnr this puqmso a circuliir trough of mercury, as
eheivn in Vig. 33, Art. 491. The current enters the troug'h through
llie vfirc AH, it id divided at B, und after flowing through the ares
BQP and BKP it uiiitc« at P, und leaves the trough through the
wire PO, the eiip of mercury 0. and a vertical wire beneath 0,
down which the eiirrent flows.
The mngnet (not fhewii in the figure), is mounted so as to be
capable of revolving about a verticul a>:is thtDUgh 0, and the wire
OP revolvea with it. The body of the inngniit. pnsisieii through the
aperture of the trough, one pole, say the north pole, Iteing benenth
the plane of the trough, and the other above it. As the magnet
and the wire OP revolve about the vertical axis, the cnrrent it
gradually transferred from the .branch of the trough which lies in
front of the magnet to that which lies behind it, so that in every
complet« revolution the ntognet passes from one side of the current
to the othiT. The north pole of the magnet revolves jibout the
deeeending current in the dirx^tion N.G.S.W. and if w, w' are the
wild angle* (irre«)ectivc of sign) subtend»>d by the circular trough
at the two poles, the work done by the electromagnetic force in a
complete revolution ix
m({47r— (0— «'),
where n is the stJ'ength of either pole, and i the strength of the
CUTTVDt ♦.
487.] L«t OS now endeavour to form a notion of the state of the
magnetic field near a linear electric circuit.
Let the value of w, the solid angle subtended by the circuit,
be found for every point of space, and let the surfaces for wlueh
■ [TU* prolilou may be diwuand u {uUoar* Referring tu Wig. SS, Art, 491,
lat u t»ke OP in uit ■mailioD uui inlndurc Jmagiiuu'y bafaiiciog piiit<<dU i aloii^
BQ Kad a, y klaug OB. Am Ihe mngno( ntuicbotl to OP i* carried chroufh ■
oenploU nvolutlon nn wotk U Anixv on Ihi inuth pule bj^ tbu cummt (. tuppoMnl to
ian timtiK AHOX, tli«l uola ilwc^ribini;' n i:ti)»'<1 i^iirr* which dfiw nol tinbnc* l]i«
cuamnt. Tbr north pola howsver clHuribiui n uluMd ourra whioh don emhooci
llio cvmmti iuhI the nrk Aowt u^od it in 4 vmi, Vfa hare now ta estimate the
•ttwU at tbo cuTTVDta i in the circuit BFO and V in the oinmic BSPO. The
pnlewllal uf th« tuirUi pula whicli ta bolciw the planon of thoui circuit* trill be
— niv^'Mnytv— Bf) ami, iif tliu »">ulh, — mia', — ra)r(— •' t b'j).
wImm t>f wkI ttf' donoto the nlid anglcn >ubteni3o] at the two >iil«a by BOF, anil ».
■* ikiaa mibtonied by the drculor trough. The rceultant poUmtUl !■
my (■> + !«')— (It ( (a>j+ ir'j).
H«nM aa OF rcrvlvM from OF in tile dirwtion NEt^W back lu OP ■c*'* <''*
poMntlail arlU cban^n by —(«((* + ■')• The work dcniB by the inuTWiU i* ihersfute
Uial {<min la th« xvxt,]
VOt. II.
ELRCTROMAOSETIC FOBCB.
M is oonsfant he deitoribed. Tliese aurfaces will Iw tlio ixjiiipottntial
surfaces. Each of tUeee surfncea will bo boiiniiwl hy the eirrmi
and nny two surfacea, wj and <»,, will miit in the oircait at
Fiifuiw X\Tn. at the ond of thin vi>lume, represents a. Bcction
of tbo eqnipotential eurfacM diip to n circular current The Rmall
rarclu rt'jiresenta n section of the conducting wire, and the hori
Mnt«1 line nt ihc bottom of the fi^ur^ ik tlie per>endien1ar to th
plane of the circular current through ito centre. Tlie eqnipotential
siir&cca, 24 of which are drawn corresponding to a series of values
arc surfocfw of revolntion, having this tine (o\
of «e dilfcrini* br ■ ,
o
their oouimon axis.
;ial
lit^d
1
inn ^
I
■4
They are eridcntly oblate Ggtirev, being Bat
tened in the direction of the »xi». T1>ev meet nob other iii the lint
of the circuit at angles of 1 S°,
The force acting on a magnetic pole place«I at any point of anj
eqnipoloiitinl MiHacw ia perpendienUr to this surface, and variMJ
invoracly a** the di^nce between consecutive surfaces. The cln«ed
eurvea surronndiog the section of the wire in Fig. X^^II are the
lines of foree. They are copinl from Sir W. Thom«»i*s Paper on
' Vortex Motion t' See also Art. 702.
Actum of am Elrrtrie GntiU om irny Magnetic ^H^m.
48S.] Wfl u« DOW able to tleduoe the action of an eU^■tnl' circuit^
on my niagnrtio aystem in it« neighlKKirhood from \\w theory of 
magnetie ahelbi. For if we eonstnict a magotfic i^>dl. whoM* 
strength is nnmerically equal to the stmngth of the cunvnl, and
whoae edge coincides in ]>oeitiiin with tha ououil, white the rheW
itedf dow not pass thron^h any part of the magnetic system, the '
action of the shell on the magnetic ayvtem will be ideetieal with
that of the electric cnrrent.
Am«M iftXt MfmHe &f»l«m m Ika Bitirit CiftmU,
489.] Kmm this, applyiag the prineiplfi that action and mction
ara niHU and opiwsite^ w« Hiooliide that the nMdianical actieo «f 1
TORCE OIT THX CIRCUTT.
B th« mftgnetio tytAeia on the electric circuit is identical witli ite
~ acttoti on a ma^octic sbell havinsr the circuit for its edge.
Th« potential energy of a magnetic ehell of sttength ^ placed
in ft field of magnetic force of wbich the potential is V, is, by
ml, m. H ua the directioncosines of the normal drawn rrom the
potutive ud« of the element dS of the shell, and the intogrntion
is extended over the surface of the shell.
H Now the suriitceintcgral
A'=JJ{ta + mit + nc)dS.
where a, (, e are the components of the magoetic induction, re.
presents the (uantity of tnagaetio induction through the shell, or.
in the Ituigoagc of FAradaf, the tiumher of lines of ma^ctic in
^daction, rackoood al^braieally, which pass throKph the shell from
^■thtt negative to the positive side, lines which pnss through the
shell in the oppowto direction boioff reckoned negative.
Kememheriiig that the shell does not helong to the magnetic
syittein to which the pot<minl V is due, and that the mngneUc
force is tlierefore equal to the magnetic induction, we have
a= — i — c'^^
6 ''^
^L^ dx' df' '~ it
^^^^Vc may write the value of JA,
If ftjp, represents any displacement of the shell, and .V, the force
acting on the shell so as to aid the displacement, then by the
principle of couEerrntion of energy,
x=^
dx
We have now det«rmined the nature of the force which cor
responds to any given displacement of the shell. It aids or reaista
that displacemeot aocordinj;ly as the displacement increaeea or
diminishes A', the nnmber of lines of induction which pass through
the shell.
Tbo same ia true of the equivalent electric circuit. Any diit
^jjutmeDt of the circuit will be aided or resisted accordingly an it
id
138
ELHCntOMAOSmC POSCB.
inoreases or diminiehcs tho Dtimber of lines of iDduction wiiich j*u
through the oircuit Id tho positive iliroctioa.
We must rciDcmWr thiiL the positiri; dircct4on of a lint* o^_
magnetic induction is thi; ilrrection in which thu polo of n magne^^
which pointE north tends to movt? along thv line, and that a litte
of induction puK»<s through thu circuit in the po«itiTC dirfctioD^_
when tLe direction of the lino of induction i» related to tl>^
direction of the current of vitreou* electricity in the circuit as
the loDgittidinal to the rotational Diotioii of a rightbanded wcrcw^H
ScttArf. 23. fl
490.] It i« matiifM tliat the force corresponding to any di*
placement of the circuit as a whole may be deduced at once fron
the tlicoiy of the magnetic shell. Dut this is not all. If a portic
of the circuit iti flexible, so that it may be displaced independent!]
of the rest, we may make the edge of the shell eapable of the sami
kind of displaceracDt by cutting up the surface of the shell int
a snfficieDt number of portions connected by flexible joints. Henc
we conclude that if by the displacement of any portion of the circiiii
in a given direction the number of lines of induction which
through the circuit can be inereu^cd, this displacement will be aided
by the electromagnetic forct> mtiug on the circtiil.
Erery portion of the cinuit tbenfore i* acted on by a force
urging it acroiw the tinea of magnetic induction m> a« lo incltidc
a greater number of the«e lines within the embrace of the circuit
and the work done by the force during this displacement
namerically equal to the number of the additional line* of
duction multiplied by the etrcngth of the current.
Let the element d» of a circuit, in which a current of strengttf
t is Sowing, be moved parallel to iteelf tlirough a space bx, it will
sweep gut an area in tho form uf ii iiarallclogram whoso sides arc
parallel and equal to </« and b» reiti>ect)vely.
If the magnetic induction is denoted by Q, and if ita directiog
inakcH an angle t with the normal to tlte parallelogram, the vali;
of the increment of jV corresponding to the displacement is found^
by multiplying the are* of the parallel<^ram by 39 cos «. The result
of this opemtion ia represented geometrically by the volume of a
pnmllelopiped whoee edges represent in magnitude and direction
6x, di, and 9, and it is to be reckoned positive If when we point
in the«c three directions in the order here given tho pointer
moves ronnd the diagonal of the panllelopiped in the dirccUo;! of thf
huDd« of a watch. The volume at this pciralleIopi{<ed is equal to Xia
arc
491]
FOKCB OS AS ELEMEST OP CIRCUIT.
139
I;
¥
If 6 is the an^le Iwtwecn tit and $, the arw of th? par!i11elo'>rain
is tls.^ ein 0, And if i; is the »n^lp nhith thi.' disphicemcnt Sx
makes with the normal to this paruJIulogrkm, the volamo of the
paralklopi^ IB
I rf* . S sin S.ixixs r} = iy. I
Now Xix = iJ.V= fV/.SdudSJTCOSi), I
and X = iJa . S sin fl cos jj '
» th« foree which nrgee d», resolved in the direction ix.
The dirwtion of this force is therefore perpendicular to the paral
lelogram, and is equal to i .ds.^Q sin 6,
Tbis is the area of a parallelogram nhose sides represent in maj[
nitude and direction idt and ® The force acting on </» is therefore
represented in magnitude hy the area of this parallelogram, and
ID direction by a normal to its plane drawn in the direction of the
loDgitudinal motion of a rig'hthanded screw, the faimdie of which
is turned from the dii'ection of the current icit to that of thv
nuignetic induction ©.
Wo may express in the language of
Quaternions, both the direction and
the magnitude of this force by saving \fitt
that it is the vector part of the result
of multiplying the vector ids, the
viemmt of the current, by the vector
^B 9, the mn^ctic induction. S*ua
491,] Wc hnvc thus completely de
■^ termiiied tlic f'lrce which acts on any
^P )ortio» of an electric circuit placed in
~ a magnetic field. If the circuit is
moved in any way fo that, after osetiming various forms and
jioaitions, it retuniH to its original place, the strength of the
eument remaining ooniitjtnt during the motion, the wliolc amount
of work done by tbe elcctm magnetic forces will he zero. Since
this is tine of any cycle of motion* of the circuit, it follows that
jjH it IB impossible to maintain by tU'cironingiielic forces a motion
^^ of eontiniioua rotation in any purt of a linear circuit of constant
strength against the resistance of friction, &c.
It is possible, however, to produce continuous rotation provided
that at some part of the courwe of the electric current the current
in, passes from one oonducl/ir whieb slides or glides over another.
^m When in a circuit there is sliding contact of a conductor over
^K the surface of a smooth solid or a fluid, the eincuit caa no longer
KtirIA
firl
Znt
Fig. 32.
KLECTBOMACSOTIC FOBOf.
mt
I
bo considered afi a single linear circuit of constant etr«n^h, but
miixt be rcK'snled as a system of two or of some ^ft^ater niimVicr
circuits of variable strength, tkc current beinj; so dtstnbut
ftmoiir tlium that those for which A' is incrensinf; have ciirrcn'
in the [luKitivu direction, while those for which iVis iliminishi
have ciirreuLd iu the negative direetion.
Thus, in the apparatuA represenUHl in Fig. 23, OP is a mov<abl6
conductor, one end of which reels in a cup of mercury 0, while the
other dips into a circular trough of mercury concentric wiUi O.
The current t enters alon^
AS, and divides in the circular
trough into two p&rts, ono of
which, X, flows along the wc
BQP, while the other, y, Bowb
along BKP. Th«eo ctirrcnts,
uniting nt P, flow along the
moveable conductor PO an
the elcctrwle 0^ to the zil
end of the Wteiy.
strength of the current ale
Fi8.83.
OP and OZ is z+y or i.
Here we have two circuita, AMQPOZ, the strcnglh of the ourre
in which is x, flowing in Uie poallve direction, and ABU POX, tli
strength of the current in which is y, flowing in the negatii
direction.
Lot 9 be the magnetic induction, and let it be in an upwa
(lircclion, normal to the plane of the circle.
While Op mov«M through an angle $ in the direction oppoaitj
to tltut of the liande of a watob, the area of the first circuit inci
by {0P*.$, and that of the second diminishes by the Esmc quanti^.
Since the strength of the current in the first circuit is «, the work
done by it is  ;r . OP^. . ®, and since the strength of the second
is — /, th« work done by it is ky.OP'.O^. The whole work done
is therefore ^^
do])cnding only on the strength of the current in PO. Hence, if
i is maintained constant, the arm OP will l»e carried round and
round the circle with a uniform force whose moment ift \ i .OP')
If, as in oorthem latitudes, $ acts downwards, and if the curreig
is inward*, the n'tatioa wiU be in the DCgative direction, that w7
in the direction I'QUS
d
493]
ACTTON BRTWEBS TWO CITRTIESTS.
t41
^1 492.] V!e are now able in pass from tlie mutnsl aclion of
^nnognels and currentii to the action of one current on another.
^Kpor we know ttuit the magnetic properties of nn electric circuit (?,,
^prith respect to any magnetic BTBtero J/,, are identical with those
of ft ma^etic efaell iS,, ivfaose edge coincidee with the circuit, and
Hwhose Btrength is mimerically equal to lliat of the electric cnnent,
Hliet the tna^etic system J/^ be a magnetic etell S^, then the
mntual action between S, and S. ie identical with that between 8^
utd a circuit (7,, eoinciJinf* with the ed^ of S.^ and eqiiiO in
nomencal BtrcD<rth, and this hitter action is identical with that
between C, and C,.
Hence tb« mut^inl action between two circuit)!. C, and Cg is
iilenlioiil with that between tbo corresponding niajfucltc shells ^
hnd S^.
Viv hare alrcjuly invi'Hltgnted, in Art. 123, the mutuiil aetioo
'two iiiagnctie shcllit whose edges are tlie cloved curvus *, and »^.
If we make
Jn . In
coae
.'o .'0
<f'l^f
where ( if the an^lc between tbe directions of the olementH /'#, and
dt^, and r is the distance between them, the integration Ining
extended onco round a, and once round «,, and if wc call M the
potential of the two closeil curves «, and 4., then the potential
encrj^y duo to the mutual a<^!tiun of two muj^nctic sbclU whose
■trengths aro ij and i^ bouudctl by the two circutts Is
and tlie force X, whteb aids any displacement ix, is
I ''•»^
The whole theory of the force acting on any portion of an electric
circuit due to the action of another electric circuit may bo deduced
^^Aom this result.
^B 493.] Thi> method which we have followed io this chapter is
^nbat of Faraday. Tnxlejid of beginning, as wc shall do, following
^T Ampere, in the next chiiplcr, with the direct action of a portion
of one circuit nn a portion of another, wc shew, firet, that a circuit
produces llie same effect on a magnet a« a magnetic shell, or, in
other wordic, wc determine the ualnrc of the magnetic field due
to Uie circuit. Wc chew, Rceondly, that • circuit when plai^ in
af magni^c field exjuiriinccs tbe mme force ek a nuignctic ehell.
Te thus determine the forec acting on tbe circuit placed in any
142
BLECrBOMAOSRTfC POBCl!.
.494
muj^ncttc field. Lustlj, 1>y snpiKiKing the in>gi)eti« field to be
to » «'corni electric circuit, we dftt>nnine the action of ouc ci
CD the whole or iiiiy iinrtion of the olhfir.
4{^.] Let iiM iipply thix method to the caee of a Rtraight current
of inlinite length acting on a portion of a parallel straight oon
dtiutor.
I^et us tFiipposi ttiut a current t in the finl conductor is flowing
vcrticnily downwnrdo. In this case the en<! of a magnet vrbi
poiiitu north will point to tlie rigbthuid of a man looking at
from the axis of the current.
Tb« lines of magnetic induction are therefore liorizontat circl
haTing their oentreti in the axis of the current, and their po«iti
direction is north, ea^t, south, west.
Let another descending vertical current be placed due west of
tbe first. The line^ of magnetic induction dne to the first current
are here directed towards the north. Tbe direction of the forci
acting on the second current is to be determined by turning t
handle of a righthanded screw from tbe nndir, the direction
tli« cnrrent, to the north, the direction of the magnetic induction.
The screw will then move towurdx the cast, that is, the force acting
on the Gccond cnrrent is directed towards the first curr«nt, or, in
general, sinou the phenomcnou dcpvnds only on the relative poniti
of the currents, two iwunUel currents in Uk snnie din?ution att
«ach otber.
la the xame way we may shew that two parallel currents
opposite directions repel one another.
499.] The intcntiity of the magnetic induction at a diiitance
from a stiaighl current of strength i is, as we have shuwn
Art. 479, i
r
Honc», a portion of a second conductor parallel to the first,
oarr^'ing a current i" in the sntne direction, will be attracted town
the first with a form „ . „«
r« 2»ri
r
where m is the length of the portion considered, and r is its dist
from the first conductor.
SiDM the ratio of n t<t r iii a nnmerieal qnantity indepcndedf
the olwolute \'«lue of eittur nf thnto Hues, the piuduct of twt)
currents measured in the elect roniagnetic srstcm most be nf
dioacsuiou of a forve, hence tbe diiiiennions of the unit euncnt are
[0 = [/'»]=[j/u»r].
*
497]
DnnWTlOK OP FORCB OH CTHCmfT
143
^P mS.] Another method of dotormintiig the direction of the force
" whioli acta on a current is to consider tli»> tclfttion of the ning^netic
' action of the current to that of other currents and magnets.
^m If on one aide of the wire which oarrieM Uie current the magnetic
action due to the carrent ie in the same or nearly the same direction
as that due to other currents, then, on tlie other aide of the wire,
tlieM forces will be in opposite or nearly opposite directions, and
the force acting on the wire wili be from the aide on which the
»£jn!ea strengthen each other to the side on which they oppoee each
other.
Thus, if a descending current is placed in a field of magnetic
foroe directed tonards the north, its magnetic action will bo to the
Dortli on the vroit side, and to the south on the east side. Hence
Pthe forces strcn^hcn each other ou the west side and op]>ose each
other on the eiut Hide, and the current will then^forc be acted on
bjr a force from woift to «ii»it. See Fig 22, p. 139.
^H In Pig. XVtl uttlicend of tbie volume the small circle repnsi'iits
^^« Roction of the wire carrying a descending current, and placed
in a uniform fivld of inagnclic force acting towards the leftbuiid
»of the figure. Tlie magnetic force i« greater below the win' than
above it. It will therefore be urged from the bottom towards the
top of the figure.
Hf 487.] If two currents are in the name plane but not parallel,
we may apply this prineii)le. Let one of the oondnctors be an
^^ iDBnit« straight wire in the plane of the paper, supposed horizontal.
^M On the right side of the current the magnetic force acta downwards
^B and on the left side it acta upwards. The same is true of the mag
" tutie force due to any short portion of a second current in tlie same
plane. If the second current is on the right side of the first, the
magnetic forces will stren^^hen each other on its right side and
op)Osc each other on its left side. Hence the second current will
be acted on by u force urging it from ita right side to its left side.
The magnitude of this furce depends only on the position of the
^_ aecond current and not on its direction. If the second current ia
^poo the left side of the fimt it will be urged from left to right.
' Uence, if the second current is in the same direction as the first
it is attracted ; if in the opposite direction it is repelled ; if it 6ow8
at right angles to the first and away from it, it is urged in the
direction of the first current ; and if it flows toward the first eiinenfc,
it ia urged in the dinctioD opposite to that in which the first
11, cnnent flows.
ELECTBOKAOJranC TORCB.
[498.
In coDsiilorin^ the mnttial action or two ourreota it in not ncces
aar} to b4.ar in inini3 t.Iic roUtioiiM betwesD el«otricity and ina^cticm^B
wliicb wclmvv (>n<luuvourvd to iUiutrat«bymeanBot'ariglit~luuidoi^P
Bcrew. Even if wc have forgotten tlieae relations we tbM anivo
at vorrvcl n*itultfi, provided we adhere consiBtenUy to one of the two
posiiiblo fnrni» of the relation.
408.] Let us now bring together the ma^etic phenomena 0:
the electric circuit so far aa we have inveetirrated them.
We may conceive the electric circuit to consist of a voltaie
battery, and a wire connectin^^ its oxtromities, or of a therinoelectrio
urrKngcmcnt, or of a charged Leydcn jar with a wire connecting ita
poNitive and ne^tive coatings, or of any other arrangement for
producing an electric current along a definite path.
The current produces magnetic phenomena in it* Deighbonrbood.
If any closod curve be drawn, and the lineint<rgral of the
magnetic force t^kcu completely round it, then, if the clo«od curva
is not linked with the circuit, the linisintGgml ix I'.ero, but if it
IE linked with the dreuit, so that llwi current > flows through tint
IUImSm bMnw tU alaetiw cnntrat u<l Ika Una ef nanu tic (nducUon ]
oUwwl cam, tlw lut^integal ts 4»i, an«l » positive if the dinwtiaa'
of integrmtion ronnd ihr elowd curre woqU coincide whh tbati
of the hands of a watch u «M>n by a {wnvn pusiiig throogh
in the dirw'twB in which the ^Wtric cnmnt Bow*. To a peiw
muTiag along Ihe clwscd curve ia the direction ^t inUsgntiaa, i
RBCAPmTLATlOS.
145
I
pnsEinf; tlirou^ the elwtric circuits Uie direction of tlie carrcnt
would appear to be th«t of the hands of a watoh. We may oxprvnt
thi» in another wjty by ea,yiag that the relation between thi liiRc
tioD8 of the two closed curves may be expressed by dcsoriliiii^ a
nght*ltund«(l screw round the electric circuit and a righthamlwi
•crew round tlw clo«>ed curve. If the direction of rotation of the
tliniid of either, aa wc pass alon$; it, coincides with the ])08itive
direction in the other, then the )ine>iutc^tal will be positive, and
in th« opposite case it will Iw ncjpitive.
49!).] AWtf. — The liui^iutc^^l 4»t depends solely on the quan
tity of tJtc current, and Dot on any other thing whatever. It
doea not dcjiend on the nature of the conductor throug'h which
the currtrnt in pitsiin^, as, for instance, whether it he a metal
Of an electrolyte, or an imperfvel conductor. Wc have niwon
for b^lievin^ that even when there i» no proper conduction, but
merely a variation of electric diaplacenient, ax in the f^loxs of a
Leyden jar during char^ or diwliurge, the magnetic ofliect of tlie
electric movement is preoiseiy the name.
Af^in, the value of the lineintegral 4;rt docs not depend on
the nature of the mediutn in which tito closed curve is drawn.
It is the ame whether the cloMtd curve is drawn entirely through
air, or pasMis through a magnet, or itofl iron, or any other sub
stance, whether pammagnctio or diamagnetie.
500] When a cinniit ia placed in a magnetic field the mutual
action t>etween the current and the other constituents of the (ield
dc]*ends on the surfaceintegral of the magnetic induction through
my surface bounded by that circuit. If W any given motion of
the cinuit. or of part of it, this surface in (^^■gral iau be irec/rdwrf,
there will be a mechanical force tending to movis the conductor
or the portion of the conductor in the given mitnner.
The kind of motion of the cotiduetor which increases the surfncc
tatc^t^ >B motion of the conductor [>erpondicular to the direction
(^the current and across the lines of induction.
If a parallelogram be drawn, whose sides are parallel and ])ro
portioual to the strength of the current at any point, and to the
magnetic indnetion at the same point, then the force on unit of
length of the eonduetor is nuraericully equal to the arw* of this
HI parallelogrum, and is {terpen diculnr lo it> pliiue, and aL'tn iu the
^1 direction in which the motion of turning the handle of a right
^f handed acrew from the direclii>n of the current to the direction
^ft of the magnetic induction would cause the screw to move.
^M VOL. n. L
us
BLECTSOMAOHBTIC FOfiCE.
[501
Hviice WA have • new eluctromngnetic definitioD of a line
magnetio iaduction. It is that line to which the force on
conductor is aUvays pi^qiendicular.
It may also he dflined as a line along which, if an electric od
be transmitted, the conductor carrying it will experience no foroe.
601.] It mustbeearefiilly rem^mhered, that the mechanical force
which urges a conductor carrying a current across the lines of
magnetic force, acts, Dot on the electrio current, but on the ood
duotor which cjirrics it. If the conductor be a rotating disk or a
fluid it will move in obedieuce to this force, and this motion may
or may not bu ni.'<'om{ianicd with a change of position of the electrio
current which it carric*. But if the current itsEtlf bu tne to choose
any path through a Rxed m>lid conductor or a network of wires,
then, wlirn a eonalant miignetic foive is made to act on the system,
the lutth of the current through the conductors is not pormancntl]
altertid, but after certaiu transient [ihenomcna, called inditetic
currents, have subsided, the diiitribtitii>a of tlie current will be fou
to be the same as if no magnetic force were in action.
The only force which acta on electric currents is eleciromotii
force, which must be distinguished from the mechanical force whic
is the Huhject of this chapter.
rt( u.
thMS riihMsMM Mm
CHAPTER II.
AlirERE's INVESTIGATION OP THE MUTUAL ACTION OP
ELECTKIC CURItENTS.
502] Wa have conBidered in the !aBt chapter the nature of the
magnetic fitld produood hy an electric curreDt> and the mcehaaical
action on a conductor carrying^ an electric current pUoed in a mag
iwtic fifld. From this we went on to couaider the artioa of one
electric circuit upon aDother, by determioin^ the action on the first
diK to the ma^riic field produced hy the second. But the action
of one circuit u]H)n another was oriirinnlly investigated in a direct
maniwr liy Amjidrtf almost immcdialcly afttr the publication of
Orated's discovery. We shall thtTcforu give iin outline of Ampere's
method, resuming the method of this treatise in the uest chapter.
The iden« which guided Ampiirc belong to the eystem which
ftdmita direct action at » distnncc, and we shall find that a ivmark
able course of tpcouhition and invCKtigstion founded on tbo»e ideiis
has been oarried on by Qau^ ^Vcl>er, J. Neumann, Itirmanii,
Bettt, C. Neumann, Lorenz, and other*, with very remarkable
results both in the diecovery of new facts and in the formation of
» theorj of electricity. See Aria, 8(6866.
The ideas which I have attempted to follow oat are those of
action through a medium from one portion to the contiguous
portion. These ideaa were much employed by Faraday, and the
development of them in a mathematical form, and the comparisi.>n of
the results with known fact«, have been my aim in several publitlicd
papers. The comparison, from a philosophical point of view, of the
resalts of two methods so completely opposed in their fmt prin
dplefl must lead to valuabli} data for the study of the conditions
of scientific Bpcctilation.
503.] Ampere's theory of the mutual action of electric currents
is founded on four experimental fucts and ono assumption.
L a
148
AMPfellE'S TBEOBT.
tsH.
Ampftro's fundamfntal experiment are nil of tliem oxamplea of
vlixt has been calltd ihv null method of e(>ni[i«riiij> foroca. Sec
Art. 214. losimd of m<'UKuring Uk^ forec liy th« dynamicml oITm^h
nf commnninling motion to a body, or tlte Rt«tioal method c«^
placing it in equilibrium with the freight of a body or the elaslicitj
of a librv, in the null method two forces, due to the same »ourc4>,
arc nuide to act simultaneously on a body already in e4ui)ibnuro,
iind 110 effect is produced, which shews that these forces are tbem
sclves in equilitiriiim. This method 19 peculiarly valuable for
comparing the etlects of the «loctric current when it passe« tbrouj^h
circuits of diSbreiit forms. By connecting all the conductors
one oontinuous wncw, we ensure that the ftreiigth of the correntl
is the Fame at every point of ita oourHC, and »ince the L*urT«ntj
begins everywhere throughout its conrxc almost at the nme instiint,
wu may prove thitt tho forces due to its aelion on a sujipend^d
body are in etjtiilihiiiim by observing that the body is not at all
affected by the starting or the stopping of the current.
804.] Anii»4re's balance consiBts of it light frame capable of
reTolving about a vertical axis, and carrying a wire which forma
two circuits of equal »r««, in the same plane or in parallel plauea,
in which the current Bows in opposite directions. The object of J
this arrangement is to get rid of the effects of terrestrial nutgDetisiti [
on the conducting wire. When an electric circuit is free to moreJ
it tends to plac« it#elf so n« to embrace the largest possible number
of the lines of indiutioo. If these lines are due to terrestrial
magnetism, this >o«itii>n, for a circuit in a vertical plane, will Im
when the plane nf the einnit is Mst and west, and when tlie
direction of the ctirrcnt is opposed to the apparent coutm of the^J
ily rigidly connecting two circuits of eqnnl am in panillel phines,
in which equal iiirrents mn in opposite directions, a eoinbinutioaj
is formed whieh is nnaffictcd by terrestrial magnetism, and isT
Iherelbre called an Aitatio Combination, see Fig. 26. It is acted
on, however, by forces ariwng fmax owrrents or magnets wtucfa
so near it that they net dilTer^'ntly on the two circuits.
506.] Aup^rs's Grsl ex)>enment is on the vfivcl of two cqinl
Fwrents close lugxlhtT in i^pposite tlin\tions. A win> rovetvd withj
iwutating matetiat is douhlt^d on itwlf, and pUced near one of the'
eitvuits of tlw aslalii' halamv. When a current is made to pun
thr\>u;;h the wirv ami Ute hatsnw, the cqnil>bnum of the Imli
nuuiUDs untlistnttied, ahevrinit that two ninnl ourTVDtaeltM>t(^«tlier]
I
1
:4
507J
TOUE EXPEBIMEKT9.
149
in opfKMite directioDs neuLriiliRo umih otlter. If, uutMd of two
wires tiido br side, a wire he iiwiilul^ i» tlie midillc of a rooUl
tube, and if the curreut paas through tlie wtte and back by the
tube, llifl action oatside the tube i» nnt only approximately but
accurnlcly null. This principle is of great impiirtance in tJie COO
structioD of iWtric apparatus, as it atTordti the means of convoying
ibo current to and from any f^alvanomctor or other iniilrumeDt in
fiiclj a WHy that no clcctroma^ctic effect is proilmcil by the current
on it« pagitagt! to and from the inEtrumcDt. In praclice it is gv&e*
I
FigM.
rally iculTicitnt to bind the wires tojjfetber, care bein^ tak«n that
they are kept perfectly insulated from each other, but where they
mu»t pa« ncM any sensitive part of the apparatus it is better to
make one of the conductors a tube and the other a win; inside H.
See ArL 6S3.
506.] In Ampere'* s»M>nd experiment one of the wires is bent
and crooked with a number of small Kinuosities, but so Uiat in
every part of its courae it remains wry ntar the straif^ht wire.
A current, flowing through the crooked wire and back again
through iJie straight wire, is found to be without iutlu^uce on tbe
■static balaooc. This proves that the effect of the current running
through any crooked part of the wire is equivalent to the same
current running in the straight lino joining its extremities, pro
vided the crooked line is in no part of its course far from the
straight one. Hence any small element of a circuit is equivalent
to two or nion^ component clcmcute, the rclulion brtwo^n tbe
component elcouenu and the resultant clement being tbe same as
tliat between eouipunent and n^sultaiit diK]>Uiccmeiit« or velocities.
fiOT.] In the tliinl exx>eriuienl a conductor capable of moving
IBO
A«P^BR*S TnBORT.
only in thft direction of itx len^h is Rnbslitiiled for tb« astatio
baUiioe. Tlie ctirrent eaten the conductor and leaves it at (iied
points of ipatfc, and it if round thut no clnit^ circuit pla(.<ed in ■,
the neighbourhood U able to moTC the cundactor.
Rj.W.
The conductor in this oxpcrimcnt is a viro in the form of a
ciroolar arc euspvndtd on a fnimo which is c>i]>ab1e of rotation
about n vcrtiwd axis. Tb* circular arc it. hori/.ontnl, atwl it« centre
ioincidw with tho vertical axis, Twio Muall trough* arc filled with ^
nicrcur}' till the convex surlace of the mercury riiceK above t]iS^
level of ihe trouffhs. The tronghB ar^ placed under tlic cirwihir^^
arc and n4^uat(>d till th«> mer<?nry tdiiches the wire, which is of
copper well auui1j^nuite<l. The current is made to enter on« of
ihcM tionj^ha, to IravcrBe the port of th« circnlar an* lietweeo the
troafHu, and to esMpc by the other trough. Thus part of the
circular ant is traversed by tho Rurrfnt, and the arc ia at the now
time capable of moving with eonxidcrmhle freedom in the dlree
tion of its length. Any closed ourrents or ma^^ncts may ttow be
BMwle to approach tho moveable condwlor without producing* the
■Ughlvist tendency to more it in the direction of its length.
soft.] In the l^^urth cxpmment with the astatie hakaea two
rireoita are employed, each joiniUr to one of those in the
but one of them, C. hannit dimensioDS ■ timn greater,
other, J, m time* less. Theee an plan<J oo opposite sides of th«'
oin'uit of the balaoKi, whtch we ehall all B, so that thrr are ^
similaHy pUml with rvtftvi to it, the distaoee of f from S hdlri
UMW tWO^I
balana^H
aod th^
>s or til* V
« tiniM gnatvr than the distauoe of if frow A. Tb« drtedaaa'
J
508.]
151
■treoglh of tiie. <nirT«nt is tbo snmo in A and C. Its direction in
B may be the samo or op]>osil«. I'ndcr the$o (;irciitnstanci>6 it is
foDud that Ji iaiu <^uilibrium under thr action o( A iind C, wbatpver
be ihe forms and diatanoea of Uie throe circuits, provided thoy Iioto
the relations igrivon above.
Since the actions between the complete circuits may be conxiOcrcd
to be dne to actions between the elementH of the oircniU, wt may
use the following: method of determininj^ the law of these actions.
Let J,, 5,, C,, Fig. 28, be corresponding elements of the three
drctiits, and let J^, R,, C^ bo nlso corresponding elements in an
otlicr part of the cirenit«. Then the dttmtion of 5, with respect
to Af is similar to tite situation of C^ with respect to B.f, but the
Sig.se.
distance and dimensions of (7, itnd 1i^ are it times the distitnce and
diroenxions of B^ and A^. roxiH'Ctivoly. If th<> Uw of electromag
netic action is a function of the distance, then the action, what
ever bo it« form or quality, between H^ and A„, may be written
and that between C, and B.^
wbere a. b, e are the strengths of the currents in A, B, C. But
nB^ = Ci, nAj = B^, hBi Ag= C, B^, and a = c. Hence
r^ n*Bi.A,f{nS^)ab,
and this is equal to F by esperiment, so that wo havo
n>/{HAtS,)=f{Aj,):
tkefartt tvrta ineerteiy aa ike aqitttre of IA« dUiaHce.
1S2
AMPKBES THEOBT.
[509.
609.] It may be observed with reference to these experitaents
that ever}' ele^ric curreDt forms a closed ci'miit. Tlie currents
usttl by Ampere, btinf* producc^l by the voltaic buttery, were of
oonrso id closed circuits. It might bo supposed that in the ess*
of the current of dtsihurgv of a conductor by a spark we mif^ht
have X current forming nn open 6nitc line, but ncconling to lb«
news of this book even this caiti; is that of a closed circait. No^f
(.'xiKriments on thi mulual action of unclosed ciimrnts liavo bevg
n1all(^ H<n<'e no sintement about the mutual uelion of two ele
ment* of circuit* can he «aid (o resA on purely exjwrimental gmiinds.!
Tt is true ne may render a portion of a circuit nioreabh, so as to '
ascertjiiii the action of the other currents upon it, but thc«e eur>J
rentH, together with that in the moveable portion, necessarily for
closed circuits, eo that the ultimate result of the experiment is t\\t
action of one or more closed currents upon the whole or a part of 1
closed current.
510.] In the analysis of the phenomena, however, we may re
g»td the act'ion of u closed (.ireiiit on an ilcmcnt of ilaelf or of
another circuit as the resultant of a numl>er of separate forces,
depending on the t>ei>uniti }<urt.ii into which the fir«t ciroutt may
be conceived, for mathematical purpowes, to be divided.
This is a men^y mathematical analysis of tbe action, and ia
therefore perfectly legitimate, whether tbe&e finoea can really ao^H
separately or not^ ^
511.] We shall beRin by considering the purely geometrical
relations between two lines in space representing the circuits, and
betweeu elementary portions of these lines.
Let there be two curves in space in each of which a Bsed point
is lalien, from which the arcs are
mcuMirvd in a defined direction
along the curve. Let J, /f ht
these poiut«. Let PQ and i^^
be elements of Ihe two curves. ^A
Let AP^t. Jp=^/, ) , ■
F%.W.
r=i.\ *"
PQ = rfi, P'Q'
and let the distance PP' be de
noted by r. Let tl»e angle P'PQ be denoted by f. and WQ*
by tf*. and let the angle between the planes of these angles be
denotisl by ij.
The relative position of Ihe two elcroenia i* suffii
their distance r and the three angles 0, jf, and ^
CEOSJETBICAL SPKCIPl CATIONS.
153
given their relative position is «» eonipletely determinnd as if tliey
ForKKHJ part of" th« same rigid body.
512.] If ue use rectangular coordinates and make if, g, t tlie
fioordinatea of P, and jp*, /, / lliose of 7**, and if we deDote by
/, Ml, n and by f, n^, »' the direetioneosines of i*Q, and of P<^
respect ivuly, thin
dx
= 1,
<y
4a
= tii,
7 = »,
U9
= «,
, = «,
■ and /(a^_a) + «(/_,)+i,('_j)= rcos^, .
■ '•(*'') + «'(/^) + «'('^) = '■cos^'. \
^B W + ntts' + tin = coat, *
B whcr« 1 18 tlie angle between the directions of the elements thom
mItm, and
Crts « = — co« 9 COS I?* 1^ sin sin ^ cos 7.
(2)
(3)
0)
(5)
Sr=(''')5"(^^)if(''')i'
= — rcoBtf.
= — reo«(f ;
(6)
rfr
mod diflerrntJuting t^ witti respect to /,
d*T dr dr
^dtd^'^ 4t dt'
did/
dtdi"'
in
djt duf dy dy
did7~d^j7'
= — COB t.
We COD therefore eiprees the three angles 6, 6', and t;, and the
auxiliary un^lv < in terms of the ditferential eoellicients of r with
respect to I and / as follows,
dr
di'
dr
ooefl = —
COB^s —
d/'
coat •= — r
nn (? 6Jn fl* COB 17 =  r
dsd^
d'r
dtdd'
d»d^'
ifi)
154
ahpIese's theort.
613.] AVc Bbnli next coiuiidor id whiit way il i* matlieinntinllf .
eoncdvmblc llutt the vlinutntM PQ nnd P'Q' might uct on «acl!^f
other, and in doing so wt? ulinll not at first XMume that th«ir mutoaf^^
actiiin is nvcf^Hiril} in the lino joining them.
Vfe have seen that we may ttappoae each elenaeot Ksolved into
other elements, proWded that these componente, when combined
according to the rule of addition of vectors, produ<^e the original _
element as their reealtant.
We bhall therefore consider dt as reeolred into ooe Odt = a in tK«1
direction of r, and ein 0Jt = fi
S^
^
r
in a direction peqiendicular to
r in the plane F'Pq.
W« shall «l*o conxidcT d.
the direetioD of r reversed
1
I
FSk 30.
aa reeoWed into ta%& di =a m ttie direetioo ol r
sin 0'coB !)<//= ^ in s direction parallel to that in whieh was
measured, and &in ff'sin i)(^/= / in a direction perpendicular to
«' and j3'.
Let us consider the action 1>etween the components a and ^ oi
the one hand, and o', f(, y' on the other.
(1) a and a'are in the same straight line. The force beiwoeD
them must therefore bo in this lino. Wc shall suppose it to
SD attiractton = j^ao' ii,
where jf ifl a function of r, and i, V arc the intensities of tlM
current* in da am) di{ rcspcetiveljr. This expression satisBes
condition of changing sign with J and with %.
(2) )3 and are parallel to each other and pcrpcndienlar to
line joining them, llie action between them may be written
This force is evidently in the line joining j3 and f(, for it most
be in the plane in which tJiey both lie, and if we were to measure
/3 and 3* '« the reversed direction, the value of this expression
would r«miuii thv same, whiih elicvrs that, if it repreHeots a force,
that foren Itiia no component in the direction of ^, and must tliere
fure be directed along r. Let ua assume that this expression, wbeo
positive, repTesenta au attnu'lion.
(3) j3 and / ore perpendicular to each other and to the lino
joining them. The only action poasihle betwM>n elrtiuntii ho related
\* a couple whose axis is parallel to f. We are al prtBtnt eng
with forces, so we shall leave this oat of accoant.
(4) The action of a and /?*, if they act on each other, most \m~
ignifOi^
not lut^^
exproBsed by
C^0
»i .
FORCES HEnVEES TWO ELEMENTS.
155
Tiiff fdgn of thi> cxprc^ion is ruvcrecd if «c rcverw the direction
in whicli we mvaKura j^. It niii»t tlivrcforc ropresent oitlicr a (one
'io the direction of ff', or n «r>u]>k! in the jilniie of a and ff. As we
are not iDvestignl ing enapUM, vc nliiill tulcc it lu a force acting
on a in the direction of ff.
I There is of course an c<iual force acting on ^ in tbe opposite
direction.
We have for the same reason a force
j Cay it
acting; oa a in the direction of /, and a force
Cfia'ii'
ting on ff in tb« opposite direction.
514.] Collecting our results, we find that the action on d» is
ampoundcd of the following forces,
X = (Aaa'+ B^a^) ii' in the direction of r, )
r= C(oj3'y;3)»i' in thciiirectionof/J,  (9)
£ = Cay'iV in the direction of /. )
■ u» cuppote that this action on d» is the rc«n1tant of three
reee, Si/iitd/ acting in the direction of r, 8i\d*<l/ acting in
direction of ds, nnd Sii'dtda' acting in the direction of dt',
hen in tenox of 0, C, and 7,
R = A cimDcosd'+^sinlJain^cosi}, 1
8 =Cco»fi', 5'=CeoBtf. i
In terms of tlic diScrontial coifiicicnts of r
(10)
^~ ^d.d/^'d^''
S=.C%.
as
00
In temu ot i,m,n, and /', «', m',
= (^ + 5ji(/f+Mii + iiO(/'f4<a'.j + ti'0 + ^(i/' + «w' + «n')/
c!(/'f+«', + i,'f), S'=ci(/f+«i,+«<).
whore f, ij, C arc written for a^—x, y—y, and /s respeetiTely
515.] We have next to calculate the fiirce witli which the finite
current / actit on the Ruitfl current «. Tlie current a extends from
(, whrre *=0, to f, where it hiw the value *. The current /
extendit from A", where «'= 0, to P", where it has the value /.
(12)
166
AMpfeKE'a THEORY.
[516
The (>oonliiiat«B of points on either current are functions of « or
of #'.
If F JB »ny function of th« position of a points then we BhaU n
the stibteript (, „] to denote the eice«« of iU valu« at P over that
at J, thus i',..rt = >V/'^.
SikU fiinciions nec«Bsarily disappear when the cirrtiit is closed.
Let llie conipoDenta of tbe total force with which A'P' acts
AP be i/X, ii'T, and tiZ. Theo the component i>ani)lel to X of
1^
lat
J
the forcD with whidt da' acta on itn will be tt'
Maue
£^=*^ *'+''■■
4*S
dt4/.
Suhetitutin^ the values of S, S, and S" from (12), mnnnbering
that ^ . dr
r^ + m'n+H'C^r
d/'
and arian^g the t«ntis with respect to /, in, 11, we find
lUik'
Siaee A, B, aod C are ftmcUona of r, we may write
P=[(A + B)^d,, q^Tcir,
tbe int«gT«tkHi b«ing tak«B betveeo r and ac becatue A, S,
vanish when r s oo.
Itcnoe
1 iP
and
Iff
I
0)
51 61] Now WB know, by Amp^'s third otae of equiUbriuin, that
when / is a ckwed circuit, the fom aotinfj: on Jt is perpradicular
to tlM diroetiaa of d*. or, in other worda. the oomponent of the fi>re»
ia tbe direetiM of dt itself is zem. Let as therefore aasome tht
direction of theaxi» of x so as to be {larallel to dt by making' f= I,
« M 0, a s 0. Equation ( I i) thm bvcomn
To Gad . , tbe focw on dt nbtnd to mut of tength, wc muiL
I.
[gi7.] icnoF or a closed cntcmr os as elemebt.
I
I
tal«{>rate this exprf«sion with ri?«]wct to /, Jnttgratiag the first
ternj by farta, we find
^:^=iPeQh.0Jj^P''^c)^^^. (19)
"Wlien y U » cloaed circuit this eiprossion must be zero. The
first term will disappear of itself. Th« Rcond tprm. however, will
not in general liiaappenr iu the case of a clowd circuit nnleae the
quantity under the taixn of intcfjrntion is alwny« zero. Hence, to
satisfy Ampere's condition, wc must put
P = rr^^+C). (20)
617.] We can now eliminate P, and 6nd the genoial value of
dX ( S + Cf ... ,, )
■ ml 
~c m'S~rn
d/
.r?
crcM's
d*'. (21)
2 r .'o 2 r
When »' is a closed circuit the first term of this espression
vaniuhes, and if we make
2 r
c rcn'(
2
d/,
d^.
d/,
(22)
2 r
' where the integration !a extended round the closed circuit /, we
I nay write jx
^=1/.^.!
Similarly
dV
d*
= »a'//,
^f=.^«a'.J
(23)
The qiiantiti^ a, ^, •/ are somctimos called the di'tvrmSnanls of
(1m! oirciiit *' referred to the point P. Their ru«iiIUint i» called l>y
Ampiirc the directrix of the eleetrodynamic action.
It id f^vidcnt' from the eqnation, that the force whose components
are ^i y, and j is perpendicular both to dt and to this
dircHrix, and is repri'4mt«d nun>cri«ally by the area of the parallel
, c^niffl wboJie sides are da and the directrix.
AVPERRS THEORT.
[518
In th« langujige of qnaternioiui, the resultant force on i/« is tlie
vector juart of the product of the directrix multiplied by dt.
Since we already know that the directrix is the eame thing
the nmgtietic force due to a uoit carreot id the circuit ^, we :
b«nceforth eiieak of the directrix as the nu^eUc force due to th
circuit.
518.] We diall now complete the cuIcuUtion of the compoaeati
of the force acting botncon two finite currcuttt, whether clo«d
open.
Let f) be a new function of r, such that
p=ij'\BC)4r,
tbeDbjr(l7)MKl(20)
And equjitJou (1 1) become
i»
JE = J««c+r~,«24..),
S's
dt
With thcee valaes of the component forces, equation (13) becomes
rf»X
dpi
^Q^r'fQ
_^ = co..^j+f35,(Q+.W^+r^.
m 00*
519.] Let
F^f'ipdt,
•i'^^^'^'S
m I'm pdt,
cr= rm'pd/.
ff=f\pd4,
w=jyp4i.
(29)
T^ptp qnaatitiM ban definite mines for any given point oT <mm.
Wti«ii live nrcuite are etowd, tbey cofw s pond to the oompotten to of
the Yoctorpotcntials of tbo cinuits.
Let £ be a new ftumtMn of r, fuch that
i=rr(«+,)*.
ami let Jtf be tli« «]«ubl« int«^ntt
/"T
fOMtdtif,
[52*]
POBCB BETWEBS TWO niHTE nTBRESTg.
wfaicli, wh«n the oircaita are closed, beootnes tlieir mutual pot«ntia],
. then (27) may be written
520.] Integrating, with respect to i aiul /, bvtwcoo the gives
limittf, wo find
IX= j^ — ^{Ipr—ZAi'—L^'e + J'Ajr),
^ +n~FUF^+FA,, (33)
whi>T« 1h« suWripts of L iuditalc the di«tsncP, r, of tvbioh tlio
quantity £ is a function, and tlxt etibscrijitv of F and /" indicate
ith« pointH at which their values are to be taken.
Tlie expressions for Y and Z may be written down from thi*.
Mnltiplying the tlirce conij>oiientfi by i/x, dy, and di respectively,
we obtain
Xdr+Yds + Zd: = J)MI){LrrLjpruf+L^j),
^ {Fdx + Gdy + IId.'\^^A^, (34)
wncre D U the eymbol of a complct* ditlbrcntial.
Since Fdx tGdy i^ Udz is not in general a complete differential of
afttnction a{x,y, z, Jrf»+ Ydy + Zds is not a complete differential
for currents cither of which is not closed.
521.] If, however, both currents are closed, the temu in L, F,
.. 0, H, r, G', 11' disappear, and
■ X<ix+ Ydff + Zrh = DM, (35)
^ where M is the mtitnal potential of two closed circuits carrying unit
currents. The quantity M expresses the work done by the electro
magnetic forces on cither conducting circuit when it is moved
parallel to itself from an infinite distance to its actual position. Any
alteration of its pcMttion, by which M \» increased, will lie axtitted by
the electromagnetic forces.
jB It may be shewn, as in Arts, 490, 596, that when the motion of
^"thfl circuit is not pamtlel to itself the forc(« acting on it are still
<let«rmin«d by the variation of M, the potential of the one circuit on
tfae other.
522.] The only cxprimental fact which we have made use of
in Uiis investigation is the fact cstaliUshcd by Am[)ftre that the
action of a closed current on any portion of another current is
perpcndicnUr to the direction of the latter, Every other part of
160
AKPfcRK's THEORT.
tbe invnti^tion depends on purely mathematical considoratione
depending on the properties of lines in space. The retuonin? there
fore limy be presented in a much more condensed snd apprupriAte
furm by the use of the Ideas and Inngnage of tbe mathematial
mctliod speciiiUy adapted to the cspression of such geometrical
relations — the QHatemwm of Hamilton.
This liaa been done by Professor Tait in the Qaart^fy Mati«_
KUtHeal /ournat, 1860, and in his treatise on Qaairrnioiu, ^ 399, fyi
Ampere's original investigation, and the titudent can t^ly adap
the same method to tbe wmenliat mora gvnerul investigation givt
hero.
623.] Hitherto vt« have madi no ajsnmption with respect to tl
qiiantiticK A, It, C, excejil that they are functions of r, the distAoe
between the elements. We have next to ascertain the form
the«e fuuetions, and for this purpose we make use of Amp&re'a"
fourth case of equilibrium. Art, 508, in which it b shewn that if
all the linear dimensions and dtstanoes of « s^'stem of two circoitt
be altered in the same proportion, tbe ctirrente remaining th« suoh,
the force between the two circuits will remain the same.
Now the force between the circoita for unit cnrrents is j— ,
MB
nns^l
Binee this i« independent of the dimennons of the system, it m
be a numerical quantity. Hciiee M itself, tlie coefficient of the
mutual potential of the eirciiitfi. mwil be a quantity of tbe dim
•iouB of a line. It follows, from equation (31), tJtat p must )>c
reeiprocal of a line, and therefore by (24), ii— Cmnst be the invei
square of a line. But since B and C are both functions of r, B—
must be tbe inverse square of r or some numerical multiple of it.
524.] The multiple we adopt depends on our system of meadure
ment. If we adopt the electromagnetic system, so called hecauae
it agrees with the system already established for magnetic measore
menls, tbe value of J/ ought to coincide with that of tbe potential
of two magnetic m1mi11« of strength unity whose boundaries are tbe
two circuit* respectively. The value of if in that case is, by
A'^*". M=(j^d,4r, (36)
the integration Uting {lerforined round both circuits in the positive
direction. Adopting this as the numcrieal value of if, and
paring with (31), we find
1
and
BC=p,.
(37)'
J
t525l
AMPBBraPOBMUtA.
525.] We may now express tlie components of the force on ti»
^tnsias fpotn the action of r// in the most general form consistent
^with oxpcrimcntiil Satds.
The force on ds is compounded of an Attraction
the direction of r.
8=: — ^ i'/iUd/ ill the dircctJoii of 44,
tnd S'^
irhere Q
: —Wdfil/ in the direction of <Ai',
m
f:
Cdr, and since C is an unknown function of r, we
luiow only that Q is some function of r.
536.] The quantity Q cannot he determined, without osEtitnp
tions of some kind, fruro exjieriments in which the active current
Ibnns a cloited eirciiit. If we suppose with Ampere that tho action
en the cU'menta d» ami d* is in the liuf joining them, then
<9' must diaapjii'ar, and Q must be coiistaut, or Ecro. Tho
ree is thon ixduced to an attraction whose value is
^i^dsd^. (39)
^^\d»d? ^'
Ampere, who made this investigation long before the magnetic
tryvtem of units had hvt.n estublishcd, uees ti formula having a
nnmerictil value half of tiiis, naniidy
Here the strength of the current is meaitured in what is colled
fleetrodrnamic measure. If i, t'art the strength)! of the currents in
leetro magnetic measure. and_;'.y the same in olcCtrodj'DUmic meu
ore, then it is pl&in that
y/=2ir, or >=^/2;. (41)
Hence the nnit current adopted in electromngnetic measure b
greater than that adopted in electrodynamic muisurc in the ratio
of v'S to 1.
The only title of tho elcctrodynamic unit to consideration is
that it was originally wlopted by .\mpiire, \\w dixcovervr of th«
law of action between eitrnnts, Tlic continual recurrence of •/^
in calculation* foundcil on it is inconvniicut, and the electro
magnetic system has W\e great udvuutugeof coioeidiug numerically
VOL. ri. u
AUPESSS TRBORT.
[527.
villi all our na^«lic rormiiloe. As it is tlifficnlt for the stadetil
to bear in mind whether he is to multiply or to divide by \^2, we
ehall hf^nccforth uso only the ekctromagnetic system, as adopted b
Weber nnd most other writers.
Since thi Torm itnd vhIuu of Q have do effect 00 any of the
eicperimente hithorto made, in which the active current at least
is always a cIokviI one, we muy, if wc please, adopt any value of
wbicli aipc3r> to us to simplify the formulae.
Tlius Ampere u^iMiiDieE that the force between two elementa
the lin« joining llum. Tliia gives Q = 0,
Gmssmann * sssumes that two elements in the nm« straight lini
have no mutual action. This givc«
laSt
I
i .=.
3 dh
^A£<^^j
We might, if we pleased, oienime that the attraction between t'
elements at a given diatanee is proportional to the ooeine of
angle between them. In thin case
., J?=^eo9.. S = ;i^. S = ^^
Finally, wc mig^it assume that tlte attraction and the obli
forces depend only on tbe angles which the elements make with the
line joining them, and then we should have
«=r'
t. S = 3
I dr rfr
S 2^^
^*^il^'
*=^£ (»:
527.] Of tltme four diffrn'ttt a»stimption« that of Aii>{>^ro ii
andoubtnlly the l*rst, since it is the only one wbioh makvs tli
forom on the two elements not only txjual and oppoaiW but in tli
•tnight line nhieh joins them.
* P<«B.. Mm. lxl>. p. 1 (IMS).
ClfAI^ER III.
THE INDCCnON OF BLECTBIC CCBBEKTS.
5S8.] Thi: <li»ci>v(>ry by Onttt^ of the magnetic notion of an
electric cnrrent led by a direot prixesa of ieasoninjj to tliat of
ma^DetizatioD by elwrtric ourrents, and of the mechanical action
between electric currents. It wo* not, however, till 1831 that
Famday, who had been for some lime endeavouring to produce
electric currents by magnetic or electric action, discovered the con
ditions of mngoetoeleciric induction. The metbod which Faraday
employed in hia researches consisted in a constant appeal to ci
imcnt as a means of testing the truth of his ideas, and a constant
iltiration of ideas under the direct influence of experimenl. In
his pnblished researehea we find these ideas expressed in language
which is all the better fitted for a nascent science, because it is
■oni«whnt alien from the style of physicists who have been aocus
tomvd to cstAblislied mathematical forms of tliought.
, The trxpcrimmtal investiffation by which AmpJrre establivhnl the
laws of the mechanical action between electric curn'ots is one of
the most brilliant achievements in science.
The whole, theory and experiment, seems as if it hJld leaped,
full grown and full armed, from the hr:iin of the ' Newton of elec
tricity.' It is perfect in form, and nnassailable in accunu^y, and
it is eammed up in a formula from which all the phenomena may
bo deduced, and which must always remain the cardinal formula of
el ectrodynam ica,
Till method of Amp^^e, however, though cast into an iiultictive
form, does not allow ns to trace the formation of the idotx which
guided Urn "We can scnroely bvliere thai Ampdre mlly diiicovered
tin' law of action by means of the experiments which he describes.
We ore led to susp4'G4i what, indttd, ho tells uk liimself •, that he
• TiMu dt§ PJIoMmAMi Bntndynamlfutt, p. 0.
K Z
Vki
[5»g
I
(liitcoTcred Ike law by aomc ])rooe68 which be bsB not shewo uii,
And that when bo had afVerwanls bniU up a perrcct lUmoD
•tntion he removed all tntces of the ecafTolding by which h«
raiaed it.
Faraday, on the other band, shews ua his nnsacoesaful aa^
aa Ilia Hiocewtibl experiotenta, and his erude ideas as well as liii
developed ones, and the reader, however inferior to him in inductive
(lower, feela sympatliy even more than admiration, and is tempted
to believe that, if be had the opportunity, he too would be a dis
coverer. Every student therefore should read AmpJrcB rwearch
as a splendid esnmplo of scientific style in the statement of a di
covery, but he should also study Faraday for the cultivation of
scientific spirit, by means of the action and reaction which
tnko plaoo between the newly discovered facts as introduced to him
by Faraday and the nnscent ideas in his own mind.
It was perhaps for the advantage of science that Faraday, though
thoroughly con«cioiis of tltc riindnmental forms of spaccj time, and
force, wiu not a profesited mntliematiciim. Ho was not tcmpt«d
to enter into the many iatcre»tinf> researdies in pure motbematM*
which his discoverifs would have suggested if tlicy had
exhibited in a mathematienl form, and he did not feel eallcd u
eillier to rorc« hi» results into a Khaie acce>tible to the null
matical taste of the time, or to express Ihem in a form wb
matbematiciAns might attack. He was thus left at letsore
do his projKr worV, to coonlinate his idea« with bia facta,
cxprttw them in natural, untevhnical langnagek
It is mainly with the hope of makini; thete ideas the basts
matbentaticiil mvtht^l that I have uudertakea this treatise.
52f>.] We are uceuKtomed to consider the universe as made up of
parts, and matbemalieiann ukudIIv he^in bv eonsiderin^ a si
particle, and then conceiving iU ivbtion to another particle, and
on. This has generally been sup)>cMctl the most natural melb
To eoneeivc of a particle, however, nHjnirex a pn>ct»i of absttactii
nnoe all our percepltuns ore rebted U> ejitt^ndoil bodies, so
tlw Mm of the 41// tluit is ra our HmacioudneM) at a given
is perhaps as primitive an idea as that of any indi\idual
Henee there may be a utatlieinatical method in which we pri<e
from the whoht to the parts itistltad of from Ihe patt« to the who'
For example, Kwlid, in his first hook, oonMives a liiw as traevd
out by a p^int, a fMrfiK.'e as swept out by a line, and a volid m
generated by a suifacr. Dut he alwi defines a snr&oi
530]
BI.BMENTABY niENOMHNA.
165
boundary of a aaMi, * line m the iHlg« vf a KurfaiM, and a point
I tbe extremity of a line.
In like manner we may coneeive th« potential of a material
■ystem as a function found by a certain prooe^ of intcg^ration with
Lreepevt to th« maaees of the bodies in the tield, or we may Hup^Kiw
lliMO masses themselves to have do other mathematical meaning
Ihan Ute volnmeintOrntlK of — V**, where * is the potential.
In electrical investigations we may use formulae in which the
]aantit!c« involved arc the dielauces of certain bodies, and the
^electrifications or currents in these bodies, or we may use formulae
which involve otlior quantities, each of which is continuous through
all epoco.
The mathematical process employed in tJie first method is in
tegration along lines, over snrfaees, and throughout finite spaeea,
thow employed in the aceond methoi:l are partial dilTcroutial equa<
_tiona and integrations throughout all Ktinoe.
The method of Faraday scema to be intimately related to the
ad of these modes of treatment. lie never oonsidera bodies
existing with nothing between them but their distauoe, and
acting on one another according to some function of that distance.
He eonccivc* all spiicc ns a field of force, the lines of force being
io general curved, arnl those due to nny body extending from it on
all sides, their dirretionn being modified by the presence of other
bodies. He even m{ks1>'h * of the linen of force belonging to a body
M in »ORM> tteiiMt part of itiielf, sen that in iU action on distant
bodies it cannot be ^id to act where it is not. Tliiii, however,
not ■ dominant idea with Faiaday. 1 think he would rather
bavc mid that the field of space is full of lines of force, whotv
I arrangement depends on that of the bodies in the field, aud that
taie mechanical and electrical action on each boily is determined by
tiona 1
^^neconc
^98 exi
^tia
PHBXOUEXl. OF MAOKIITOBLIXTRIC IKDUCTION t<
580.] 1. Muciioit hi/ Farialion of the Primary Ciirrmt.
Let there he two conducting circuits, the Primary and the
idary circuit. The primarj* circuit is connected with a voltai«
• iSip. A>., ii p. 903 ; iii. p. 417
t Bud Pandky'* JSK/nrimatal Jtofordtn, mtim I ud IL
160
MAGSETOELECTEIC HTOUOTIOS.
[53c
bntlety hy wliich thn primaiy carrent may be produced, tiiamtauiL><l,
Ht«{>]wd, or Tovoraed. The wcondaiy circuit includea a fpil^no
m«Uir to indicate any ciiri«ntd which msy W formed in it. This
galviinoRi«UT iR placed at such a dietance from all jtarts ot the
I>riinury circuit that the primaty current hs8 no sensible direct
iiitliwnw OH ita indications.
Let part of the primary circiiit consist of a stTaifjlit wire, and
part of the secotidury circuit of n strai(;ht wire near, and pnmUel to
the first', the otlicr parts of Ibu circuit* bein^ at a ^^Jitor difitanoc
from each other.
It is found that at the instant of Miidin); a current through
the straiffht wire of the primary circuit the galii'aDonHlcr of the
secondary circuit imlicut^Jt ■ ciirreut in the fivondary straight wire
iu the (tppotUe direction. Hiis in culled Uie induced current. If
the primary cnrrenl '\* maintained L'onatant, the induced current soon
ditapjieam, and tlio primary current appears to produce no elTeet
on the Micoiiilary eiivuiu If now the primary current is stopped,
a aeoondary current is observed, which ta in the an>« direction as
the primary cnrreut Every variation of the primary currvnt
produecA electromotive force in the seoondary circuit. When the
primary current inerawco, the electromotive foroe is in tite opposite
direction to the current. When it diminishes, the electromotive
force is in the same direction as the current. When the primaiy
aurrent is conttunt, there is do electromotive force.
IImmo effectn of indudion are inereasvd by brini^ing tlte two wires
ncanr ton^llier. They are alw iocrvoeed by forming them into
two oirouUr or spiral coils placed doeu together, and still more by
placing an iron rod or a buntUe of iron wires inside tfaa ooilsi.
3. Imimetiom if Motam V Ur Primaty CinaU.
We ba^e seen that wbm the primary enrrent is maiBtained
vonatant and at rest tbe aeeotkdary •.urrmt rapidly disappears.
Now let tbe primary cum>nt be maintained constaiDt, bat let tbe
primary' slrnight nire be made tin approach tbe seeoadary ittaigfat
wire. IHiriiig the approach there will be a smiodaiT comnt in
tbe vfpMtt* dit«vlii4i tkvm tbe pfiiaary.
If tbe >riuiary eiivall bo moved away from the svcoDdaiT', tbd*
will be a secondary ourrvnt in tbe mmr dirvetiua as tbe prnnarv
If tbe senudary eimni W mvn^ tba saooadb^ ettmat m
531.]
ELKUBXTABT PHKNOUE.V^.
167
opposite to tha priinar3r when the E««ondar)' wire U approKohin^
the primary wire, nad in the giimu (lircctiou when it is rocodiag
Ktrora it.
In nil cofca the dircctioii of the sccondury current is ttiie)) that
the mechanical action between the two conductors is oppoKite to
the direction of motion, iming u repiilHion when the wtre« are ajt
prouehintf, and an attmctiou when tliey nro receding. This wry
importBiil fact vmit eetJihlixhed by Leiix*.
I
4. JnduelioH by tie Behlire Motion <^ a Magnet and Ike Stcondary
Circuit.
If we inih«t1itute for the primary ctrcait a ma^etic shell, whose
ke(]g« coincides with the circuit, nhose strength is Qumfricutly c([uU
ho that of the current in the circuit, and nlioae auHtml fuio oor
KspoMlM \o the positive face of the circuit, then the phpnomena
produced hy the relative motion of this shell and the gecoiidary
circuit are the same as those observed in the cose of the primary'
eirouit.
531.] The whole of these phenomena may be summeil up in one
>iw. When the nnmher of lines of magnetic induction which puss
fUiroug'h the seoondary circuit in the positive direction is altered,
in el<'elrnmotive force aet« round the circuit, which ix measured
]ty the rate iX decreikse of the magnetic induction throu{>^h tb«
^rcuit.
532.] For inalanee, let the raits of a railway tn* insulated from
^hc earth, hut connected at one terniitius tlu'ough s galvanometer,
id let the circuit he oampleteil by the wheels and axle of a rail
ray carriage at a distance x from the terminus. Negleiting the
eight of the axle above the level of the rails, the induction
lirough the secondary circuit is due to the vertical component of
the earth's magnetic Ibrcc. which in northern latitudes is directed
lownwards. Hence, if i is the Bffluge of the railway, the horizontal
area of the circuit is bx, and the surfaceintegral of the mugnetia
iaductioD through it is Zbx, where Z is the vertical component of
the magnetic force of the earth. Since Z i« downwanls. the tower
face of the circuit is to be reckoned positive, nnd the positive
direction of the circuit it*clf \t> north, vnA, iwuth, wcstv thai in, in
direction of the sun's apparent diurnal mur^e.
S'ow let tlie carriage be set in motion, then x will vary, and
" P«g., Am. uL 403 <1834)l
168
MAGNETOELKCTRIO ISDUCTIOIT.
tiiere tvill be an electromotiTe force in tlie circuit whose valoe
u
<■
n
If « is increa^Dg, that is, if the earring \» movingr away from
the tcnninufl, this electromotive force is in the negative direetinn,
or north, weet, south, east. Hence the direction of this force
through the ax]e ia froro ri^ht to left. If x were diminishing;, tJte
absolute direction of the force would be reversed, but since the
direction of the motion of the carriage is also reversed, the eleotto
motivc force on the axle is still from right to leR, the observer
io thff carriage being always supposed to move face forwards. In
southern Intitudcs, where the south end of the needle dips, the
elcctromotivu force on a moving body is from left to right.
Hence we have the following rule for determining the eleottx>
motivc force on n wire moving through a field of magnetic force,
Ftaoc, in imagination, your head and feet in tbe position occupied
\>y Uie emlH of a coropuss needle which point north and south respec
tively ; turn yrmr fuet in the fonvard direction of motion, tlicn tho
electromotive I'oree due to the motion will be from left to right*
S33.] As thnra dircctionu) ivlatioits are important, let us tike
another illuKtrntion. Suppose s metal ginllc laid round the earth
at thu «utttor, and n mitiil wire
laid along the meridian of Clrccn
wich from the eijtwtor to tlic north
pole.
Let a gmt quadrantnl arch of
metal be constructed, of which one
extremity is pivoted on tlio north
pole, while the other is durieJ round
tho equator, sliding on tJio great
girdle of the earth, and following
the sun in his duily ooiir»e. Tht^re
will then be an electromotive fc
along the moving qnadranl, actio,
fn<ni the pole towards the equator.
The electromotive ror<« will be the saaie whether we Eoppooe
the earlh at re*it and the ijuadnint niovpd from cant to we§t, or
whetWr we «ippo»c llie iiuadnml ul rnst and the earth tnmed from
west to east. If we suppose the earth to rotnte, the elcctromotiv
force will be tlie same whatovcr \\v the form uf the part of lb
circuit fixed in space of which 004 end touches od0 of Iho po
here «j
roTo4^
tio^
tor. ^
5S^1
EXPEKIMESTS OP 0OMPABI8OS.
169
Band
^^BDi
the other the equator. The cnrront in this {Hirt of tho circuit
is from the pole to the equator.
The other part of the circuit, which is (iscd with respect ta the
eartli, may abo he of noy form, snd eithtr withiu or uithoiit the
earth. In this part the current ib from thf iquntor bo cither pole.
534.] The int^neityof the cktitromntivc forco of maffnctocli^ctrio
iutlactioD is entirely imkpeudi'nt of the nntiirv of tli4i KiiliKlance
of th« ooDiiuctor in which it acts, nnil also of tlie nature of the
condiKtor which carries tho indiicin^^ current.
To shew this> Faradsy * made a conductor of two wircK of different
letals in«nlat«d from one another hy a silk vovurinj'', but twiiftiMl
together, and solileied together at one end. Tlic other cods of the
wirvs were connected with a ffalvanomctiT. In this way the wires
were similariy situated with rcspiet to the primary circuit, hut if
the clet'trwmotirc ibrco were ptronffcr in the one wire than in tho
other it would produce u current wliich woidd he indicated hy the
gialra no meter. He fonnd, howcvir, tluit *uch a eomhination may
be ex>osed to th« most powerful elwtromotive forces due to in
ducliiin without tha galvauomcter Wing iilfccted. He also found
that whether the two branches of the eomjiound eonductior conMsted
of two metals, or of a niotal and an electrolyte, the galvanometer
,»«• not aflVcted t.
Hence the clectromotiTe force on any conductor deiend8 only on
tlie fonn and tlie motion of that conductor, together with the
Htrength, form, and niotioit of the electric currentn in the field.
r»3o.] Another negative property of electromotive force i» that
ha» of itxelf no tendency to caui«e the niccliauioal motion of any
ly, hut only to cause a cunynt of electricity within it.
If it aetnaJly produces a current in the body, there will be
mechanical action due to that current, but if we prevent the
current from beio^ formed, there will be no mechanical action on
e body itself. If the body is electrified, however, the electro
JTO force will move the body, as we have described in Electro
statics.
bSQ.] Tlie experimental investif^tionof the laws of the induction
of electric currents in fixed circuits may be conducted with
oonaiderabic accuracy by methods in which the electromotive force,
and therefore the current, in the ffalvanometcr circuit is rendered
zero.
instance, if we wish to shew that the induelioit of the coil
1
mec
cun
r^ Ua.. 19S.
t W>., WO.
170
MAGXETOSLECTRIC ISDUCTION.
A on the «oil X is equal to tliat of B upon T, we place the fintl
]«iir of coiU A and X at a sufficknt distance from the seeoDd pair
fig. 3S.
^
B and Y. We then coniwct A and 5 wHb a voIt*ic Taltwry, ao
that W(! CAD make the E&nu: primary current flow through A in liie
powitivL direction nnd thi'D tlii'oii{;h B in th*. mjpitivt: diri.Htion.
We aim connect X and }'willi ngalvanomilci', ho tluit tl^; Mvondary
onrrvDt, if it eitsts, ehsll flow in the >umc direction tkrxiugb Xaad
1' in series.
Then, if tho induction of A on A' \* equal lo Uiat of .ff on }',
the f^vani>nirt^M \\\\\ tiidkatj^ no taductioD current whin tl
batlprjf circuit it) «Ui«cd or bniljMi.
The accunKy of this method iiicn.'ai>e« with tlie ftren^tli of the
primnry furnMit and tbi' nennilivcneja of the gmlvanomcter to iij
stunlaneous currcuts, and the ex>eriin<:utN are much ninrv easily
pcrfomied thau those relating to elect roma^tetic attrnction*, wltero
tho conduflor ilKrlf haa to ke delicatelv nuRpendid.
A very instructive series of welldevivcd esperimenta of this kind
is de0eril(id by Professor l'elici of W.^a '.
I shall only indicate brielly some of tlic Uws vi tiicb may be proratl
in this way. ^^
(1) The electromotive force of the loduotioa of one eircuit on^
auothcr is ittdcpendent of tlw area of the section of the conductora
and of the malcrial of nliioli tht^y nrc made. ^l
For wc cjin excbangv any one of tlic circuits id the experimental
for aiiotlHT of a difTcrent section and roalcnal, but of the same fnrni,
without altering the resall.
• Aa*ak* it CUmU, xxiir. p. M (t8»), and Kmtn OfaiinMi ix. p. 34i (IfvV).
S37]
rBLICIS EXPERIUEN'TS.
171
I
I
I
»
(2) The indudion of the cireait A on the circuit X is equal to
that of X upon A.
For if we (mt .1 in llu cr»IvaDOinet«r cireuit. and X in the battery
circuit, Itie 4^(11 iijlirium of flovtromotiri.' force is not disturbed.
(3) Tlte indu«tiuo is propnrtional to the inducing current.
For if we Have aswrtaincd t}i;it the induction of J on X is cqnfti
to that of H on }', and nliio to thiit of C on Z, wo mny make the
battery current Unt How through .4, and then divide ittrvlf in unjr
proportion bLttveeu /i and C. Tiit^n if viv. connect A' rcvcnted, Y
and /direct, nil in «erie«, nith the galvanometer, tUo cli;ctroniotive
force in X will balance the sum of the electromotive forces in Y
and J?.
(4) In pairs of ciretiitit forming systems geometrically niinilur
the iodoctioD is proportional to their linear diuienaiona.
For if the three pairs of circuits above mentioned are all itimilnr,
but if the linear dimension of the first pair is the mim of th«
oorrespondiot; linear dimensions of the second and third pairs, then,
if A, £, and C are connected in series with the battery, and .V
reversed, faud H are in series with the galvanometer, there will
be equilibrium.
(5) ITms electromotive force produced in a ooil of n windings by
• current in a ooil of m windingrs is proportional to the product mn.
637.] For ezi^riments of the kind we have been considering the
galvanometer should be as sensitive as possible, and its needle as
lif^t as poasibte, so as to give a sensible indication of a very
small transient current. The experiments on induction due to
motion require the needle to have a somewhat longer ]i«r!o(l of
vibration, so that there may be time to efTccl certain motions
of the condnctors while the needle is not far from ilK position
of Ci(nilibrium. la the former expirimoiit*, the eloctromoUvo
forcM in the galvanometer circuit wore in equilibrium during
the whole time, so that no currtnt panwd tliroufjh tho gnlvano
meter coil. In those now to be described, the elctrnmotive forces
act lintt in one direction and then in the other, so as to produce
in Kucccjwion two currents in oppnditc directions through the ^1
vanonietvr, and we have U) shew Unit the iuipulBes on the galvano
meter needle due to these successive currents are in ecTtaiu cases
equal and opposite.
llie theory of the application of the galvanometer to the
mcoMaremcnt of trannient currents will be considered more at
IcDgith in Art. 718. At prvveDb it a sufficient for oar purpoi>« to
172
MAQSETOELECTBIC INDVCTJOS.
[53S.
4
observe that as lon^ se the galvanometer needle is Dear its position
of oquilibrinm the deflecting force of the current is proportiooal
to the current itself, and if the whole time of action of tlie current
is small compared with the period of vibration of the needle, the
final velocity of the magnet vUl be proportional t« th« total
quantity of electricity In tlie current. Hence, if two currents pass
in rapid succession, conveyinq; equal quantities of electricity in
opposite directions, the oeoillc will be left without any final
velocity.
Tlius, to shew Mint tive inductioncurrents in the secondary circuit
due to the closing nni] the breaking of the primary circuit. »re
eqiiul in totiil ijimTitily hul' oppoxiU' in direction, wc may arriinjpt
the priniuiy cin,'uit in connexion with the battery, »o tliat by
touching tt ]n}y the current may be m.'ut through the primary circuit,
or by ruitioviiig the finger tito contact may be broken at pleasure.
If the key id preaiscd down for some time, the ^Ivanometcr in
the Kccondary circuit indicates, at the time of making contact, a
tr.intiient current in the opponitt? direction to the primary current.
Ifcaiitatt be maintained, the induction current simply pai^ses and
disappears. If wc now break contact, another transient current
passes in the opi>oaite direction through the eeoondary circuit,
and the galvanometer needle receives an impulse in the opfioaite
direction.
But if we make contact only for an instant, and then break
contact, the two induced currents pass through the gnlvnnomete^H
in Bueh rapid succession that the needle, when acted on by the firs^H
current, has not time tt> move a sensilile distance from its positioo
of equilibrium before it is stopped by the second, and, on account
of the exact equality between the quantities of these transient^
cnrrents, the needle is stopped dead.
If the needle is watcheil carefully, it appears bo ha jerked suddc
froin one position of rest to another )io«ition of rest very
the first.
Ill this way wo prove that the quantity of electricity in th^
induction current, when contact is broken, is exactly iqiinl an
opposite to that in the induction current when oonlaet is ina<lK.
538.] .\iiotlier application of (hisnuthcNl in the following, wUk
is given by Fclici in the second veries of his lle*ftreh/Ji.
II is always possible lo find mnny dilTervDl positions of U
secondary coil it, such tliat the making or the breaking of contoi
in the primary coil A produces no induction currvnl in li. 'It
539]
173
I
positions of tbo two coils arc in such cases said to bo em^Kgal* to
cnch othor.
Let Sj asd S, be two of these positions. If the eoil B be sud
denly moved from the position ^ to the position B.j, Uk al^braicnl
sum of the transient currents iti ttio eoil fl is exactly zero, eo
that the galvanometer needle is left at rest whuii the motion of £ is
completed.
This is true in whatever way the coil Ji is mov«d from B, tn B.,,
and also nhcthcr the current in the pnmiiry coil A bo continued
Iconstant, or mmlo to vary during the motion.
A^in, let ]f \k any other pofition of B not ooujiigati* to A^
BO that the makiiif* or brvakin^ of eonfjict in A produces an in
duction ctirront when B a in thi. position 7f .
H Let the contact be mnile whtn B is in the eonjiigale [wsition j?,,
, there will l)c no induction current. Movp B Ut B", theio will be
an induetJon current due to the motion, hut if U is moved nipidly
^rto B^. and the primary contact then broken, the induction cu^c^ut
■ due to breaking con1iu;t will exactly nnnnl the viFeet of that due to
the motii'ii, ho that Uie galvanometer needle will be \vh at rest.
» Hence Ihe current due to the motion from a conjugate position
to any other position iii ci)ua1 and opposite to the current due to
breaking contact in the latter position.
Since ihe effect of making cootact is equal and opposite to that
of breaking it, it follows that the effect of making rontact when the
coil B is in any position J? is equal to that of bringing the coil
from any conjugat« position Bi lo £" while the current is flowing
through A.
If the change of the relative position of the coile Is made by
moving the primary circuit instead of the secondary, the result is
found to be the same.
639.] It follows from these experiments that thv total induction
current in fidnn'ng thesimultimcous motion of /^ from Aj to ^,,, and
of B from Bf to B.J, while the current in A cliungM from y, to y,,
dci>om]K only on the initial i;tatc A,, B^, y,, and the final state
Aj, B., yj, and not at all on Die nature of the intGnnediate statea
tiirough which the Kyxt<m may jMss.
Hence tJie value tif tho total induction current uiujtt be of the
form hy^. B„ y,)>'('^i. A Vi)
where /* ia a function of A, B, aud y.
[h With reicpcct to tlie form of tins function, we know, by Art. 53il,
^wthat when Utere is no motion, and therefore Ai= A^ and if, = //,,
174
re ISDUC
the iodueLion ctirroat » >roportional to the primary' cunrnt
HcDc* y enter* simply iw a furtor, th« olh«r fnrtor l»cing a fuoc
tion of the form and pnitition of the circuito A mul B.
Vio iilso know thnt Wvit Tulue of this ftinction <I«pcn<Is on the
relative nod not on the nlisohito povitions of A imd B, so that
it must be cHpiiblc of bciti^f cxprcsMd as a fonction of tttc dutAtice*
of th« diliercnt cl«in«nt« of which the circuits nrv composed, and
of the aii<>:l(.)t wliieh these rlements malie with eaoh other.
Let M be this fuiiction, thon the total induction current may
written C{J/, y.i^y,},
where C is the conductirity of the secondary circnit, and J/,,
arc the oriffinal, and ,1/,, y^ the final \*Iue8 of Jf and y.
These ciprrimenU, therefore, shew that the total current
induction depends on the chan^ which takes p1ac« in a certain
quantity, My, and that this change may arise either from variatioa
of the primnTy current y, or from any motion of the primary or^^
eecondiiry circuit which alters M. ^^
540.] TliceonoeptionofHiich acjtiantify. on the changes of which,
and not on its absolute magnitude, the indtiction current depinitii,
occurred to Fnnulay at an early sfugw of his iwearohcs*. He .
ohserved that the t^ccondary circuit, when at rest in an e1«etfo*^
magiiclio field which rentainD of conittant intennity, doc« not shew^^
itny eIietrio»l efleet, whereas, if the Miinc xliite of the field hud Ven
suddenly produced, there would have been a current. Again, if the
primarj' circuit \a removed from the field, or the magnetic furees
abolished, there is a current of the opj>o«ite hind. He therefore ^i
recognised in the secondary circuit, when in the electroinagiietia^
field, a 'peculiar electrical condition of matter,' to which he gave^^
the name of the Electrotonic State. He ufterwaids found that he
could difipensG with this idea by means of considerations founiled on
the !inf> of mn^'netic force f, but even in his latest researched
he snyK, ' Agiiin imd iigain Uie idea of an tltctrolome state §
been Ibrced upon my mind.'
The whole history of this idea in Uic mind ofT^raday, as shewn
in his published researches, is welt worthy of study. By a course
of experiments, guided by intense application of thnnght, but
without the aid of mathematical calculations, he nas ImI to recog^
nine the existence of something which we now know to be a matb&
taalical quantity, and which may even be called the fandameatal
Hip. lUt, Hriei i. CO.
t H>„
U. {.U2).
t lU, 32<9.
I lb., SO, 1114, 1<«1, 17», 17S3.
S4(l
LI!
175
quantity in the theory of e1ecfroina);;nettem. But ua he was led
up to lliia conwptioD l>y a purely experimental path, he awribed
to it a physical existence, and euppoaed it to bo a peculiar con
dition of matter, thotigh ho was ready to shaodon this theory at
soon as be could explain the phenomena hy any more familiar forms
of thought
I Other iiivestijiratore were lonp afterwards led up to the same
idea hy a pun?ly m:ithematio:il path, but, so far as I know, none
of them recn(;ni6cd, in the roRnod mathcmntical iden of the polentiat
f two circuits, FaTaday'e hold hypothesis of nu uloctrotonic state.
Th<«e. therefore, who have approached this subject in the way
pointed out by thow eminent in vest if^a tors who first reduced ita
laws to it mathematical form, have sometimes found it diSicult
to sppnctato the sricntific accuracy of the statements of laws which
Faisdiiy, in the Gist two series of his Hrtcarcien, has given trilh
Buch wonderfnl oomplctiness.
Ute seientilic value of Faniday'* conception of an electrotonic
c consists in its direelintf the mind to lay hoW of a certnin
(jiuintity, on the changY'^ of which the a<'tuiil [ihcimmcna depend.
Without a much gn»ter degree of devel(>iment than Kiirailay jfavo
it, this conticptinii di>e« not easily lend jtaelf to the explanation of (ho
!ie»om<nta We .ihall return to this eiubjeet again in Art. 594.
S4l.] A method which, in Faraday's hands, was far more powerful
u tliat in which he makes use of those lines of magnetic force
which were always in hie mind's eye when contemplating hia
magoeta or electric currents, and the delineation of which by means
of iron filinge he rightly regaided * as a most valuable aid to the
xperiineataliet.
Ftiraday looked on these lines as expressing, not only by their
direction that of the magnetic force, but by their number and
concentration the intensity of that force, and in his later re
se«rchc6t he shews how to conceive of unit lines of force, I have
explained in various parts of this treatise the relation between the
properties which Faraday recognised in the linejg of force and the
matlieniatical conditions of electric and magnetic forces, and how
Faraday'e notion of unit lines and of the number of lines within
certain limits may be made mathematically precise. See Arte. 62,
404, 490.
In the 6rst series of his ScManiei % he shows clearly how the
direction of the current in a conducting cirouiti part of which ia
BUCi
■ qiui
W
it.
• e*p. ttm..sai.
t IK 31«.
: lb., lu
176
fAOSBTOBrECTBIC ISDPCTIOK.
moveable, dopendti on the mode in ivliich tlie moving parL cute
tUrougl) the linen of tnagnetio force.
In th« seeond Rones* he sbewa how the phenomena >rodaMd
by variation of the strength of a enrrent or a magnet may be
explained, by enpposin^ the system of lines of force to exjKind from
or contract towards the wire or magnet as its power rises or fallM.
I am not certain with what degree of clearness he then held the
doctrine afterwards so difilinetly laid down by himf, that the
moving conductor, as it cuts the lines of force, sums up tJie action
due to an area or section of the lines of force. This, however,
appears no new view of the cose after the inTeetigrntions of the
•ocond series J liave hcen taken into account,
Tlic conception which Faniday had of the continuity of the lines
of force prcchides the jirts^ibility of their suddenly startin;* into
existence iii a place where there wore none before. If, tborcfor^H
the number of lines which pusB throii;;h a conducting circuit ^^
made to vary, it can only he by the circuit moving iivroKs the lines
of force, or elite by the line* of forre moving iicroos the rarcuit.
In either i'usf a (.tirrent in generated in the circuit.
The niimberortlieline«of'foree which nt any instaat patw through
the cireuit is mathematttuilly e<iuivulent to Fanidfty'K earlier con
cci)tion of the eleetrotonic stat*.' of that circHil, and it is repreMrnted
by the quantity .ffy. ^M
It is only ainoe the definilioiw of eleetromotive force, Arta. 69.
274, and itii mefisuretntnt have I>»en made more precise, that we
am enunciate completely the true law of magnetoele«tric induction
in the folh)wing terms :■ —
The total electromotive force acting iwmd a circuit at any
instant i» mi.«sured by the rate of decrejise of the number of lines
of magnetic forire which pu« through it.
When integrated with respect to the lime this statement
comes : — •
The tinwintegral of Oie total electromotive force acting roui
any circuit, together with the number of lines of miig^etic for
which pass through tlie circuit, is a constaat qnantity.
Instead of speaking of the number of lines of magnetic force,
may speak of the magnetic induction through Ihu circuit, or ll
BUrface>integtal of ma;[^ctic induction extended over any anrfiicc
bounded by the circuit. ■
■ £n. ifM^ S>8. t Ik, S0S3, 3037, 31U. ^H
: lb, SIT, ftCL ^H
HBLSnOLTZ ARD TnOHSOlT.
I
»
I
We dUall rttuni again to this method of Kanda/. In the mean
time we must enumerate tlie theories of induction which are
founds! on other considerations.
Leas'* Law.
542.] In 1834, I^ienz* cuuueiatcd the followinf* remarkable
niUtioo Ixrtwevn the ptieiiomcnii of the mechanical action of eloctiie
current*. M defined by Ampferv's formula, and the induction of
electric ciirrenls hy the nIiitivi motion of conductors. An vurlicr
utlcmpt at a statement of KU(th a rulaiion vim given by Ritchie in
ill* PkUoMjiiieal Mugasine for January of iho Nimc year, but the
direct ion of the induced current, viva in every cn«e stated wrongly.
Ijtvx'* taw is as follows.—
If a conttaxl current JfoKt in fhe primary circuit A, and i/\ bg tie
notion of A, or of Ihf lecniidiirij/ circuit h, a current in iniluceiliu Jl,fie
direelioH of thii induced curreitt tuUl be tac/i Hot, bjf its eleetromaguetie
aeiioH on A, it fends to oppose tie retatice motion of the eircuita.
On this law J. Neumann t founded bia mathematical theory of
induction, in which he established the mathematical laws of the
induced currents due to the motion of the primary or secondary
conductor. He shewed that the quantity M, which we have called
the >ot«nt!a1 of the one circuit on tho other, is the same aa the
eleclromaj>nctic potential of the one circuit on the other, which
we have already investi^tod in connexion with Ampere's formula.
We may regard J. Neumann, therefore, as having completed for
the induction of currents the mathemiktical treatment which Am[>£re
applied to their meehanicul action.
.] A ftcp of still greater Btientific important was soon nPtcr
le by rielnihoitz in his Kitay on fie Conaervaliim oj Force ;. and
Sir W. Thomson ^ working somewhat later, but iiidepeudently
elRibollK. They nlHtNX'd that thy induction of elect ric current*
vcred by Faraday could be mathemattcally deduced from the
ekctroougnctic actions discovered by Orstcd and Ampere by the
appliration of tho principle of the CoiiBcrvation of Energy.
Uelmholtx takes the cosc of a conducting circuit of reststanco R,
in which an electiomotivo force A, arisb^ from a voltaic or thcrrao
• P(« , Jm. KiKi. leS (18811.
t BaiiB Af«d.. Mih and tSI7.
* K«ad Ului« Ui» fhyitaJ HneldT xf B^tIId, JuIj 3S, 1S47. TmuUtoa in
Tft)ti>'> ' 8ciMitif{^ Mnmrin^* fttU ii. p. I14>.
I Trim. lint. ia.. 184«. uia Fhit. Mtn.. Dm. ISSl. Sn klto LU jutpw on
"IVaaiiimt Ehetrie ConcM*,' fkit Has, IbCi
VOL. It. K
178
MaoiraTOELKTrRio isDtrcriOK,
vl«ctric Anangomcnt, nets. The current in the circuit at any
insUiDt i* /. He Ktipposes that u magnet is rn motion in t)i«
n«ighbourliood ol' Uie cirouit, ivtid Umt it« potontial wiUi respect to
the conductor \» V, bo tliat, during any xuiaII inUnal or tiioe dt, tho
encr^ communicated to the magnet \>y the eleetromitgDetic action
Thv work done in generating heat in llie circnit is, by Jouh
law, Art. 242, 1Jtdt, and the work Hjicnt by the eteotrontoti
force /f, in ntaiutaiiiing the current / during Uie time dt, ia Aldt.
Hence, eioce the total work done must be ecjual to the work spent.
I
ncc we find Uie intensity of the current
rff
di
A
1 =
R
Now the value of d may bo what wo pleoso. Let, therefor
A = 0, and then
. \dV
^^lidi'
or. there will be a current due to the motion of the ma^et, equal
jzr
to that due to an electromotive force ~j7'
The whole iudticcd current during tlw motion of tltv magD<
from a place where its potential is T, to a place where it« potei
IB
a*
IB
^r/r
/^'''=i/^'" = i<^'^*)
antl therefore the total current is independent of (he velocity or
tli« path of the magnet, and depends only on its initial and final
positions.
Ilelmholtz in hia original investigation adopted a system of
units founded on tlie mcnsuremcnt of the heat generated in the
conductor by the current. Con6i<Ienng the unit of current as
arbitrary, tho unit of resistance is that of a conductor in which this
unit carrcnt gnnentea unit of heat in unit of time. The ooit of
electromotive force in (his Kviitem im (hat required to produce the
unit of current in the oondiietor of unit resistance. The ndoptioo
of this lyrtem of units nevcMitates the introduction into the equa
tions of a quantity a, which is the mcchauioal tfjui viiU'nt of t
unit of heat. As wc invariably adopt eitlicr the elect roxtutic
WEBEB.
179
I
the elcctromngnctic system of units, this factor docs not occur in
tho <y]u»lions here giveu.
541.] Holinholtz also deduces the current of induction when n
condtK'ting circuit and a circuit cArryiog a conittanl current are
made to move relntivel}' to one another.
Let It,, /^, bs the rcaist^nceH, /,, I^ the currents, v/,, J^ th«
external electromotive furces, and F the potential of the one circuit
OD the other doe to unit current in each, then we have, ae before.
If wc suppose / to be the primniy current, and /^ so much lens
ftlian /, that it does not by it« induction produce sny sensible
^
[•Iteration in /j, so that we may put /, = ^
then
h =
II,
reiiult which may be interpreted exactly as in the case of the
/,=
A T
we siippouc /; to be the ]>rimary current, and /, to bu very
Ifiiuch MUiullcr than /,, we get for /,,
Tliis fhcwe that for eqtial currents the electromotive force of the
' first circuit on the second is equal to that of the second on the first,
. whatever be the forms of the circuits.
Hclmholtz does not in this memoir discuss the case of induction
due to the strcngtbeninp or weakening of the primnn, current, or
the induction of a cnrrent on itself. Thomson * applied the same
principle to the determination of the mcchaniml value of a current,
ind pointed ont that when work is done by the niuluul action of
two constant currents, their mechanical value is iuentitxt by the
nme amount, fc that the battery hax to supply tioulle that smonnt
of work, in addition to tliat required to maintain tJie currents
Bgainat the resistance of the circuiu f,
645.] The introduction, by W, \Vcber, of » system of absolute
* MochiMkal TIifotj of Elprtnilnit. Phil. Um., Deo. 18S1.
t NkUuT* <'ycioiHiAV>Vi «/ riiytlnil Scinet, A IMO, ArUi^ ' UsgoetUm. T>yiw
K a
180 MAOSBTOELECTRIC INDUCTION. [545,
unite for the meaBnremeut of electrical qoantitiee is one of the moEt
important steps in the progress of the Eoience. Having already, in
conjunction with Qauss, placed the measarement of magnetic quan
tities in the first rank of methods of precision, Weber proceeded
in his Electrodynamic Metuuremenft not only to lay down sound
principles for fixing the units to be employed, but to make de
terminations of particular electrical quantities in terms of these
nnits, with a degree of accuracy previously unattempted. Both the
electromsgaetic and the electrostatic systems of aoits owe theii
development and practical application to these researches.
Weber has also formed a general theory of electric action from
which he deduces both electrostatic and electromagnetic force, and
also the induction of electric currents. We shall consider this
theory, with some of its more recent developments, in a separate
chapter. See Art, 846.
CHAPTER IV.
OV THE IKDVCTION OP A CDBRBST ON ITSELF.
>.J Fabaday )ifl£ devoted thv ninth G«ri(>e of his Resfarciet to
the inv«8li»»tion of a class of phenomena exhibited by the current
in a wire which forms the coil of an ehclixiTiiagnet.
Mr. Jeiikin had ob«crvLHl that, Blthoii{;h it Js impossiblv to prn
duoe a nenxiblc shock by the direct action of a voltaic system
teonsiftliiig^ of only one pair of plat(4, yet, if the current is made
lo poM through the coil of an elect romagnet, and if contact is
then broken between the extremitieif »f two wires htld one in ' ach
hand, a smart xhoek will be telt. No «ucb nhock is felt ou making:
t contact.
Faraday shewed that this and other phenomena, friiich he de
scribee, are due to the i>ame inductive action which he had ulrendy
observed the cnrrent to exert on neig'hbouring eomlactors. In thin
oaae, however, the indtictive action is rxurtcd on the same uouihiclor
which carries the current, and it is so niueh the more powerful
ad the wire it«elf i* nearer to the diOerent elements of the current
thaa any other wire can he.
■ 647.] Ue obserres, however*, that ' the (IrMt thought that arises
in the mind is that tlie electricity circulates with something like
momentum or inertia in Uie wire.' Indeed, when we consider one
partieuUir wire »nly, the phenomena are exactly analo^us to those
of a pipe full of water (lowing in a continued stream. If while
the stream la flowing we suddenly close the end of tlie tube, the
iiHnnentam of the water produces a sudden pressure, which is mtieti
greater than that due to the bead of water, and may ho sufGcicnt
to burst the pipe,
If the water has the means of escaping tbrongh a narrow jet
• Etg. Jtm., Iffr.
^
182
SELFITOOCTIOS.
nhen the principal aperture is closed, it will be pR>jwt«d witb n
velocity much ({reater than tbat iluu to th« head of wator, nnd
if it can escape Uiroug:b a valvo into » chamber, it will do »,
even when the pressure in the chamber is greater thui that due
to the bead of water.
It is on this principle that tlie hydraulic rem \» constructed,
by which » stRBll quantity of water may be niiiEicd to a great height
by means of a lar^ quantity flowing down from a much low«r
level.
548.] Titoc ellWts of the inertia of the flnid in ilie tube depend
solely on the quantity of fluid running throogh the tube, on its
length, and on ita section in diOerent parts of its length. They
do not doiHnd on anytliing outride the tube, nor on the form into
which the tube may be bent, provided its len^h remains tlie
same.
With a wire conveying a current this is not the case, for
if a long wire is doubled on itself tlie effect ia very small, if
tlie two parts are separated from each other it i* preater, if it
IK coiled up into a helix it is still greater, and greatest of all i^
wlien so coiled, a piece of soft iron is placed inside the coil. j
Again, if a second wire is coiled ap with the 6tsi, but insulated
from it, then, if the second wire docs not form a closed circuit,
the phenomena arc as before, but if the second wire forms a closed
circuit, an induction current i« formed in the second wire, and
the ciTects of nelfindocliou in tlic first wtiv are retarded.
549.] Tlteie results shew dnHy that, if the pbeDomena arc dite
to momentum, ihe momentum it ceHainly not that of the eleclrieityi
in the wire, beeauw the same wire, cenvenng the nme current,'
rxhibits eflecis which difler according to its form ; and erea when
its form remains the Esme, the pneeence of other bodies, sncb we
a piece of iron or a closed metallic ciieait, affects the romlt.
530.] It is difficult, bowerer, for the mind nhich has onea
recognised the analogy Wtwren the pbenonena of telfinductioii
and thocc of the motion of material bodies, to abandon aitogcther
the help of this analogy, or to admit that it ts mtiidy mpofieU
and misleading. The fundamental dynamical idea of matter, ae
capable by its auitiuo of becoming the reetptent of momentum anil
of energy, b so interwoven with oar fonna of thoogfat that, whrn
rver we catch a gUmp*e of it in any part uf natiuv, we fed that
a path is beforv m leading, wooer or l^er. to tha complete under
vtandiaf of the aal(i«et.
BLECTSOKlilOTIC EyHROT.
[83
H 551,] In the case of the elwtric carrent, we fiod that, when the
eleciromotiTC force begins to act, it does not at once produce t!ie
^_full current, btit tbut thu current rites ^nduitUy. What is the
^ntlcctromotive force doicig (luring the Lime thnt the oppo&iujtr ^
^Bsistancc is not able to Iialunce it? It is incn^ming the electric
^kca Trent.
^g Now an ordinary force, acting on n body in the direction of it6
motion, increase* it» momentum, and comQiuuieiites to it kinetic
» energy, or the power of doin^ work on account of its motion.
In like manner the unresisted part of tbe electronKitive force hsia
lieen employi,d in inorea^ing^ thfi electric current. Has the elcctrio
current, when tlnui ]irmluoed, either momentum or kinetic energy ?
We have already ahewn that it has something very like mo
^knentum, that it resists being suddt^nly stopped, and that it can
" exert, for a nhort time, a great electromotive Ibice.
But a conducting circuit in which a earreut has been set up
has the power of doing work in virtue of tlits current^ and this
power cannot be said to be Bometliing very like energy, for it
i^j« really and truly energy.
^B Thus, if the current be left to itself, it will continue to circulate
till it is stopped by the resietanec of the circuit. Before it in
stopped, liowever, it will have generated » certain ijuantity of
heat, and the amount of this heat in dynamical measure i* equal
to the energy originally existing in the current.
Again, when the current i« left to itself, it may be made to
do mechanical work by moving magnets, and thu inductive effeet
of these motions will, by Lonz's law, stop the current eooner than
tbe resistance of the circuit alone would have stopped it. In this
way part of the energy of th« current may he transformed into
mechanical work inst«ad of heat.
552.] It appears, therefore, that a system containing an electric
current is a seat of eoerg) of some kind ; and since we can form
^■no conception of an electric current except as a kinetic pheno>
^^menon*, its energy must be kinetic energy, that is to say, the
^^ energy which a moving Iwdy has in virtue of its motion.
^P We have alrcAdy shewn tliat the elfctricity in the wire cannot
fiT»e considcied ns the moving body in which we art^ to find this
energy, for the energy of a moving body dois not dei>cnd on
anything external to ititclf, whoreas U»e pK«enec of oUier bodie«
near the cuircnt alters its energy.
11^ • ITjkntUjr, Sjrp. Bo. {iSi}.
tSRhTIVDVCnOTS.
We are th«rerore led to enquire whetber tber« may Dot be some
motion jKoin^ oa in the space ontaide the vrire, wliich is not occnpic
hy the electric current, but in which the electrom»gi>ctic effects
the current are manifested.
I Kliall nnt lit present enter on the reasons for looking in onal
place mtlit^r tliiin another for such motions, or for regarding th<
■DOtiotiH tut of one kind rather than anotJier.
What I propose now to do is to examine the oon»eqaenc«a of
tlic asaamption that the phenomena of the electric current are those
of » moving system, the motion bein^ communicated from one part
ttf the system to another by forces, the nature and laws of whiol^l
we do not yet even attempt to define, because we can eliminate™
these forces from the tsjuations of motion by the method given
by Lagrange for any connected system.
In tjie next five chapters of this treatise I propose to dcdti'
tile main strticture of the theory of tlcctricity from a dynamical
hypotlicsis of this kind, insind of following the path which has
led Weber ami othor iuv<i«tignU>rs to many rumarkahle di««ovcri
and vxperintent^i, nud to conctptions, some of which are aa bciiulii'i
u tbey are bold. 1 hare chosen this method because I wixh to
■how that there are other ways of viewing the plienomena wh
appear to me more sati^bctory, and at the xanie time are moi
consistent with the methodit followed in the preceding partjt of thi
book than those which proceed on the hypothous of direct acU'
at a distance.
4
»lfl
CHAPTER V.
OK THK EQUATIONS OP MOTION OF A CONSECTKD SYSTEM.
i
653.] Ik tlip fourth section of tlie second part of bi« JSfAattljue
daalyltqnc, Lagran^ has ffiven a method of reducing the ordi'nury
dynmniral equations of tlie motion of fbo parts of a conncctid
ij9t«m to a number equal to that of the degrees of freedom of
be ^if«m,
Tbe c<)uations of motion of a connected sj'stem have been (riven
in a diirerent form by Hamilton, and have kd to a great cxt«.>usioQ
^^of tbe hightT pnrt of [iiiro djnaroics*.
As »c iibrtll find it neecHwry, in our cndoavonrs to bring eleetrieal
phenomena within the province of dynumicii, to haro our dynamical
idnnn in a stxit* fit for direct nppbeation to physical qiiestioos, we
■halt devote this chapter to an exposition of those dynamical ideas
from a phmcal point of view.
554.] The aim of liUginn^ waa to bring dynamics under the
power of the ealunluK. lie brgnn by csprcssiug the elementary
dynamical relation.i in tirniN of the corrcKponding relutionit of pure
algebniienl ipiantities, luid from the equation* thus obtained be
diduct^ hi.i final e<)iiaUon« by a purely nlgebniical proeess. Certain
i)uantiti4>» (exprwwing the reattionv between the parts of the .*y«t«m
^■Called into play by '\i» phyitioni connexione) appear in the eijtiations
^^f motion of the component parts of the svotcm, and Lagrange's
investigation, bm mxju from a inatiieniatical point of view, i« a
method of eliminating tkeae (uantitie)i from tbe final cquatiou.t.
In following the stepii of lliiH tilimituttion tbe mind is cxereitMl
in calculation, and should therefore hi? kept free from the intrusion
or dynamical xA^as. Oar aim, on tbe other hand, is to cultivate
. • See VtvXrmar't Cajl^'i ' Ueoan on ThdONtical Djnunic*.* BrtliA Auotiittiva,
■SAT i and TbooiMti Mid lUt'* AiMunt FUloioplkf.
m
KISETIC3.
onr dynamical ideas. We therefore avail ouraiJves of tlie labour
of the mathematiciani), and retranslate their rcfultti Troin the Ian*
^iig« of the calculus into the langiuf^ of dynamtcit, »o tliitt our
words m^ call up the mental image, not of some algcbnucat
process, but of some property of moving bodies.
The language of dynamics has been considerably eactftnded bjr
those who have expounded in popular t«nns the doctrine of the
Conservation of Energ}, and it will be seen that much of tfa«
following statement is su^^sted by the investigation in Titoraaon
and Tait's Natural PiUosofAj/, especially the method of beginnin g ,
with the theory of impnlaive forces, ^H
I have a>ptied this method so as to avoid the explicit oon^1
sidcration of the motion of any part of the syetcm except the
coordinates or variables, on nhich the motion of the whole depends.
It is doubtless important that the student should be able to trace
the connexion of the motion of each part of the system with tJiat
of the variables, but it is by no means necessary to do this >^H
the process of obtaining the final equations, which are independent^^
of the particular form of these connexions.
J
to
u uiese
1 itself^^
Tie Fariallet.
555.] The number of degrees of freedom of n system ts
number of data which mast be given in order completely to
detirrminc it« position. Dilferent forms may be given to these
data, but thvir number depends on the nature of the xyHtem itself
and cannot be altered.
To Gs our ideas wc may conceive the system connected by
of suitable mechanism with a nnmbvr of moveable pieee», caeb
capable of motion along a straight line, and of no other kind of
motion. The imaginary mechanism which coimeda each of th
pieces with the vystem must be conceived to Iw free from frictioi
destitote of inertia, and iucajathle of being strained by the artJo:
of the applied foroos. The use of this roecliauiitm is merely '
Bssbt the imagination in ascribing iwwitiou, velocity, and moinentn
to what appear, in Lagrange 'm investigation, as pure algibraical
quantities.
Let ^denote the position of one of the moveable pieces a» define
by its distance from a fixed point in ila line of motion. We k1:
<li^inguish the values of ; corresponding to the diflercnt pi<
bv the suffixes ., ,, &c. When we are dealing with a set
!■ 3>
quantities belonging to one piece only wc may omit the suCBz.
ss*
iMPFiJtE AND MomnrrcM.
^P When the taIu^s of all the varinbles (</) are given, the position
~ of each of the moveable pieces is known, and, in virtue of the
ima^innry mcvlioiUBiii, the coufiguratioa of the entire ayatem is
determined.
H Jie VeloeifitB.
H 556.] During tho motion of the system the configuration changes
" in «>nK' tUfiniUr miinner. and BJncc the confiiiHration iit each instant
I i« fully d<^riii><l \>y till values of the varinhk'S [q), the velocity of
^■«very part of the eystem, ns well ns its confii^uration, will he com
" pl«t«ly defined if we know the values of the rariablea {q), together
with tJieir veiocitiM (^, or, according to Newton's notation,^) ■
I
t
t
The Force*.
537.] By a proper regulation of the motion of the variables, any
motion of tlio Ky«U>m, consistent with the nature of the connexions,
nuiy ho produced. Id order to produce thiM motion by moving
lie variable pieoen, forces must he apiilied to these pieees.
We shall denote the force which must he applied to any variable
J, by F,. The system of forees {F) ia meehanically equivalent (in
virtue of the connexions of the system) to the system of forces,
whatever it may be, which really produces the motion.
(The Momenta.
558.] When a body mores in such a way that its configuration,
0^ respect to the foroo which acts on it, remains always the same,
BR, for iiiKtam^, in tJiu cut^e of n force acting on a single particle in
the line of ito motion,) the moving furce is measured by the rato
^br tncreoKC of the momentum. If F tg the moving force, and p the
f'lnomcnttun.
F =
dp
4i
vncnee
=/
Pdi.
The time^integrat of a force is called the Impulw of the force ;
ao that we may assert that the momentum is the impulse of the
force which would bring the body from a state of rest into the given
Btat« of motion.
In the caw of a connected system in motion, the configuration ix
continually changing; at a rale depending on the v«locitic» (^), to
KINBTTCa.
th»t vrv ean no longer astmnie thitt tlie mom^ntuin iii the time
integral of the force whioli acts on it.
Dul tlio increment 2} of any vari&ble cannot be greater thin
^it, wtierc hi in tite time duriog nhich the increment takes place,
and ^ is the greatest value of the velocit}' during that time. In the
cue of a system moving from rest under the action of forces ilnraj^J
in the same direction, this is evidently the tiDal velocity. ^M
If tbe final veloeity and configumtion of the system are given,
we may conceive the velocity to be communicated to the system
io a very small time tt, the origioal coaiij^uration diflcring from
the final conOgnration by quantities 8 j,, i^^, kc, which aro leas
than ^,B^ j,&', &c,, respectively.
The smaller we suppose the iucrement of time it, the grenbeT
must be the impressed forces, but the timc^integnil. or impulse,
of each force will remain finit«. The limiting value of the iinpitliw,
when the time is diminished and ultimately vantahos, is <Ii;linvd
as tbe inatantaticoKi impulse, and the momentiun p, corrcfponding
to any variable q, is dcGucd nit the impulse corresponding to that
variable, when the system is brought instantaneously from a state
of rest into the given state of motion.
This conception, that the momenta are capable of being produood
by instantaneous impulses on the systMu at rest, is introduced only
as a method of defining the magnitude of the momenta, for the
momenta of (he system depend only on the tnstantancotu stnt*
of motion of the system, and not on tlie prooe^ by which that atate
was produced.
In a oonncet«d syrtcm the momentum corresponding to anf
variable is in geneml a linear function of tJie velocities of all the
varinbicK, instead of being, as in the dynamics of a particle, simply
proportional to the velocity.
The impulses required to change the velocities of the system
suddenly from ^p j,. &e. to yi', j/, &c. are evidently equal to
Pi'—pi, fia~Pt' ^^^ changes of momentum of tho scversi variablos.
Wori done by a Small ImpuUe.
B30.] The work done by the force F^ during the impulse is Iha
ftpac«intcgral of tlie force, or
=jP^i^4t.
56ol
mcREirauT OP KXTxvnc enrhqt.
If ^,' U the givatesi and ^," tbe l«st value of tbo velocity ji
luring tbe action of tbo force, ^' must be less tban
ad greater tbaa
4i"J
Fdt or
Fit or
%
If we Doiv Etippose the impulse / ¥df lo be diminii^ed without
limit, the values of ^,' and ^," will approach and ultimateiy coiucide
with that of <y,, and we nay write /»i'— jo, = ijoj, so tbat tie work
done is ultimately g ^^ = ?, 8ft .
ir, W« »ey>*it ifoii* fiy a triy *OTaW impiihe it uUimaittjf (At prvduel
qftic mjinltt and the veiocily.
^M iHerement ^ tAe Kinetic Energy.
^ 560.] When work ie done in settiuj* a conservative sjstcm in
jnotioii, energy ii; coramimicuted to it, iind the system hccomes
cupshlt! of doing an eqmd iimonnt of work ugainst resistanoes
L'bcfore it is reduced to rest.
The energy which n syst«Ri poE6«'ssGH in virtue of its motion
cullitd its Kinetic Encrg}', and is communicated to it in tlie form
of ihe work dontr hy the forces which Bct it in motion.
If T Vmj thu kinetic enirgj' of the system, and if it hecomei
^+ir, on account of the action of an inliniteKimnl iinpulte who»o
[eomponenttt are hjp^, ip^, &e., the increment hT muit he tlie Mum
of the <]aantilica of work done hy the components of the impulse,
>r in symbols, ^j.^ ^,ip,^j,bp,+ko.,
= 2{?M 0)
The instantaneous state of the system is completely defined if
the variables and tbe momenta ar« given. Hence tbo kinetic
energy, which depends on the iastantaoeous state of the system,
CIO he ex])ressed in tenns of the variables (^), and tbe momenta (p).
This is the mode of expreesing T intioduced hy Hamilton. When
T is expressed in this way we shall distinguish it by the suffix ,,
thus. r,.
lite complete variation of T^ is
»n = S(^'e;>)HS(J'5,).
dp
^dq
(2)
1 90 KIKBTIC8.
Tltc Ust term mny be written
= (S*»0'
[561
4
which dimiDiBh^ with 6 1, and uUimately vanishes with it when ih^
impulee becomes iDBtantaneoiis.
Hence, cqoating the cottfiicients of bp la equations (1) and (s1
wre obtain rfT", ,.
or, tie veioeily eorretpondit^ fo fAe tnricMe j it tie different
efitjicient of T^ wi/i retpevt to tAe carr^tponding tnomenfum p.
We hare arrived at this result by the consideration nf im]iuln«
forcce. By this method we have avoided the oon.tideration of tt
chan^ of configuration during the aefion of the forces. But th^
instantaneous state of the system is in all respects tlie same, whctli<
the ^atem was brought from a state of rest to the gircD at
of motion by the transient application of impulsive force*,
wbctlier it arrived at that state in any manner, however gradual
In other words, tlw variables, and the corresponding velociti4
and momenta, depend on the actual state of motion of the system
it the given instaut, and not on its previous history. ^A
Hence, the equation (3) is equally valid, whether the atate <^
motion of the syst«m is supposed doe to impulsive foroe^ or t«
forces aeting in any maniter w1uite%'er.
We may now therefore dismiss the consideration of impulsive
forces, together with the limitationii imposed on their tJm«
actioD, and on the chnngn of coufiguratton during their action.
HamiliiM't Eqnafiont ^ Motto*,
&61.] We have already shewn that
LiCt the system move in any arbitrary way, subject to the
ditious imposed by it« connexions, then the variations of^ and q are
rf/.
hp=H, hq^qht
Henoe
Hi
dT.
(»)
<*?.
«^=:f*H
di
if
dt
'?.
5623
BAUILTOn's BQITATtOXS.
191
and the complete variation of T^ is
'^ = M^*^ + 5»0
=^((:
,ip
(n
h~ ~ \'<d( ^ dq
at the increment of ihc kinetic energy arises from Uio work
Idoae by Uie impreBsed forces, or
hT, = S(Fh^). (8)
In these two espreesions the Tariaticn» ig are all independent of
eooh oilier, so that we are entitled to c<uatc the cocffioicnts of each
of them iu the two expressions (7) and (;g). Wo thuit obtain
' '. = #^'^' (»)
where the momentum p, and th« forci; F, belong to the variahle ^,,
There are as many equations of thin form as there are variablcv.
These eqnations were given by Ilnmilton. They shew that the
force corresponding to any variable is the eum of two partsi. The
first part is the rate of increase of the momentum of that Vkriuble
with respect to the time. The second piirt is the rate of increuK;
it the kinetic energy per unit of increment of the variable, ths
other variables and all the momenta bciog constant.
Tie Kinetic Sneryy cxj/rrxted in Tenas of the Mom^nla and
I'tiacilien.
562.] Let p^, Pg, &c. be the momenta, and j,, y,, &c. the
rdoeitiea at a given instant, and lot i\. p^, &c., q,, q^, &c. be
Ftnother system of momenta and velocitiex, itueh that
Pi =«Pu qi = «y,.&«. (10)
It i)> manifeit that the systems p, <[ will be consistent with each
other if the systems p, q are so.
H Now let « vary by In, The work done by the force F^ is
I F^h<\^=i^^h■[>^ = 4^p,nhn. (IIJ
1^ Let n incnase from to 1, then the system ts brought fmm
A itate of rest into the stotc of motion (^qp), and the whole work
eipOMled in prodncing tliis motion is
Bnt /'*"'" = 1>
aiiil tli« work fpciit in producing the motion is eqairalent to tlv
kiuetic cni»igy. Ilcnco
wUero 7^ cIeDot«« the kinetic cnei^ expressed in terms nf tJiu
motnmtu and velocities, llic \anab)cs ;, , ;,, &g. do nofc enter inl')
UiiK cxjireftiion.
'Die kinetic enorjfjr is tlierefore half tJie sum of ibo product* of
till' mnmi'nta into their corresponding velocities.
When the kinetio energy is expressed in this way we shall denotf
it hy UiL> symbol T^. It Is a function of the momeota aod vr)n
cities only, and docs not involve the vartalilcs thetnsclvett.
DGfl.] Tht'io is H third method of expr(>!i!iiDg the kinetic energy,
which \» generally, indeed, regiinhtl lui the fundaunental one. By
wJving the (K{tutionB (3) w« may cxprtM the momenta in terins
ofthe velooitiea^and thfn, introducing theM vnlne* in (13), wv Khali
have au expTvoaion fur T invoKing only tlie velocities and the
nriahles. When T m expr^j^ in this form vt shall indicate it
hf tlM aymhol T^. This is the form in which the kinetic energy b
Mtprtncd in the equations of lAgrang^
SGi] It is roanifftti that, sime T^, 7, and T^ are three diflerfjit
wrpnwions for the same thing,
or i;+'*AftAf,ftco. (H)
titwt. if all the tjiiaatitira ^, f, ud ^ vary.
Ttw wriMtKws A^ »n net
t^ •» tiat wv cKBnol at vac*
wriatim in IW njwbiw W kva.
IV wi — i»>> f VMnAana tf
»lUl wv tM^ttf •^MtMhgt*
B«t w« knov. boa
(15)
If ud
^*<u
t«)
mobakoe's equations.
193
or, tie eompweiti* ^ mcmenluim are t^ diffe/entiat coej^ienit <^ T^
with resfiect to He corresj^ndimg vticeiliet,
Agaiiij by equating to i«ro tli« ooefficients of S^i, &c.,
dT, dT. „
^1 'hi
(18)
or, tie differential ccejficieat of the kinetic enrrgi/ witA retpecf to any
tarialte 7, M equal in ma/^niiuile but opposite in lign when T it
atpreued at a Jnnetion of He velociliea iniiead <if a« a /imctictt of
tie momenta.
In virtnc ofequktion (18) wc may write the equktion of motion (9),
rf/., dT^
^'dT'df,'
' dl dq^ dq^
(19)
(20)
v
^■wbJch is the fonn in whicb tbe equations of motion wore gireo by
HliBgrange.
^^ ^^] ^° ^B procedioff iDvestigntion we have avoided the con
eideratiou of the form of tho function which expresses th« kinetic
(enerjry in terms cither of the \clocities or of the momenta. The
only explicit form which we have assigned to it is
in which it is expressed as half tho mim of the produottt of the
momenta each into its corresponding velocity.
We may expresa the v«Iocttien in terms of the difTcrential co^
ofiicients of 7*, with iesi>ect to the momenta, as in equation (3),
n=*(/.S4.p/^..&c.)
(22)
This shews thut T^ is a homogeneous function of the second
' of the momenta /J, ,/^, &c.
We may also txprcM the moinenUi in terms of T^, and wc find
^=*o/4';V4f^H
(23)
[which shews that 7 is a homogeneous function of the second degree
nth rrspcct to the velocities q^, ;,, &c.
If wc write
P., for 44^ , &«.
P„ for ^i.
u
^1 YOI.. II.
For  j ~ / , &c. i
CnfBTICS.
then, siooe both T^ and T^ are functionE of tlie second dc{;re«
^ and of p respect! vol}*, both the P'e :ind tbo Q'e will bv funrtion*
of the variables q only, and indcpndcnt of the Tclocitii;* uotl t
tnomenta.
We thus obtain the pxprcsdous for T,
^
{2i)
(26)
llic motrwDt* are exprosiwd in terms of the velocities by
linmriimtioM ^, = />„,, + p,^^^+&c., ('^
and the relocitJes aru expressed in t«nua of the (oomcnta hy
linear equations ^^ ^ Q„;..+ fi^+fcc. (2
In treatises on the dynumin of a r!(>id body, the oovSivieDi
«orrvKpiindt»g ta P„, iu whioh the HuS!s«(i tkn tlto same, arc called
MoiDCiitH of Inertia, and Uio»o corri'»iioiidin{r to P^, in whiili
the Kiiffixeit are ditltTeut, arc called I'roiinetM of Inertia. \Ye may
extend lliese names to the more gentinil problvin which in now
before u^, in which these quantities are not, as in the cdk of a
rigiil I'lxly, absolute constants, but art; fuuetions of the vari
In like n)iiim<T we may call the «oe(Bcknt« of the fonD
Momenta of Mobility, and thote of the form 9,j, Product*
Mobility. It w not oAen, however, that wc shall haw occowion
to H[)fal; of the coefficients of mobility.
S6(>.] TLie kinetic energy of the system is a quantity msential
po«itive or zero. Hence, whether it be expnwsed in terms of
TOlocitiw, or in terms of the momenta, the (Mxrfficit'nts mast
vnch that no rwd valnes of the variableif can make T ncgativv.
There are thus a wt of ncceKKury conditions which tlic rnlom
the eooflicicnt« /' must xatisfy. TheM! conditions are a« follows:
Thu (jnantities /'„, i'^, &c. must nil be pointivc.
! 01 a
■iJjlM
Tbti M — 1 detemiinantfl formed
jninant
iu mieccvsion from tho detci^
111
u>
Pi.
a
»*•
13,.
' 3i'
p..
■ A.
A A
by tlie omission of terms with suffix l.dien of terma with either
or 2 in tbeir suSii, and bo on, must< all be positive.
The number of conditiotts for a variables is therefore 3 b— 1.
5671
irOUESTS IKD PBODDCTS OF ISBBTIA.
195
I The cocfDrlrnls Q arc 8ul>j<>ct to oonditioDs of the Rime titid.
667.] In tliia outline of Iltf runcluniontnl principles of ihe Ay
imic* of a oonnrt'ted njsUm, we Iwvo kept out of vi«w the
ecliaiiiiim by which the pari« of tlw syrtt^m arc connected. We
itav« not even written doivn a »ei of eijimtions to indicate how
this motion of any part of the xystem depends on the variation
of the variables We have confined our attoution to the vnriahlce,
their Telocities and momenta, and the forono whidi act on the
pieoes representing the variables. Our only a9«iim)tioiiK arc, that
llie connexions of the system are eueh that the time in not explicitly
contuined in the equations of condition, and that the principle of
the conservation of energy is applicable to the system.
Such a description of the metbo«U of pure dynamics is not nn
oeoeesary, because LajirrRDge and most of his followers, to whom
we are indebted for tbeHc methods, have in general cou6ned tbem>
elves to a demoDstnition of them, and, is order to devote their
Attention to the symbols before them, they have endeavoured to
all ideas except thoise of pore qnantity, so as not only to
'dispense with diagrams, but even to get rid of the ideas of velocity,
momentum, and energy, after they have been once for all eop
planted by symbols in the original equations. In order to be able
to refer to Uie results of this analysis in ordinary dynamical hin
II g uage, we have endeavoured to retranslate the principal equntions
^fef tlte method into lan^a^ which may be intelligible without the
^T»e of symliols.
Ah the development of the ideas and methods of pore mathe
matics has rendered it possible, by forming a uatbemntieal thcoiy
of dynamics, to bring to light many truths which could not have
been diseovered withotit mathematical training, eo. if vk are to
fofin dynamieal Uteorica of other sciences, we must have our minds
unboed with these dynamical truOis as well as with mathematical
Kpetbods.
la forming the ideas and words relating to any science, which,
ike electricity, deals witli forces and their effects, we must keep
^^noatantly in mind the idea* appropriate to the fundamental Kienoe
^■f dynamics, ao that we may, during the first development of the
^Bcienoe, avoid inconiiHleiicy with uhat is already establish itl, and
^^ftlio that when our views become clearer, the tankage we have
adopted may be a help to us and not a liindrance.
CIIAPTEE VI.
DYNAMICAL THEORY OF ELECTBOMXGNETISM.
568.] We liavo shewn, in Art. GS3, lliat, when an elcctrio omrenl
Kiiets in a conducting; circuit, it Iihs ii i.apacity for doin;^ b cei
lUDOunt of mvchniiical work, unil this imicjiendeiitly of any cxtc
elMtroinolivv foroc muintuiiiin;; tlie ctirrent. Now cakpocity ft
performin;; work i^ nothing eivt; thun onorgy, in wbateTer wa]
it arises, and all eactgy is the same in kind, however it may difler
in fonu. The energy of an electric current is either of that form
which eonsi«ta in t!ie aotual motion of matter, or of that which
consist* in the capacity for being set in motion, arising from forces
■ding between bodies placed in certain positions relative to each
other.
Tlie first kind of energy, that of motion, is caJled Kinetic eoer^,
And when once understood it appears so Ibndaineiita] a fact of
nature that we can hardly conceive the poutibility of resolving
it into anything elne. The second kind of energy, that depending
on poiition, is called Potential energy, and is due to the action
of what we call foccos, that is to eay, tendencies towards change
of relativo position. With re*pcct to these forces, tbougli we may
aoocpt their vxiElcnce as a diinoostrated fact, yet we always feel
that every explanation of the mechanism by which bodies are set
in motion forms a rent addition to oar knowlcd^.
569.] The electric current cannot bo conceived except as a kineii
phenomenon. Even Faraday, who constantly endeavoured to ei
cipkle hii( mirul from (ho influonc« of those suggestioDS which
words 'electric current' nnd 'electric fluid' ace too apt to carr
witli them, speaks of tl>c electric current as * something prc^ivssi'
and Dot a mere arrangement.' *
• £ip. Am., 2SI.
ineii^l
nuu^l
h th"
KINETIC ENEROT.
197
H The eBecis of the current such as eIi>otroly«i8, and the translet
of electrification from one body to anothiT, ur« all progressive
actions which require time for their accom^ilishment, and xre there*
fore of the nature of motion§.
H As to the velocity of the current, we liiive shown that we know
HliothinK' shout it, it m»y be the t«ntb of an inch in nn hour, or
^K^undrvd thousand miles in a second *. So f«r arc we from
HScnowtn;* its absolute value in any case, that we do not even know
whether what we call the positive diiectioD is the actual direction
of the motion or the reverse.
But all that wo assume here is that the electric current involvCM
1 of some kind. Tliat which is the cause of electric ciirrentu
I'lMb Mllcd Electromotive Force. This name has* long been
n»d with frrcat advanta]>e, and has never led to any inconsistency
in the IaTi!;uft(ji' of science. Electromotive force is always to be
•understood to act on electricity only, not on the bodies in which
bbe electricity lesidee. It is never to be confounded with ordinary
mechanical force, which acts on bodies only, not on the electricity
in them. If wc ever come to know the formal relation between
electricity and ordinary matter, we shall probably also know the
fetation between cl<,ctromottvo force and ordinary force.
570.] When ordinary force acts on a body, and when the body
yields to the force, the work done by the force is mco^tired by the
product of the force into the amount by which the body yields.
BThus, in the caso of water forced through a pi>c, tlic work done
Bat any section is nwisunH) by the Quid pressure iit the «cvtioa
ntuUipIted into the quantity of water which crosses the xoction,
In the sime way the work done by an clcetromotivc force is
_faeaaure<l hy the product of the electromotive force into the quantity
^f electricity which croxxes a section of the conductor under the
'action of the cleclromotive force.
The work done hy an electromotive force iM of exactly llie same
kind as the work done by an ordinary force, and both are meamred
hy the same Ktondards or units.
Part of the work done by an ehctromotive force acting on a
conducting circuit is spent in overcoming the resistance of the
circuit, and Ihic part of the work ib thereby converted into heat.
Another part of the work is spent in producing tJie electromag
netic phenomena observed by Ampere, in which conductors are
modo to move by electromagnetic forces. The rest of the work
• £fr Hf. >«ie
198
ELECTROKlSFTtCS.
it spent in incmsing tlm kiD«U« energy of the carrent. and
efteats of this [)art of th« action are ahewn in tbe pbenomoui of L
indoction of cuminbi whwrved by Faraday.
Wc thi'irforv know ctKXigb about electric coirents to
in a sTEbptn of outerial conductors carrying currents, a dynami
system which i* the eeat of eaetgy, part of which may be kin
nnJ part {totcntial.
Ilic Datura of th« connexions of the parts of tliis system
nnknown to tw, but an wc have dynamical methods of investigation
which do not require a knowledge of the mechanism of the i^atcm,
we shall apply them to this case.
We shall Grst examine the consequences of ossnmin^ the Ui
^neral form for the function which expresses the kinetic eaergy
the systcni.
571.] Let the system oottsist of a number of conducting circuits^
the form aDd position of whidi arc ilctmnineal by the raluea
a system of vflriabU>s ar, , r,, &c, the nombcr of which is equal
to the number of drgrvcs of fRHtlom of the system.
If the whole kiiutic energy of the systi>m were llial doe to tbi
motion of tbcso conductors, it would be expressMl in the form
r = 1 (*, «,) i,' + &c. + (x^ jg xj ^j 4 &c.,
wlicre the symbols (a*, Xi), &c. denote tlio quantities which we ha'
oalled moments of inertia, and (2, jJ, kc denote the pro<luct« of
inertia.
If X' is the impmsod force, tcndin^f to increase the coordinate
whkb i» muircd to produeo the actual motion, then, by Lagrangv'
equation, ^H_il— r'
(/( di dx ~
When T denotes the energy' due to the visible motion only, w«
shall indicate it by the su0ix ^i thux, T^.
But in a Kyittcm of conductors carrying «)ictric currents, part
the kinetic energy is due to the existenec of Oicw currents.
the motion of the electricity, and of anything wliuso motitm
goTemed by that of the electricity, b« detcrmimd by another
of coordinates jr, , y^, &c., then T will be a homo^uwius Uinctioi
of squares and products of all the velocritics of the two «ct« of
coordinate*. We may therefore divide 7' into three portions, in tJ:
first of which, T^, the veli:idlies of the CMirdiiuilcs « only ocei
white in the second, T„ the velocities of the ooortUnates j onl;.
oceur, and in the third, T^, each term eoutains the pnMlitel of the
^locities of two ooortlinutes of which ouu is x and the other >.
Jl
of ,
I
of
I
J
BLECTHOMAONKTIC FORCE. 109
We have therefor« T = T +T iTH,,
n, where 7, = i(»,a!,) i,* + &c.+(a,jEjjf,jrj+&c,
B ?'* = ('.5'i)j^yi+&«
^ ST^] I" tbp CPiieral ilvnaiiiiciil tUeon', tho cneffioienU of every
I t<Tm msy be functions of all tlie coordinatcis, both j: and y. Id
^kthfl case of electric currents, however, it is easy to see that the
coordinates of the cln« y do not enter into the coeflicienta.
^_ For, if all the electric ciirrentii are mnintiiin«d consliuit, and the
^poondiictors at mt, the whole state of the field will remain constant.
Bnt in this ease the coordinates y are variable, tliough the velocities
jr arc constant. Hence the coordinates y cannot enter into the
expression for T, or into any other exjirMuiou of what actually takes
place.
Besides this, in virtoB of the equation of oontinnity, if the con
ductors are of the nature of linear circuits, only one variable is
Pe(uired to express the strength of the current in each conductor,
jet the velocitieA y, , ^,, &c. represent the strm^hs of the currents
n the several eonduetors.
All this would 1>e true, if, inatoad of electric currents, we had
currents of an inoonijirestiible fluid running in flexible tube*;. In
tills CTJsc the velocities of tliwc eiimnto woidd enter into the
expre«i(ion for T, but the coefficients would dcjwnd only on the
vanablr« i, which detenninc the form and position of tlic tubes.
In Ihe case of the fluid, the motion of the fluid iu one tube does
not directly HlTeet that of any other ttibe, or of the llnid in it.
Hence, in the value of 7"^, only the xquares of the velocities^, and
not their products, oocur, and in 7'm„ any velocity y is associated
ouly with those velocities of the form i^ which belong to its own
tul>e.
In the case of electrical currents we know that this restriction
does not hold> for (ho currents in different eircnits act on each other.
Ilt'Oce we most admit the existence of terms involving products
of the form ^i.r,, and this invotvt« the existence of something in
SH>tion, whose motion depends on the strength of both electric
cartmts ^, and y,. This movinff niatler, whatever it is, is not
confini<<I to the interior of the condiM^tors currying the two current*
but prol»I)ly extcnils thnHighout the wImIc space surronndinf; them.
S7$.] Let us next consider tlie form which I^agraoge's equations
of tnotiott a«sume in this cose, het X' be the impressad force
200
I0K1HBTICB.
[573
corr«]}onding to tlie coordinate x, one of those which determine
the form and position of the coinln<?tin;y drcuits. Tina ia a force
in the ordinary ^nse, a tendency uivards change of >oeitioD. It
it> &:iveti by Uie equation
' lit dx ^
We may consider this force as the stun of three parte, corte
Bpondin^ to tbe three parts into which we divided the kinetic
energy of tbe system, and we may distingaieh them by tbe aame
suffixes. Thofl i'= JC'^ ^ X',+X'^ .
The part A"„ is that whieh depends on ordinary dynamical con
nderations, and we need tint attend to it.
Since 7*, does not contain x, the fint term of the expreciioil
for X', is xero, and its value ia reduced to
* dx
This i« the expression for the raechanica] force which mnst be
applied to a conductor to liabinoc tbe elect romngnetie force, and it
imertit that it is measured by Ihe rate of dimtnatitm of the purely
elect roki net ic energy duo to the variation of tbe coordinate «. Tbe
electromagnetic force, X^, which bring« this extcnial miclianical
force into play, is c<]iial and opposite to A"^ , and is therefore mejMured
by (he rate of encreate of the electrokinctio energy corre«pontling
to an increase of tbe coordinate x. Tbe value of .V^, »ince it depends
on squares and produeta of the currents, reuairu the same if we
reverse the directions of all the eanenls.
The third part of AT' is
I " <ff "35" dig
The qaantity T^i contains only products of the form x^, so that
■ j^ » a linear function of tbe sf lengths of tbe currenta j>. TTie
first term, therefore, depends on the rate of variation of the
elrcni;ths of the currents, and iodicntcs a mechanical force on
the conductor, which is zero when the ctirrents are constant, and
whieb is positive or negative according as the currents are in
creasing or d<.'Ciia»ing in strength.
Tbe second tt^rm depends, not on the variation of the canents,
but ou their actoal strcngtlm. As it is a linear IVinction with
respect to these currents, il chiing<s si^n wbin the cnmntj? chnng
rign. Since every term involvs » v.!"citr i, it is zero when
5 74] WAS AN ETECTBTC CTRllimT TOIJE MOMKNTCM ? 201
f
»:
coitductors are at re*t. Then; iirc aUo tvmiH arising; from the tunc
ilT
variations of the coefficients of/ in y™ : these remarks apply tiso
to t)i«n).
Wc may th«Teforc iiivp«t.ig»tc Ihcse t*nat Beparately. If the
conductora are »t rt'itt, we bnvc only the first term to deal with,
f the currents uro condtitnt, «c liuve only tlie second.
574.] A* it IK of groat imporlnnce to dftorniine whether any part
of the kiniltP rtiergy in of the form ?*,«, eonnijiting of products of or
dinary velocities and strtnf^hs of vlcetric currents, it is desimhle
that ex[icriinciitM should W made on thin «ul}ject with ^reat cnre.
The dctenni nation of the forces noting on bodies in rapid motion
ia diffionlt. Let u* therefore attend to th<^ lirnt term, which depends
on the variation of the strength of the current.
If any part of tlio kinetic energy depends on the product of
•n ordinary velocity and the strength of a
current, it will pri.lmbly be moBt easily oI>
Mrved when tlic velocity and the current are
in the same or in uppo»ite dinctions. We
therefore lake a circular coil of a great many
windings, an<l wispind it hy n fine vcrticid wire,
ao that its windingit arc horizontul, and the
coil is capable of rotating about a vcrticid axis,
either in the same dircetion on the current in
the coil, or in tbe opposite direction.
i We shall suppow the current to be conveyed
into the coil by means of the eusjKDding wire,
and, after paxsing round the windings to com
pleto its circuit by piwxing downwards through
wire in the jame line with the 8UK])ending
wire and dipping into » eup of inerenn.
Since tJte action of the horizontal component
of terrestrial magnctiKm would tend to turn this
coil round a horizontal axis when the current
flows througli it. we ^hiill suppose that the horiEootal component
of terreatrin) niugnrtism is exactly neulralized by means of fixed
vatgaeU, or that the exj>erimi,nt is made at the majjnclic pole, A
vertical mirrorLiuttach^lotheooil to detect any motion in azimuth.
Now let a current be made to pafs through the coil in the
direction X.ES.W. If eloctricity were a fluid like water, flowing
the wire, then, at the moment of etartiitg tlie eurrcut, and as
R».W.
S02
ElECTROKINKTlCa.
[574
lonar OM its velocity in increaiting, » force notikl rcqaiiie to ht
supplied U> produce ihn nngiilar mnmcntiiin of tlio fliti<i in passin^f
roiiDd the coil, and lU thi.i niiuil W Htijitlio<] by tlie elasticity of
the BUBpendin^ wire, tlie coil would at firrt rotate in tbe o]>(weit4^
direction orW.S.E.N., and this would be detectod hy mcnnti of
the mirror. On Htoppiiig the current t^crc wduld be anotbei
inoveiuent of tbe mirror, this time in tJic mmc direction as thai
of tbe current.
No phenomenon of this kind hatt yet l>ocn oWn'od. Such nn
action, if it existed, might be eonily diittin^ti«hcd from the alrouly
known actions of the current hy the following peculiaritieii.
(1) It would occur only when the ntrcn^lh of the current raries,
a8 when contact b made or broken, and not when tlie current is
constant.
All the known mfekanieat actions of the otirrent depend on the
strength of the currents, and not on tite rate of vsriation. Itie
electromotive action in tbe case of inditced currcntd canbot be
confounded with this electroougnetie action.
(2) The direction of this action would be reversed when that
of all the currents in the Geld is reversed.
All the known mechanical aetioiui of the curri>nt remain the same
when »ll the aiirrentd are reverwed, siuce tliey dejiend on Kiiuares
and products of these current*.
If any action of this kind were (litcovered, we should be able
to regard one of tbe socatled kinds of ctectricity, either the ]>o«itivo
or tlie negative kind, as a real xubiitance, and we should be able
to dcAcribe the electric cum^it ii» k true motion of this substance
in a particular direction. In fact, if elMtrical motions were in any
way comiJarable with the motions of ordinary matter, terms of the
form 7^ would exist, and Ihcir cxisteaoc would be manifested by
the mechanical force X^.
Aecording to Fechner's hypotb<«is, that an electric corrent con
sists of two equal currents of positive and n^ati^e electricity,
flowing in opposite directions through the same conductor, the
terms of tbe second class T„ would vanish, each term belonging
to the positive current being accompanied by an equal term
opposite sign belonging to the negative current, and tbe pbN
Bomeoa depending on these terms would have uo existence.
It appears to me, however, that while we derive great advan(s»^
from the recognition of the many analogies between the i'IpcItm
current and a current of a material lluid, we must carefully M
575]
BXPKRIMEJTT Oy BOTATIOK.
203
makiti(> any aasuinptioD not warrant^tl hy cxporinionlal oviilenco,
»Rn<l that there is, aa yet, no pxperimeutat t.vi<i«iict> i/a shew whetlier
thf clcvtric current is really a current of a matorbl substance, or
In double current^ or whether ita velocity is gn^t or amM aa mea
■ured in feet per second.
A know]cd};r of theee thin;^ vrould amount to at least the be^D
DiD^ of a complete dj^amical theory of electricity, in which we
vbould rc^rd electrical action, not, as in this treatise, as a phe
nomenon due to an unknown c&use, subject only to the (general
\&vi» of dynamics, hut as the result of known motions of known
portion* of mutter, in which not only the total efTeuts and final
rcAilts, but tht' whok intermediate mcclianism and details of th«
^^ motion, aro taken ■» the objects of study,
^B 57S.] I^iv experimental inveetigation of the second term of X,^,
^^HlKly —j^, is more diflieult, a» it involves the observation of
the eiTvet of foroeit on a body in rapid motion.
The apparatus shewn in Vig. 3(, which I had eonrtmctod in
■ iSfil, is intended to t««l the existence of » force of this kind.
BLECTKOKISBTICS.
31
I
ia
The el«ctitHiiBgnet A is capable of rotating abont tbe horizontal
axis Blf, within a ring which itaelf revolves about a vortical
axis.
Let J, B, C he the moments of inertia of the electromagnet
abont the axis of the coil, the horizontal axis BB". and a third axis
CC pespeclivply.
Let bo the angle which CC' makes with the Tcrti<«l. ^ tbf
azimuth of Die axis BB, and ^ a variable on which the motion of
electricity in the coil depends.
Then the kinetic cnei^ of the electromagnet may be written
2T = J4'' sin' S+B4' + C4>*coaH + S{4> an 0+^)*,
where £ is a ([uuntity which may be called tiio m»m»)t of inertia
of the eWtricity in the coil,
If & i» the moment of the impa^cd force tending to locreaM
we have, by the cquationg of dynamics,
& = B'^[(AC)4fi»ia0coBO+£^txn$(4>f\n0+'if)\.
By making 4', the imprc^xed force tending to increase ^, eq ml ,,
to xero, we obliun ^H
^8ind4^ = y, ^^
a oonxtant, which wc may consider as representing the strength o^
the current in the coil.
If C is somewhat greater than A, €t will be zero, and the
lihrium abont the axis BIf will be stable when
This value of depends on that of y, the electric cnrrenf,'
is positive or negative according to the direction of the current.
The current is passed throngh the ooil by its bearings at S
and B', which are connected witJi the battery by means of sprin^H
rubbing on metal rings placed on the vertical axis. ^^
To determine the value of 0, a di«k of {uper is placed at C,
divided by a diameter parallel to BB" into two parts, one of which
lA pniiit<d red and the other gran.
\^')un the instrument is in motion a red cirde is seen at C
when $ is positive, the radius of which iitilioate« roughly the valne
of S. When is negntivc. a green circle w *een at C.
By mcanic of niit« working on ecrews attached to the dec!
magnet, the axis CC' is adjust«d to be a principal axis liavin
its moment of inertia jnst exceeding that round tlie axis A, to
sin 9 =
ZLECTEOMOTIVE FORCE.
inuk« tlic instrament very Kcuiiilile to the Action of the force
if it itxiNtM.
^ Tlic chief difficulty in the ex])i;rim<!iit8 arose from the ilietiiTbing
^kction of the earth'^ ma<;tietic f<irce, whiob caused th< ekctro
^unagnct to itct liki> a dipiiet'dle. 'Vha rt^ulla obtained were on this
'iwcounl very rough, but no evideuce of any ehange in d could be
obtaiocd even when an iron core ivas ioverted in the eoti, so as
tu make it a )Owitrful electromagnet.
If, tlierefore, a m^net containii matter in rapid rotation, the
angular momentum nf tliiii ixitatiDQ must be very umall compared
Iwilh any <)uaDtitieii which we citn measure, and ne have as yet no
lence of the existence of the terms 2*^ derived from tiieir
aanical action.
9T6.} Let us next consider the forces acting on the currents
fof electricity, that is, the electromotive farces;
Let }' bo the eSeetive electromotive force due to induction, the
[cleetromotive force wliich must act on the circuit from without
to balanoe it is y = — }', and, bv Lngrangc'a cquntion,
r=r =+•
ttl </^ dy
Since there are no terms in T involving the coordinntti y, th«
lod terra is zero, and }' is reduced U) its firxt t«rm. Hence,
romotive force caonot exiet in a itystcm at rcKt, and with cou
flt«nt currents.
Again, if we divide I' into tlireo part«, I'„, f,, and T^, cor
ie»l>onding to the thmc parte of T, wo 6ad that» since 7", does not
contain y, r_ = 0.
Weilsofind r = ~'^fi.
' at dv
dT
Herw 5. is a linear function of the currents, and this part of
Uw eJectromotive force is etjuul to the rata of chan^ of this
fiinelion. This is the electromotive force of induetiim discovered
by I'liraday. Wc shall conifidcr it more at lt>iigtli afterwards.
577.] From the i)art of 7', depending on velocities multiplied by
currents, we find i_ = —
I
I
I
Now
dt .Jj,
is a liocnr function of the velocities of the conductors.
If, thcrirforc, any terms of T^ have an actual exiatenee, it woidd
be postible to produce an electromotive force independently of all
existiDg carreitte by aimply altering the velocities of the conducton.
206
ELECTHOKnJETICa.
[57/.
For inetance, in t,h« ca«c of the «ii«pcui]c<l coil at ArU A59, if, wIicd
the ooil is at rest, w« suddenly met it in rotntioa nboiit the vcrticail
lucis, nn «1l"cI roiiiotivc force woiiW U>«ill<^ into action proportional
to tlic n«Ci^leniLi»ii of thiit motioo. It vroiild vanish whim lite
Diotion becume tiDifonn, and be rovenod when the motion waa
ruturdtKl.
Now few a?ientilio oWrvationg can be maile with greater pre
cision than thai which determine* the existence or noifCxiitten<c of
a etirrcut by ineans nf & galvanometer. Thedelicacy of thiit method
far i'xi^e<l.<< tluit of luoot of tlte arrangemenlK for miJiMiiriiig the
mochaiiical foici^ acting on a body. If, Uii>rcfore, any currents eould
be produced in tliis way they would be detected, even if liiey were
very feeble. They would be distinguished from ordinary carreota
of induction by tbe following clinracterictics.
(1) Tlii'v would depend entirely on the motions of the conductore,
and ill no degive on the strength of currents or tnagnetic fo
already in the tietd.
(2) They would depend not on the abjioloto velocities of the co
ductors, but on tlieir accelerations, and nn squares and products
velocities, and they would change sign wlien tbe acceleration be
comes a retardation, though tbe absolute velocity is the same.
Now in all the canea actually observed, the Induced currenta
depend altogether on the strength and the variation of current* in
tbe field, and cannot be excited in a field devoid of magnetic force
and of cunentB. In so far as thoy depend on the motion of coa
doctors, the) depend on tlie absolute velocity, and not on the chango
of velodty of these motions.
We have thus three methods of detecting the existence of the
tenns of tbe torm T^,, none of which have hitherto led to any
positive result. 1 have pointed them out with the greater care
because it ap[>enr« to mr important that we should attain the
greatest amount of certitude within our reach on a point
eo strongly on the true theory of electricity.
Sincf, however, no evidence has yet been obtained of such
1 Ehall now proceed on the assumption that they do not cxmi
or at least that thoy produce no sensible 00*001, an assuniptioa whii
will oonstdembly simplifj our dynamical theory. AVe nliall ha
occasion, howi;ver, >n discussing the relation of ninguetism to Ugl
to shew tlukt the motion which constitutes light may enter u« a
lactoT into tcnni invoiring the motion which oooatitnles mu^
nctism'
1
CHAPl'ER VII.
THEORY OF ELECTEIC CIUCKIT8.
578.] Wr may now oon6ne our attention to that part of the
EiDetic energy of the system which deponcU on wiuaree and products
of the strenglhii of the electric currents. Wo may call tbia the
Elcctrokinetic Knei^ of the system. The part depending on the
motion of the conductors belongs to ordinary dynamics, and we
have shewn thai the part depending on products of velocities Mid
teorrents docs not exist,
IjCt yf,, Jj, &c, denote the different conducting circaits. Let
their form and relative position be expressed in terms of the variables
4*1, x^, &c., the number of which is equal to the number of degrees
of &eedom of the mechanical sjstem. We shall call these the
Geometrical Variables.
Let y, denote the quantity of electricity which has CTO6ae«I a given
, section of the conductor .(, ainoe the beginning of the time /. The
trength of the current will be denoted byj/,, the fluxion of this
^quantity.
We shall call jr, the actual omrent, and y, the integral current.
There ifr one variable of this kind for each circuit in tho system.
Let T denote the electrokinetic energy of the eyetem. It is
a Itom^^neous function of the second degree with respect to the
,.^ rixe ngths of the currents, and is of the form
^Brnere the ooeflicients A, M, &c. are functions of the geometrical
l^tarlables Xi, r., &c. The electrical variables y,, y^ do not enter
into the exprcMtion.
We may call A,, Z.,, &c. the elnctrie moments of inertia of the
circait« /fj, J.^, &c., and ifj^ the electric product of inertia of tlic
two circuits J, and .4^, When wc wish to avoid tho languugi of
l9K
4
aHM
LIITBAB CIRCUTT8.
the dynamical theory, we shall call J/, Uie coefllcient orsalfinducli
of the circuit y/j, and J/^, the coefficient of mutual induction of i
circuits ^, and A^. M^^ is also called the jiotenUal of lite cirLiili
^j with ree])ect to Aj^. These qaantitics' degiend only on the form
Aiitl relative position of the circuits. We shall find that in the
electro magnetic system of measurement they arc quantities of the
dimension of a line. Sec Art. 627.
By diiTercntiatinf: Z" with respect to j^, we obtain the quantity />,,
whieh, in the dj'naroical theory, may be called the momentum
corresponding to y,. In the electric theory we shall call pi the
clectrokinotie momvnlum of the circuit A, . Its ralue is
The electrotdnetic rocroentum of the circuit J, is therefore mode
up of the product of its own current into ita coefficient of self
induction, together nith ttie sum of the products of the currents
in the othor circuits, each into the coefficient of mutual induction
of Aj and that other circuit.
Electromotive Force.
579.] Let fi' l)c the impressed clectromotiTe force in the circuit A,
an»in{> from some caoac, such as a voltaic or tliermoelectric b«tteiy,
which would produce a current independently of mngnetoelectric
induction.
Let R be the resi^noe of the circuit, then, by Ohm's law, an
electromotive force 7lf is required to overcome tlw resistance,
leaving an electromotive force £—Jly available for changing the
tnomeulnm of the circuit. Calling this force J", wc have, by the
general equations, ^^ Jp JT
but einoc T'doos not involve y, the laxt temi disappoors.
Hence, the equation of clectromolivo force is
TTie impressed electromotive force /' is then'fore the sum of '
parts. The first, /iyr, is required to maintain the current jr ugaii
the redstenoe R, The second part is rcquinti to increaw the i
ttomagnetio momeDtum ji. This is the electromotive fom* whict
must bo supplied from sources independent of magtwtoelect
TWO CI1ICUIT8.
)9
induction. The electromotiTeforce arising from magDCtovleutric
Dduction aloae is evid^Dtly — n < or, tAe rate of decrease of the
fUctrokinttic momentHst of the circuit.
(I
f Efeetroma^netic Foret.
580.] Lrt X' be the impreBsed mecluoical fwrne arising from
kxtvrDiil cauBC», tuid tending to increase tJie variable z. Hy the
gencnil equations ^ 4 dT dT
~ dt dx dx
Sin<e the expression for the clectrokinetic oncrgy does not contain
th« velocity [i), the iirst t«rm of thu second member disappears,
und wc find ^ ^f
Here X' h lite fxt^rnal force required to balance the forces arising
from electrical iituscs. It ie nsiial to eoimider this force us the
resclion against the elect to tna^etic force, which we shall call A',
^^ad which is equal and opposite to X',
I
Hence
dT
Win,
; tie elftiroinaffnelie /orcf ttnding 'o incrfaM any rarialln iV equal
io fit ratt of ihCTfase of the tleelrokinelic energy jtcr uiH increase of
tiat variaiU, He cwrratte being mainlained exmttant.
If the cnrrenta are tnaintained consljinl by a liattery during a
displacement in which a (juantity, W, of work is done by electro
motive force, the eleolrokinetic energy of the system will be at tho
Mine time increased by W. Hence the battery will be drawn apon
for a double quantity of energy, or 2 W, in addition to that which it
■pent in generating beat in the circuit. This was first pointed out
ly Sir W. Thomson*. Compare this residt with the olcctroirtatio
perty in Art. 93.
Catt of Tko Circttite.
681.] Let Ji he called the Primary Circuit, and A.j the Secondary
Circuit Tbo dectrokinetic energy of the system may be written
where L and .V arc the coo<Iicicnt« of »eirindncUon of the primary
> N>d>..l'« Crtop<Hili» 1/ Njoinil Stitntt, (d. 1840, Artldo, ' MignMim. Dy
■tmkal livUliuna «f.'
k
TOL. II.
\AufM
USB4H CIRCUITS.
and Eecon^MTy circuits rccpt'clivcly, iind 3f is tbfi coefTicicnt of
mutual induction.
Lot as suppose that no ilcctromotive foroo *ci» oa th« second,
dicuit except that duo to the inductioQ of the primal}* curreDl
We have then .
Integrating^ this equation with reepeot to /, ire have
If,^a + Mfj + Nf^ = C, A constant,
vrhrrc y, i« the integral current in the secondary' circuit.
'Hie mvttiod of nitiasuring an integral current of sliort d
will be dviMsribed in Art. 746, and it is easy in moat cnaoii to «i
that the duration of the secondary cmreDt shall bo very short.
Let the values of the variable i^uantities in the equation at
end of the time i be acccntid. then, if y^ is the integral current,'
or the whole quantity of electricity which flows through a section
of the secondary circuit during the time t.
If the secondary current arisirs entirely from induction, its initial
value jTf must be zero if the primary current is oonstsot, aod the
con'luctors are at rest before the bej^innin^ of the time i.
If the time / is sufBcieat to allovr the secondary current to dis
awav, y/, its final value, is also Kcro, so that the equation bcvomo
The integral current of the secondary circuit dDpends io
on the iuitiul and (iiuil values of J/y,.
Indueei CmrrenU.
S83.] Let us begin by supposing the primary circuit
or y, = and let a current y,' be eiUbli«hed io it when oont
is made.
The eijuutioa which delermioes the BMondary integral current i
When the eireoib «tt pUoed side by side, and in the same direct
tion, J/ is a positive quantity. Hence, when contact is made io
the primary circuit, a negative current is induced in the soooodaty
circuit.
When the contact is broken in the primary circuit, the ;
current ceases, and tlie induced current isy, where
Hie Moondary enrrent is in this ease positive.
fe
TWO CIBCUITB.
211
I
If the primary current U muintaimid coostant, and the form or
relative position of the drcuils altered so that M becomes M', the
int^ral secondary current in^^, wh<re
In Div cn«c of two circuits placed side by side and in the same
diruotion M dimiiuahes as the distance lH'tw«cn the circuits in
creases. Hence, the iiiducoit current is piwitive when this distuooc
ia increased and ne^tivc when it is diminiehud.
t These are the elementary ca«es of induced ciirreiite described in
:t. 530.
6S
cl
I
I
I •
I
Mechanieat Action heltcerit Ihc Two Cireuiti.
583.) Let X be any one of the geometrical variables on which
i« Ibrm and relative position of the circuits depend, the clectro
nuignotiv foreu tending to increase x is
dL dM , ,dN
If the motion of the system corresponding to the variation of x
IS Fiich that each circuit moves as a rigid body, L and .V will be
indt^pcndvnt of x, and the equation will bo reduced to the form
dM
J^=i>>''^+^./.^4U/'
■i" = hh
dx
Heooej if (he primary and secondary ourrents are of the same
sign, the force X, which acts between the circuits, will tend to
move them so as to increase M.
If the circuits are placed side by side, and the cnrrcnts Dow in
the same direction, M will be increased by their being brought
DeiU¥r together. Hence the force X is in this eai<e an attraction.
684.] TIio whole of the phenomena of the mutual notion of two
CircuiU, whether the induction of currents or the mechanical fore*?
betwMn them, dejicnd on the quantify V, which we have called the
eoeflieient of mutuiil induction. The method of calculating this
itity from the geometrical relations of the drenita is given in
524, but in the inveKtigations of the next chapter we shall not
me a knotvledge of the mathematical form of this quantity,
sliall consider it as deduced from experiments on induction,
for instance, by observing , the integral cuiTcnt when tlie
secondary circuit is suddenly moved from a t,'iven poKition t« an
lite distaote, or to any position in which wo know tliat J/s 0.
F a
CHAPTER Vin.
bxpijORation of thr field by ubaxs of thb secokdakt
CIBCUIT.
vnit
I
:tn>
hese
My
«(] in
■M
585.] We have proved in Arts. 582, 583, 5B1 that the eleotro
augnetic uction botwoen the primary and the accondary oircait
depends on tlie quantity deuoted by if, which is a fuactioa of
form and relative position of the two circuits.
Although this quantity ,V is in fact the same as the pot«ntii
of the two circuits, the msthcmattcal form and proportice of whici
we deduced in Arts. 423, 492, 521, 639 from magnetic and electro
magnotic plienomena, ne shall faorc make no reference to thew
r«Bu]tB, but begin again from ■ new foundation, without any
assumptions except thoM of the dynamical thc<ory aa slutcd in
Chnptcr VII.
Tlic clei;tr»kinetic momentum of the secondary' circuit conna
of two part* (Art, 578), one, J/<i, depending on the primary eur
»',, while the other, ?i\, depends on tJie ttecondaiy current i',. We
arc now to investigate the first of theae parts, which we alialt
denote by 7», wbiTC /> = .l/t,. (l)
We idiall alao suppose the primary circuit fixed, and the primary
current coitxtant. The quantity jb, the electrokinetJo momentum of
tJie secondary circuit, will in thiti case depend only on tJie form
•nd position of the veeondary circuit, so tliat if any dosed cunre
he taken for the iteooodaty circuit, and if the direction along this
ctirTO, which is to be reckoned positive, be chosen, the valiM of ^h
for this closed curve is determinate. If the opposite direction aloo^l
the curve bad been chosen aa the positive direction, the sign of
the quantity /i would have been reversed.
S8€.] Sinc« the quantity p depcnda on the form and positiii
of the cireuit, wu may sappo^ that each portioD nf the circnt
588.]
ADDITION OP CIECUIT3.
213
■jlribi]
oontributas somethinji^ to the value of p. and tHat the part con
bated by each portion of the cireuit depends on the form and
ntion of that portion only, and not on the position of other parte
of the circuit.
This asstimpUon is legitimate, hucause we are not now considcrinf^
current, the pari* of which niuy, and indeed do, act on oiw an
other, but a mere arcntt, that ix, a closed curve alon^ wliieli a
eurreDt nnr^ How, and thie is a puiyly ^somutrical fifj^ure, the iart«
of which cannot be conceived to have any physical uetton on eoelt
_other.
We may therefore aMume that the part contributed by tlie
element */* of the circuit i* Jd*. whirr / in a fjtiuntily dc))ending
on the portion and direction of the clinicnt J». Henee, tbe valne
of J) may be expTt««ed as a lineiiit«gnil
fp=jJd>. (2)
1ier« the integration ix to be extended once round the circuit.
887.] We hav(^ next to determine the form of the quantity J,
a the finti, place, if tta is reversed in direction, J is reversed in
uga. Hence, if two cireuita ABCE and AKCD
hare tbe arc AEC common, hut reckoned in
opposite directions in the two circuits, the sum
tof the values of ^ for the two circnits ABCE
and AECIf will be e(iuat io the value of yi for
the circuit ABCD, which is made up of the two circuits.
For the parts of the lincintcgrul deppndinff on the arc AEC ai«
equal but of opposite sign in the twd partial circuits, ao that they
destroy cndi other when the sum is taken, leaving only those {torts of
the linointegral which depend on the external boundary of ABCD.
^P In the fame way we may shew that if a surface hmindcd by a
eloeol curve be divided into any number of part.N, and if the
bonndary of each nf thc^e jtarts be coti»idered u« ii ein^iiit, the
positive direction round every circuit I>eing the «arae an that round
the external closed curve, then the value of'jO for the cUibo<1 curve in
^^qual to the sum of the values of /t for all the circuits. See .\rt. 483.
^P 68B.1 Let ns now consider a portion of a aurfaee, the dimensions
"of which are so small with respect tolheprinciial radii of curvature
of the sarliu« that the variutioii of the direction of tbe normal
within this portion may be neglected. We shall also suppose tha',
if any very small circuit be earned parallel to itself from one part
if this surface to another, the value of p for the small circuit in
1 If
Fig. 36.
I
2U
fiLfiflHOMAOSETIC ?IEI.R.
not aenaibly altered. This will cviilcntly be tbc cam U* tlm diuen
aiona of the portiou of surfacu arc ridhH onoug^i eomiured wit
ita distance from the priman circuit,
f arty closed carve be drawn on Mi* jiorfhit of He tvrfaee, tie
raltre of p will 6e proportional to tV* arta.
For the nrtas of any two circuit* may be divided into sma
elements all of tbo :^tmc dimenxioni*, and having the same valv
of p. The area* of the two circuit» are as the nurobera of theM
elements which they contftin, and the values of p for the two eircoiu
are aLui in t.)ic same proportion.
Hence, the value of ji for the circntt which hounds any elemental
lis of a iiurface is of the form IdS,
where / is a quantity dei>ending; on the poeition of d$ and on the
direction of its normal. We have therefore a new expreasiOQ for jo,
=1
tie
aalH
tlaeV
. =jjld8,
1«
Kg. S8.
where th« double integral is extended over any surftce bounded b;
the circuit.
589.] Ijct AliCD be a circuit, of which AC is an ilonientary
portion, m> small tbnt it may he conHHlen^ Htruigh
Let APH and CQB \x nmall ctjual areuit in tl
same plane, then tbc value of p will be the MOM
for the small circuits APB and C<i,lt, or
p{APIi) = p{CqB).
IIcDce p{APBQCI/) =p{ABQCD) + p{APiri,
=p{Asqcii)+p(cqs),
= p{A£CD),
or the value of p is not altered by the substitution of the crool
line APQC for the straight line AC, provided the area of the circuit
is not sensibly altered. l"hia. in fact, is the principle established
by Ampin's second experiment (Art. 506), in which a crooked
portion of a circuit i* shewn to be equivalent to a etrsigbt portion
providod no ]>art of the Crookeil portion is at a sensihle dist>n<
from the straight portion.
If therefore wc substitute for the element i/t three small elemeni
lie, Jjf, and tl:, drawn in succession, so as to form a c«nlinuoi
patli from the be^nning to the end of the element i/#, and
Fdje, Gdjf, and ffdx denote the elements of tlic lineintegral eo:
tesponiling to dx, djr, and dg respeetively, then
Jdt = FdxiGdyiIIde,
ILTCTBOKISETIC KOyCEVrtUV.
590.] Wc «re now able to determine the mode in which tbe
quantity / (Icxwnds on the direction of the elemont ds. For,
by (4J.
4* <lf da
(5)
This is the cxpnsdon Tor the resolvetl i>art', in tho direction oft/*,
of a vector, thv components of which, rceolved in thti directions of
the ttXCT of », ^, nnd i, mv f, (r, nud tf rcepuctirely,
Tf this victor be denoted l>y 91, nnd the visctor from the orif^Jn
to a point of the eirciiit by p, the (element of the circnit wiil be dft,
^—»ud the quitemion expreaaion for Jd* will be
■ S3ldp.
H Wc may now write eqiiatioD (2) ia the form
Moh^sy
or fi
=fsndp.
lie vector ?[ and its conptitiiont^ F, C TT depend on the position
I of dt in the field, and not on the direction in whieh it is drawn,
rhcy arfl therefore functions of «, y, t, the coordinates of d», and
not of I, m, n, its directioncosines.
Th« vector 91 represents in direction and magnitude the time
hitcgnd of the electromotive force which a pitrticio placed at th«
point (*, y, ;) would cuperiencc if the primary current were sud
denly stopped. We shall therefore call it the Electrokinetic Mo
mentum at tif point, {x, y, t). It is identical with the quantity
which we investigated in Art. 403 tinder the name of the veetor
potvntiul of mof^ietic induction.
The elootrokinetic momentum of any finito line or eirouit is th«
iineinlegnl, extended along th« line or circuit, of the resolved
port of the eloctrokinctic momentum at faeb point of the some.
501.] Let U!i next dttermino the value of
p for the elementary rectangle ABCD, of
which the sides are Jy and dz, the positive
direction bein;; from the direction of the
axis of ^ to that of t.
Let the coordinates of 0, the centre of
gravity of the clement, be J'ofJ'u. o ""d '"^
On, Hn be the values of G and of //at thia
point.
Fig. W.
lie coordinates of A, thn middle point of the first side of Uie
BI^ECTROMAOIi
[59*
rectangle, are y, and t^—  dz. Tlie correeponding value of Q '»
(
M
and the part of the value of p which arises from the side A u
.pproxin.ut.ly G^J^'/PyJz. (9)
Similarly, fra JS^ B^dx^  r dgdz.
For 5, H^dz^Y^dgdz.
Adding these four quantities, we flod tlie valne of j> for
rectanel»^, viz: ^if dG. , , ,
If we now Asciime three new <aantitie8, a, 6, e, such that
^d/l_dO
rff dc
dF
di
dG
dU
(^
dz
dj_
dm djf '
', ^g^ conitidfr these um the eonstitucnt* of a ncn vcdor 9, tlien,
TiieorDin IV, Art. 21, we may express the limint^'gral of SI roue
any circuit iu the form of the 8tirEaec>iut£^7i>l of 3 over a »u ft'acc
hounded by the circitit, thus
p=JT%cta*dt=jJT'^cMiidS,
wbere t a the angle between ?l iind dt, and ij that between © and
the norma] to dS, wlio»e <liri>ctii)n«otine* aru /, m, n, and T% 79
denote the numerical vatuea nf 'Jl awd '8.
CompunD{> this result with equation (3), it is evident that the
qunntity / in that equation ia equal to 9 coa >}, or the resolved part
of $ nonual to dS.
592.] We have alimdy seen (Arts. 'I'JO, 54 1 ) that, acoordiny :
lVraday's thcoiy, the phenomena of elcctroma^etio fbrea.
or
MAOSCTIC INDOCTIOJT.
indoctioa in a circuit depend on Utu vaHattoa of tlio number of
lines of ma^etic indactioD which pass ihrouf^h the circuit. Now
the number of these lines is exprcfsi.'d mathemHtically by the
sarbccinte^ral of the maf^nctic induction throug'h any eurfnco
bounded by tbo circuit, tlencc, wc mii«t rrfjard the vector S
and its components a, 6, c ok representing what wc arc ulrcudy
ucquuintcd with ue the mngtictic induction and it* components.
In llie present investijfation we propone to deduce thi projierties
[of thiM VLctor fn)ni iho dynrtmieul principles stated in the Xust,
chapl«r, with .i* few nppoiilx to experiment as possible.
In klentifyiiig this vector, whinli has iip}eared as the result of
'a matliematical inveati^ution, with the magnetic iniluclion, the
Ipropertien of which we learned from experiments on magnets, we
do not depart from this method, for wc introduce no new lact into
the theory, we only give a name to a mathematical quantity, and
the propriety of bo doing is to be judged by the ngreement of the
relations of the mathematical quantity with those of the physical
quantity indicated by the name.
nic vwtor 33, since it occurs in a surfaeeintegml, belongs
I evidently to the eatepory of Huxes described in Art. 13. Tho
I vector SI, en the other huud^ belongs to the category of forces,
^nnc« it appears in a lineintegral.
593.] Wc roust here recall to mind the conventions about positive
' and negative qiuintities and diicctions, «otne of which were stated
in Art. 23. We adopt the rigiithumled syslero of axes, so that if
a righthanded screw is placed in the direction of the axis of a>,
aad A nut on tliis screw is turned in the positive direction of
rotation, tliat i^ from the direction of y to that of :, it will more
alon^ the screw in the positive direction of a*.
■ We also consider vitreous electricity and austral magnetism as
positive. The ]>0Bitivc direction of an electric current, or of a line
of electric iuduction, is the direction in which positive electricity
Imoves or tends to move, and the positive direction of a line of
'magnetie induction is the direction in which a compass needle
points with the end which turns to the north. See l"ig, 24, ^Vrt.
«8. and Fig. 25, Art. 501.
jl^ The student is recommended to select whatever method appears
^Kto him most elfcctual in order to fii these conventions securely in
^Bbts memory, for it ts far more diflicult to remember a rule which
^^det«rmincs in which of two previously indiflerent ways a stotcmecit
ia to be made, than a rule which selects one way out of many.
»i
IIJCTROMAONETIC FIELD.
[594
594.] We have next to dcduoe from d^amical principle*
pxpnsMiona for llie elct;tri>magn<)tio force actiu^ on a coadac'
currjing an electric cuirciit tbroufi^li the mai^netic ficM, and
the electromotive force acting on the electricity within • body
moving in the magnetic field. The mathemiitimi method which
we flhall adopt may be compared with the cxiwriinenlal metliod
used hy Faraday * in explorinfr the field by meanit of a wire, and
with what we have already done in Art. 490, by a method founded
on experiments. What wo have now to do is to determine the
effect on the value of p, the electrokinetie momentum of the
secondary circuit, due to given alterations of the form of that
circuit.
Let AA', BB" be two parallel straiglit conductors coaneotcd I^
the conducting arc C, which may he of any form, and by a strai^t
■^
ong
tio9
F<(. Da.
conductor AB, which is capable of sliding parallel to itself along
the conducting rails A A' and B&.
Let the circuit thus formed he considered as the secondary
cuit, and let the din^rtion ABC be assumed as the poativc directioi
round it.
Let the sliding piece more parallel to itKcIf from the postion AB
to the position ^Jf, Wc have to determine the variation of /r, t1
cicctrohinetic momentum of the circuit, due to this dieplacimcn
of the gliding piece.
Tlie aecondary circuit i« dian^i^ed from ABC to jtlfC, hence, b;
Art. 587, j>{ArS'C)p{A£C)= piAA^BTB). (i.h)
We have tlierefore to determine the value of p for the iaralUI*M
ogram AJ^ffB. If this puralielogram ia so small that wc mayH
neglect the variations of the direction and magnitude of the mag
netic induction at difiVrent points of itd pluiic, tJje value of p
by Art. 59 1 , © 00a ij . AA'IfB, where S is the tnagnolio inductio
• Bip. ft*, SOte lusr. 3lW.
596.] SLinixo PiKCE. 2X9
I
^bnd >) the angle <iThich it ranker with the positive direction of the
^piormal to the parallelogram AA'lfB.
^ We may represent the result georoetrioallj by the volume of the
pamllelcpiped, whose base is the parallelogram AA'JfB, and one of
whose edges is the line AM, which ropresenta in direction and
^maivnttiido the magnetic induction ^8. If the parallelogram is in
Ijo plane of the paper, and \i AM is drawn upwards from the paper,
tie Tolumc of the parallelepiped ia to be taken positively, or mora
cnorall}'. if the direotions of the circuit AJi, of the magnetic in
^duetion AM, and of the displacement AA', form a righthanded
syBtem when taken in this cyclical order.
The volume of this parallelepiped represents the increment of
the ^alne of /> for the secondary circuit due to the displacvment
>f the eliding pieci! from AB to d'iJ'.
Sleetromolive F(»ve acting on tie Sliding Piece.
595.] The electromotive force jirotluccd in tlie Mconduy circuit
by the motion of the sliding piece is, by Art. 879,
If we suppose AJ' to be the displacement in unit of time, Uiea
iA' will represent the velocity, and the parallelepiped will rojiresent
R^, and therefore, by equation (H), the electromotive force in tlia
negative direction BA.
I^L Honco, the electromotive force acting on tha sliding piece AB,
^Kn consMitienoe of it^ motiuu through the mAgnetio Geld, is rcprw
^Bent«d by the volume of the pnrnlKIepiped, whose edges represent
^^in direction and magnitude — the velocity, the magnetic induction,
and the aliding piece itaelf, and is ponitive when these three direc
tions are in righthanded cyclical order.
^
EUeli^ftagnelie Force aH'mg on the Slliiing Pieee.
S96.] Let f, denote the current in the ecconilury circuit in the
positive direction ABC, then the work done by the electromagnetic
forre oo AB while it slides from the position AB to the position
A'W is (Jf'— J/) /, 1^, where M and M' are the values of 3/,^ in
the initial and final positions of AB. But (.V— J/)' is e(ual
to p'~p, and this is rcprcwntwl by the volume of the parallclepti>ed
AB, AM, and AA'. Hcnoe, if wc dmw a line par»Il«l to AB
EtBCTBOMAOlTEnC nB!.T>.
to rc])n»eot the qaantity JB.ij, th« {laralldcpiped contained
this lino, by AM, llic mnf^oliv indnotion, SDi] by AA', Ihu displiwt
ment, will represent the work done during this diHpbct'invnt.
For ii ^von dietunci' of difplaocrovnt this will be gnulxict when
tlio di^placfmont i« perpendicidsr t» the pnrBllcIcigmm whom? «dw
an AB iwwi AAf. The electroiuagiwtic force in ihcrvfoTv rc{>n.>»fiilnl
by the arcu of thi> pHivllelognun on Alt and AV multiplied by ^,
Knd i« in the direction of th« normal to thin parallelogram, drawn >o
tliHt AS, AM, and the aonuat are in rtghthan<led cyclical order.
Four DffinUiona of a Line of Mapietie iHduelioH.
697.] If the direction AA^, in which the motion of the sliding
piece takes place, coincides with AJif, the direction of the magiietfc
indaelion, the motion of the sliding pioec will not call electmmotive
force into action, whatever be the direction of AB, and if ^// carries
an ilcctric current there will be no tendency to slide along Ait.
Again, if Ali, the sliding piece, coincides in direction with AM,
the direction of magnetio induction, there will be no electromotive
forru ciillt^ into action by any motion otAli, und a current through
All will not cause AB to be acted on by mechiuiical force.
We may therefore define a line of magnetic induction ia four
different ways. It is a line mich that
(1) If a oonductor be moved along it paraUe) to iuelf it will
experienoe no electromotive force.
(2) If a conductor carrying a current be free to move aloni* a
line of magnetic induction it will experience no tendency to do so.
(3) If a linear conductor coincide in direcUon with a lin« of
magnetic induction, and l>c moved parallel to itGl^ll' in any direction,
it will experience no electromotive force in the direction of it«
length.
(4) If a linear conductor carryinff an electric current coincide
in direction with a liitc of magnetic induction it will not cxiwrtenoe
any mechanical force.
OenVTtil EquatiMt g/" Elteir9m«Utt Force.
&9B.] Wo have Been that M, the electromotive foiro duo to in
duction acting on the secondary circuit, ia equal to — ~, where
dt
J
ELECTHOMOTITE FOBCB.
221
IN) determine the value of E, let us (liiterentiat« the quantity
fnnder the integral sign with respect to I, remembering that if the
«ecoD(laT} circuit is in motion, x, if, and s are functions of the time.
1 "We obtain
,rf^^ rfO^ dUdz^
^dt th'*' di ds"^ de d»^
^ = ~ J Ut di'*' dl t,"^ M T»)'^
_ \ c^^ ^'k. 'i^^\ ^j
J^dx di '*' dJ dt '*' d9 ds^ di
M
dy dt'^ ds^'^ dy <U' dl
J Us tU"^ dt'di^ d: dt^dt
■/c:
dsdl
'^^s^ + ^.^.+'^^>
dJdl^
f«)
Now consider the second tcnn of the integral, and siibotitute
ocn equations (A), Art. 501, the values of j and ^ . This term
th«n becomes,
flFdy dFdx^dx
/('
of 1 "' dF dx iir Of ar ai\ ax ,
"di' di'*"^di'^d^di'*"didi^Tf'^'
TOiMic© we may write
^df , dz dFx dst ,
Treating the tJiird and fourth terms in the same way, and col
dx df , dz
ikcUng the terxns in ji ~, and ^*, remembering that
J(didi'^^didt)'^' = ^di'
(3)
therefore that the integral, when taken round the closed
vantahca,
f, dz dx dG\dy,
f,,dx dy dir.dz .
{*)
EI.ECTBOMAOKCTIC FIELD.
We nifty writ« tli» expression in tike Torm
w
where
„ <fy .''' d^ *'*
q = a
ds das
dt~^4l
g_.dx d^ dif d^
dt "di dt di
BquUeAaof
Blrctnaiiotiv*
Furo*.
I
The tenns involving tbe new quantity * are introiluced for tin
sake of giving generality to tbe expressions for i*, Q, S. Tbey
disappoar from the integral when extended round the closed circuit.
The <)uaiitity "^ ie therefore indet<irmniutc as fur aK reg^nls tli«
prublem now before as, in which tbe totul electromotive force round
tbo circuit is to be determined. Wc shall tinil, howevor, that when
we know all the circumstancca of the prohlcm, wc can asvigu a
dcfiniU vnltio to 4*, and that it reprcMnU, ncoording to a certain
defmitigti, the electric potential at the pmnt (r, jr, .).
The ijrmntit)' under the integral nign i» etjuation (."S) r«prcsenta
the electromotive force jicr unit length acttnf on the element da of
the circuit.
If we denote by T(S, tlie numerical value of tlie resultant of P,
Q, and li, and by *, the angle between the direction of tbia re>
sultant and that of tbe element lU, wc may write equation (5),
E^jmwttdt. (6)
The vector ff is the electromotive force at tbe moving element
d». Its direction and magnitude depend on the podtion and
molioD of da, and on the variation of the magnvtic field, but not
on the direction of dt. Hence we may now disregard the circum
stance that da forms part of a circuit, and consider it nimply aa a
portion of a moving body, acted on by the electromotive force tf.
The electromotive force at a jmint ha* already been dt^fiucd in
Art. G8. It ia also called tbe reaullant elrctriea) force, Wiug th« _
force which would be experienced by a unit of positive electricity^
pluu^I at that point, We have now obtained the moet general
value of this quantity in the ca« of a body moving in a magnetic^
Geld due to a variable electric Hy»tcm. ^
If the body is a conductor, the electromotive force will produce a '
current ; if it is a dielectric, the electromotive force will produce
only electnc diq>Ucem«nt.
d
599]
ASALTSia OF F.I.ECTBOMOTITE POItCE.
223
^P 'ni« «)ectromotiv« force nt a point, or on a patiiclt;, must be
carefully dUtinguUlied from the electromotive force nlon» an arc
of a curre, the latt«r ijuftBtity being the lineintogral of the former.
I 81* Art. 69.
^p 599.] The electromotive force, the components of which are
defined \>y equationH (K), depends on tliree circumstances. The firet
I of U>ese ii the motion of the particle through the magnetic Held.
H^e put of the force depending on this motion \e expressed by the
first two terms on the right of each equation. It depends on this
t Telocity of the particle transverse to the lines of magnetic iaduction.
If IB is a vector representing the velocity, and © another repre
senting the magnetic induction, then if 6, is the part of the clec
^ tromotivo force depending on the motioo,
P e,= r.®©, (7)
OFi t])e electromotive force is tJie vector part of the product of the
magnetic indiKtion multiplied by the velocity, that iti to my, the
^ magnitude of the electromotive force is represented by the urea
B of the parallelogram, whose sides represent the velocity and the
magnetic induction, and its direction is the normal to this pamllel
ogram, drawn so that the velocity, tho magnetic induction, and the
electromotive force are in righthanded cyclical order.
I The third term in each of the equations (B) depends on the timo
variation of the magnetic field. Tliii< may bo due either to the
timevariation of the electric current in the primary circuit, or to
motion of the primary circuit. Let (S^ !>« the part of the eleetro
, motive force which depends ou these term*. Its components are
4F
dl'
dG
dt'
and —
dll
JSI
^B mod'
^Faad tlictte are the components of the vector, — "^ or —SI. Hence,
^^ @, = 9I. («}
^M The last term of rach equation (B) w due to the variation of the
^ fanctiott 4" in diScrvnt iiar1,» ijf thi; field. \V« may write the third
part of the electromotive force, which is due to this cause,
m fs, — V*. (9)
Tlie electromotive force, as defined by equations ( B), may therefore
be written in the quaternion form.
224
EtEarEOMAOKETlC FIELD.
[600.
Om tht JfodificaticH o/tie BqaafUnt of Eketfom^t'irt FQret lehm tie
Am* to nkich tiey are nferred are moctHg in Space.
600.] Let a<, /, / be the ooonliQates of a point referrwi
systeiQ of rectangular axes moving: in space, and let x, y, z be
ODoixIinatee of the same point referred to fised axee.
Let the componente of the velocity of the ori(;in of the roo'
eystem be u, v, k, and those of it« angular velocily w,, <■,, mj
referred to the fixed syet^m of axes, and let us choose the fixed
ax«s so a» to coinoido ul the f^iren instant with the movioff odM)
then the only quantities whieh will be different for the two syBteKl
of axes will be those diir«rcutiitt«d witJi respect to thv time.
ftjr
i
r— denotes a component vciocitr of a point moTinff in rigid Ci
nexion with the moving axes, and j and y those of any mnvin^f
point, having the same iostantaneous position, referred to the fixed
and the moving axes respectively, tbca fl
with similar VfoHaui for the other components.
By the theory of tl>e motion of a body of invariable form,
8*
8? = •+»*» J"'
Since .F is a com]>onent of a direct«d ({aaDtity parallel to x,^
if jT be the value of r: referred to the moving axes, it may he
dt
shewn tiint
dt
dr dVhx ^ dFhg . d¥hx „<//■ ,,.
dV dV
Substituting for j and ^ their valuM as deduced &&m the
dz
eqnatione (A) of magnetic induction, and reroeubering that, by {2)
rf B* d h}f d it _
df
dt ~ dxbt'^ dxbt * 'dx it* dyit "•" S" 8« ■'' rf» U
^87 +
M.dF
^7 + 37*
(*>
If we now pat
ELKCTROMAOIfETlC FORCB.
_^=^»£,<,»J,^,
225
(«)
^
The eqwttion for P, the component of the electrorootire forw
parallel to «, is, by (B),
■ ''' '^ '* is)
i>=/^_i_:^.^,
dl dt dt dc
referred to the lixed axes. Siib»1ittitinfr the tuIucs of the quanti
^^in tis nfernid to tJie moving axvo, uo hjivtt
(9)
^
W the valac of P referred to the movin^r axes.
601,"! It api^eara from this that the eUctroniotiTe force is cx
iressed by a formula of the same type, whether the motions of tlie
Gondactoni be referred to fixed ases or to axea movin)^ in apace, the
only ditference bctneen the formala bein^ that in (he case of
moving axe« the electric potential 4' must be changvd into 'i' + 'V.
In all cat«s in vrbich a cuTreDt is produced in a condnctiu^ eir
loit, th« eJevtromotire forve is the lineintegral
'^M<^t'>'
(10)
Icen round the curve. The value of 4' disappenra (Vom this'
ntegrai, so tJtat the introduction of 4^ ha« no influence on its
Lvalue. Id all phenomena, therefore, relating to oIim^ eircuit« and
he currents in them, it ia indilferent whether the axes to which we
Er the sy«tem be at rest or in motioD. See Art. 668.
^B O* tJie Eiairotaagndic Force aeititg on a (hndnelor kUcA cttrriet
^B a» EUcirie Otrreni ikrough a ifagHttie FieU.
^^^^pS.j We have aecn in the gt^nornl invc»l.i(^(irin. Art. .Sfl3, that if
^HP> one of tlie Tariablex wbici) delf rmine tlie [M>!(ition and form of
^Hlie secomlar} drcnit, and if .V, is thfi force acting on the necoudary
^circuit tending to increOM this variable, then
X.=
dM
d^
'ih
Sine* fi u independent ofj*], wo may write
Hi
'P
=/(
'fy
dt
^iJ*'
(1)
(2)
TOLII.
II.BCTROMAOKl
FIELD.
[6o
and wfl have for tbe value or J
'. = .i,/('
p
fix
»
.A
(>)
Now IH 1)8 «uppo»! that the displaccmcDt conni4« in movins
every point of the circuit through n distance bx to tlie dirccti
of X, ir bcinff any oontiiiuous function of #, eo that tim difle
parts of th<! circuit move independently of each other, while t
ciretiit n^RiuinK continuoUH and closed.
AW let .V h<^ the total force in the direction of dl vsting
the part of the eirrait from * = to * = », then the (MiTt
aponding to Uie element dt will be  Jt. We shall then have
followinff expression for tbe work done by the force during th^^
dieplACi'ment, ^H
wher^ the integration is to he extended round the closed cutvi^j
remembering that tr is an arbitrary fanotion of*. Wo may then^^
fore perform the differentiation with respect to hx in the mmr
way that we dilfercDtiatcd with respect to V in Art. 598, romcm
beriug that dx d^ , dt
. = '. j'\~ = 0, and ^7^3 = 0. (j
dbx
Wc thus find
dix
The la«t terra vnniehee when the intoftration is extended round
the closed curve, and since the equation mnet bold for all fonnt
of tbe function bx, wc must have
di^'i'di^d;)* '
nil ijijuntion which gives the force panllel to « on any clement
the circuit. The forces parallel to y and ; am
dY . , da dx\
dZ . /, dx dv^
llie reaultant force on tJie element » given in direction and
magnitude by the ijualcmion rxpre*Hon i^Fdp^. whore f, is the
nDn»ric*l mcauirc of the current, and dp and S are reoi
(
ioc^
ILKTBOMAOITBTIC FORCB.
trG])rc(MMit.ing th« etomcnt of tlie cireuJt and tlie magnetic in>
duotion, and tlw mtiltiptication is to be nndorrtood m Uic Hamil
jtooian wn«e.
r 603.] If the conductor in to bn treated not as n line but tus a
bod}', we must express the force on tlie el<^mcnt of li'iigth, iintl tlie
■ current through the complete section, in terms of K)'mbiils denoting
'tbe force per unit of volume, and the current per unit of area.
I^t X, }', H now represent the oomponenta of the force referred to
unit of volume, and u, r, la those of the current referred to unit of
area. Hien, if S represents the section of the conductor, which wi
shaU suppose small, the volume of the element dt will be Sd», and
I, dx
Hence, cqtuition (7) will become
XSd*
= S{vck6),
{Equaclont nf
Foroc.)
(10)
(C)
■ similarly T = ua— «c,
and ^ = «{ — ffl.
^ Here X, Y, Z are the components of the eleetromngnetie foroo on
^fun element of a conductor diviiUtl by the volume of that element ;
>, r, v arc the oomponents of the electric current throngh Uie
element referred to unit of nren, and a, i, t arc the oomponents
of the magnetic induction at the element, which arc itleo referred
to unit of area.
If the vector % reprosenbt in magnitude and direction the force
< acting on unit of volumt of thv couductor, and if (£ reprueents the
^■electric current flowing through it,
B g = r.(£S. (11)
^^^^ft« •quMtmw (B) of An. DflS nuiy ba jinrfod by Ihe following m*thwl, derived
ftttn ProfoMir M>i<t«S'4 HMBoir nn A [tynkmicol Theory of tbc ElcotrDiiuigtigtic
VMd. Fkit Tnn^ ISS4.
Tbe lime rartaUan of — ii nujr be Ukvii in two (]»rt>, imn of wlildi dopMiiU aiiil th«
«lliv doa Bot depend on Uic motion of the irirouit. The latMr fon la deul;
/'
■if, ifO. ifff .
I find the former let ua coniider >n tre It fonuins put of » dTciilt. *ni] let u*
i" tbie •!« t« move klong rula, which inav Iw uVnn u ;ianll*l, with velodcj *
hoae oon^Mmcoit* m i, jf, k, the rart vF i)im i:in:ii>i livioK meanwhile (uppiieMl
[■tolJiinBr;. We may then euppoeo Uut > mull pBnUc'ogrBin ia gUMnted bjr tb>
[inovn^ an, Ui» ilinnlloa'Ctwioea oTtbe niunuJ to which are
flymf Un* n.J^tfr
*■'*''■•* nin»"" eirtn*' ».ift#
a, Handle dfaectiau«o*ianiifJlf and 6 Ulh«M>sUl'oliroon e uul >«.
ELECTBOMAGiTETIO FIELD.
±0 B «ign« of X, ;i, >■ wo may put m — —1, r = t ; thejr then became
0. 0,  ..v'j oiryht to do with a riahtluiidwi syalcin of alee.
Now Itit a, h, c be the componenU of magDCbio inJitclJon we then hare, due to tbc
motluD of Ii in time Sc
If we tuppoee eftcb put of the circuit to move in a eimilu' manner the reaultaiit
Mt vi'lU be the uiutiun of the circuit aa a whulu, the cuireats in tlie rails forming »
I in each case of two adjacent arcs. The tiinu varialioQ of — p due tu tlu
of the circuit is therefore
— y{a(nj— Biil + twosimilai tenni} dt
taken round the tnrcuit
—J'iry — M) dx *■ two rimilar tenni.
The result* in Art. fl02 for tbe componente of eledramagnetic force m»y bo dedims!
lioni the above exprestion for tp; viz. let the arc 3( b« dinpluced in the dJreetim
I', m', n' through a diatance ti', then
Sji = {^(_cla — bB)^ > similar terms} tiS»'.
Now let £ be the :ecamponeat of tbe farce pon the arc I, tliea for unit ouirenl we
d bj Art, fifiS, rfx dp
dt dx
•« cai — bn,']
CHAPTER IX.
GESBRAL EQUATIONS OP THE ELECTKO MAGNETIC FIELD.
k
6OI1.] Xx our UiGOTOtical tlt^cussion of «Iectro(lynamios wo lic^n
hy assuming that a ejiiiem of circuits cnrrjing vItK;tnc currcnU
is a dynamic^] syatem, in which tlie currents mtiv l>e ro^mlvd as
velocitiea, and in which tJifi coordiuaUs corresponding lo thvse
velocities do not them!U>]v«« appear in the oquationti. It followi;
irora thin that tlie kinetic energy of UKiRVHtcra, ineofarasitdopemlc
on the currento, is a homogeneous quadratic function of the currcnUi,
in which the coeSicients ilciwiid only on the form and relative
position of the cireuit.H. Assuming these coefliaents to be known,
by experiment or otherwise, we deduced, by purely dynamical rca
sooing, the bws of the induction of currents, and of electromagnetic
attraction. In this investigation we introduced the conceptions
of the elect rokinetio energy of a system of currents, of the electro
magnetic momentum of a circuit, and of the mutual potential of
two cirouits.
We then proceeded to explore the field by mcansof various con
figurations of the eeeondary circuit, aud were thus led to the
couception of a vector 91. having a determiuato magnitude and
direction at any given point of the field. We called t.Iiit! vector the
electromagnetic momi'ntum at that jKiint. This qtmntity may be
conHidercd as the timointt^gral of the tlcctro motive force wliich
would ho produced at that point by the utiddcn removal of all the
currentfi from the Sold. It it! idviiticnt with the quantity alrcjidy
tnTeetigated in Art. 405 as the vectorpotential of magnetic in
duction. Ibt components parallel to ic, y, and z are F, 0, and //.
The electroma^ctic momentum of « drcuit i« the Iine>integral
of tl round tlie circuit.
W« then, by meant) of Theorem IV, Art. 24, Irausformei tlie
liMI
230
GE.VEilAL EtiUATIOSS,
[605.
I
he
M
»i1
linciotegral of 91 into tbe sarTaceintogral of aootber vector, ^1
vihoBe compoDbnta are a, i, c, and we found ihat the ]>henoineiu
of induction Hue to motion of a coodtivtor, and tfaoee of elrttro
ma^ctic force can be expressed in t^'nus of >£. We gsre ta IB
tiie namo of tbe Majfnctic induction, einoe it« properties are iden
tical witb those of tbe lines of magnetic induction as iaveetigat«d
\>y Faraday.
We aUo estnljIislRtl three eets of equations : tbe first set, (A),
are those of inaguetic induction, «x])rc»8inn: it in terms of tbe elei>
tromagnetic raoRientum. Tlio second set, (1)). aro those of electro
motive force, expressing it in terms of tbo motimi of the condoctor
ucrnea the lines of magnetic induction, and of thu latv ofTariationj
of the electrtjmagnetic moni«ntuni. The tbinl set, (C), are tbr
cciiations of electromagnetic force, wcprtwsing it in terms of tbe
current nnil the magnetic induetiott. i
Tbe current in till tlicvc nucs is to be nudcrstootl as (he sctiul
current, which includen not only the current of conductiou, but the_
crurrent due to variation of the cleetric diifjilacement.
The magnetic induction 9 i^ the quantity nbiefa ire have Blreadji
considered in Art. 400. In an unmngnet.ixed body it in identical
with the force on a unit magnetic pole, but if the l>ody is ma^
mtized, cither permanently or by induction, it io tin* force which
wt^uld be exerted on a unit pole, ii' plaoeil in a narrow cTvvasse in
the body, the walls of which are perpendicular to the direction of
magnetization. Tlie components of ® are a, b, e.
It follows from the equations (A), by which a, 6, c mw defined,
that da di tk
Thi* WW riuwB nt Art. 403 to be a property of tbe
imiuction. c
606.] We have deiined the inaguetic force within a magnet, >^
distinguished from tbe magnetic induction, to be the force on a"
unit polo placed in a narrow crevasse cut parallel to the direction of
UMignetizntion. Tliis quantity is denoted by ^, and ita compoDGOta
by 0, fi, y. Sec Art. 398.
If 3 ia tbe intensity of magnetization, and A, S, C its com
poDcntd, then, by Art. 400,
fi = ^ + 4sJ9, J (BiuatloM of M>««Mli«Uon.} (D)'
^ = '»
r defined,
nugmo^i
I
H W« iwiy cull these the equations of magrnctization, and they
HiniltcuiU (hat iu the electxomagiietic Bvetem the matr'ietic indwction
V S), ciniMdiTed tts a vector, is the eum, in the [I;imiltonun sense, of
two viHtow, the ma^netifi force ^, and the magnetiKatioQ 3 multi
plied by 4it, or S9 = ^ + 4ii3.
la oertaiu «iib»1ance«, the magnetization depends on the mai^nclie
force, and this is expressed hy the system of equations of indtiood
ma^iHism frivco at Arts. 42G and 435.
—^ GOfi] Up lo this point of oar investigution we have deduced
Be\'iTything from purely dynnmicul eouaiderations, without any
nferoiici; to qiiantitittive experiments in electricity or ma^ctism.
The only nst we have miidi.' of experimental knowledge is to rc
cognifc, in the nhstrjicl quantities deduced from the theory, the
t^nctete quantities discovered by experiment, and to donoti tliem
I by namex whieh imliaik: their physictil relations rather than their
[ntatbemattcal ^ncMition,
In this way we have painted out the exidtonee of the electro
momentum 31 as a voetor who:^e dinntion and magnitude
Dm one part of njuiec lo another, and froni thin we have
deduced, by a mathematical procettt, the magnetic induction, j*, as
a derived veetor. We have not, however, olitained any data for
_detennining either SI or © from the distribution of currents in the
Por this purpoi»e we must find the mathematical connexion
rfweiMt tbne qniuitities and the currents.
We begin by admitting the existence of permanent magnets, the
lutiml action of which satisfies the principle of the conservation
of energy. We make no assumption with respect to the Inws of
magnetic force except that which follows from this principle,
nftinely, that the force acting on a magnetic pole must be capable
of being derived from a potential.
We then observe the action between currents and magnets, and
^■we iind that a cnrrent acts on a magnet in a manner apparently the
same as another magnet would act if its strength, form, and position
were properly adjusted, and that the magnet iict« on tlte etirrcnt
in the same w^y as another current. These observations need not
be supposed to be necompanied with actual m<.ai«iireroenU of the
foTOOB. They are not therefore to bo conBiilcn'<l a* furnishing
ntunericsl data, but are useful only in suggesting queations for
our consideration.
He question these observations su^^'st is, whether the magnetic
field produced by electric curreuta, as it is ainiUar to that [iroduced
:we^
232 MNEBAL EQUATIOm. [607. "
Ity permanent magneU in many respects, rettembles it abo in beinjH
related to a ])vt«nliAl ? ~
The. cvkiJ«Duc tli«t aa electric cireait produces, in tlie Kpooe cur*
rouiiiling it, magnctiv cflVct« precisoly the name u tfaoae produced
by a niiign<tic mIivII buundetl by Uie circuit, baa been »tatcd In
Art*. 482 *85.
W« know that in the ca»e of th« nu^iietic Bh«U there is •
potential, which has a dttcrtninato value for all jioinlii oulitide tl
«ubBtttUC« of the shell, but that th« valueit of the ]x>teutjal at twtr'
nvi^hbouriug points, on opi>oiiit« eidea of the shell, difler by a flniu
quantity.
If the macnetic field in the neighbourbood of nn electric current
rcftemliles that in the Dei^bbourbood of a tnagnetio shell, the
tnftgnetic potential, as fooud by a lineiate^iatioD of tJie magnetic
force, will be the same for any two Unes of integration, provided
one of these lines can be trangformLii into the other by continuous
motion without cutting the oleetric current. ■
If, however, one lino of inttfration cannot be transformed into
the other without ciittinfj the current, the linc'intcf^.kl of thij_
magnetic force along the one line will ditfer from that along tli
otliLr by a quantity dependitifj wti the strength of the current, Th^
magnetic potential due to au electric curn.>ut is therefore a fuuctic
having an infinite Hcrics of values with a common difTereacei,
particular vuIik' depending on the course of the line of integratioaj
^'ithin the siibntancc of the condiwtor, then is no such thing
a ntngnetic jwleiitial.
607.] Asniming tliat the magnetic action of a current has a
mo^iiotic )otentJal of this kind, we proceed to exprvas this rwal^^
tnathematically. ^M
In the itnrt place, the lineintegral of the magnetic force round
any closed curve is zero, provided the cloised curve does not suiTonnd
the electric cnrront.
In the next place, if the current paeees once, and only once,
through the closed ciirvo in the positive direction, the line*tntegnl
has a determinate value, which may he used as a meaaure of the
strength of the ctirrtnt. For if Utu closed curve altera its form
in any continuous manner without cutting the current^ the lin^
integral will remain the same.
In electromagnetic measure, the lineiutegml of tlw mAgoe
foroc round a olnsttd curve is numerically e<iual to tb« cuiraol
tlirough the closed curve mulliplicd by iv.
ET.ECTBIC CURRENTS.
If WO take for the olostHl curvu tho psrallelo^m vrho«c niim
are tljf and ilz, iha liue'inUgnil of the magiietic force ronud th«
ad \{ u, e, «t aro Uie comjioncntB of the flow of electricity, the
irrent through thp {Uiralklo^rram is
Mii!ti(i]ymg this hy 4ir, and equating the result to the line
ategral, we ohtain the eriintion
dy dz
ith tbi' similnr oquations
4a dy
at dx
d& da
■\nw= J i
dx dy
(BquBtion* of
Bloetrio Cumntf.)
(E)
Jwhco
Icondi
Fhioh determine tho mafiiiitutle and direction of the electric cnrrente
tJiu mu^nctic force at every point \» given.
When tliere is no current, these equations are equivaleDt to the
lition that adx\^ds>rydz=hQ.,
>r (hat the magnetic force is derivable from a magnetic potential
I all (mints of the lield whtre there are no currents.
By dilTeretitiating the equations (E) with nspect to a, j, and *
rcsp«:tive)y, and adding the results, wv obtain the e([uation
iu dv ^"^ _n
di'*'d^'*"dl~^'
i» that the current whose cdmpoiieiits are u, v, le is
the eondition of motion of an iiici'mprcssihle fluid, and
that it must nect«»irily How in closed circuitM.
Thia equation is true only if wc take u, v, and ir as the com
ta of that oledfie flow which is due to the variation of electric
displacement as well n* ti> true conduction.
We have verj* little experimental evidence relating to the direct
electromagnetic action of cumcDta due to the variation of electric
diiplaoement in dielectriea, hut the extreme difliculty of reconciling
the laws of elect romagoetjsm with the existence of electric currents
which are not clowed i» one ncaM^n among many why we mu«t admit
the exi.'teQce of transient currviit» due to the variation of displuce
menl. Itieir importance will be seen when we come to the olectro
^ Tht
^^Msnen
^magai
etic theory of light.
GEKEnAT, EQFATIOyS. f6o&
608.] We have now d«t«nnin«J the nUtioni of th* princiial
(uantitiefl cx>iicemed in fhc pliouomctiu dii^ovcreil hy Orctod, Am
pirc, Kod Faraday. To coiinwt tlicw" with the [ilK>iinniL''na <U«cribctl
in tlic former parts of thi» treatiNe, Home additioiuil rclutions ■«
jwcvesary.
When electromotive force scIk od a material body, it gimduce*
in it two electric&l effects, called by Faraday Induction and Con
duction, the 6rst being moat conspicuous in dielectiics, and tlie
socoiid in conductors.
In this treatise, static electric induction is mea&ured by what we
iutvv cuUvd the electric <!i8plaoement, a directed quantity or vector
which wc IiavD denoted by X, and its components by J", g, h.
In isotrupic suliatance^ the displacement is in the same direction
0.1 the electromotive force which produces it, and is proportional
to it, at leiut for small values of this force. This nay be expressed
by the vquatioa , ^ . , _. .
iT> «'fe (Eqn»U"n of Eleotno /p.
where A' i« the dielectric capacity of the snbstanoe. See Art. 6!>.
In «ubstanccs which ore not iiotropio, the components/;^, k of
tlic electric displacement £ are liiieur functions of the comjwnents
P, (2> R of the electromotive force ^.
The form of the (^nations uf electric diaptacement is simiUr to
thiit of the (spiationx of conduction lut given in Art. 296.
Thene relations may be cxprevscd by xaying that K is, in isotmpic
bodies, a scalar quautity, but in other bodies it is a linear and v
function, operating' on the vector (S.
609.] The other dfcct of electromotive force is oondnotion.
laws of conduction as the result of electromotive force were estai
blished by Ohm, and an> explained in the second part of Uii
treatise, Art. 211. They may be summed up in the equation
where S is the intensity of the electromotive force at the poin
j[ is the density of tJie current of wind iicl ion, tlie oomj^neuts
which are p, j, r , and C is the conductivity of the Bubstinco, wbich^
in the case of isotropic subefances, ta a simple scalar quantity, but
in other mbstanecs becomes s linear and vectAf function operatin]^
on the vector 9. The form of this function is given in
eoordinatcs in Art. 29B.
610,] One of the chief poculiaritien of thi» treatise is the doctri
which it Bseeits, that the true electric currvnt (5i thnt on which
I
I
^
614]
CUREESTS OP niSPLACEMEKT.
235
lectromngDetio phenomena dctpend, ut not the same thing as j^, the
current of conduction, but tluit th« timeTariution of 'S>, the elet^trio
displacement, must he taken into aocount in estimating tlie total
oiovement of electricity, so that we miiKt write,
6 = ft + S), (BiumW uTTVub Cumiilt.) (H)
r, in terms of the components,
do
("*j
ill,] Since toth St and [D deiwnil on Uie electromotive force 6,
twe may express tlie true current 6 in terms of the electromotive
force, thus , 1 j ^
(I*)
»r, in the coeo in which C and A' are constants,
Alt dl
612.] The volumedcneity of the free electricity at nny point
u fon»d from the components of electric displacement liy the
,«l'»t'«i> df da . dA
Pdi + d^
+ S
fJ)
618.
The surfacedensity of electricity is
< = f/+ «,!f + nJi + lT + *«'/ 4 n'A', (K)
(where i, m, n are the directioaeosinea of the normal drawn from
the xiirfnce into the mcdinm ia which f,g,h are the eomponenta of
I dinpliiecnient, and V, m', n' are those of the normal drawn from
I Burfiict into the medium in which they arc f, y', A',
614.] When the m^netizaticn of the medium is entirely induced
liy the magnetic force acting on it, we may write tlie equation of
'ioiluced magTMtiuttion, @ = uJS, (L)
where fi IK the coefficient of ms^etic pennouhility, which may
bo considered a scalar quantity, or a linear and vector fnnetioa
ittn^ on ^, according ns the medium is itotropio or not.
236
GENERAL EQCATIOKS.
[615
<
615.] Tliiise may be rcgatded aa the principal reUtioos anions
the (juanUties we have bvcn oonsidering. T1i«y may b« combined
so a» to eliminate Aomo of tboM quantities, but our oltjcct at prwral
is Dot to oI>t«in oompftctAeM in tbe mathenuit iuil Torniutac, \M
to ezpren every relation of which wc have any knowledge. To
eliminittv a quantity which ex]irc»tc:t a uovftil idea woulii b« ntber
u lo9S ilian a gain in this stugv of our cni]uiry.
There is one remit, however, which we may obtain by cnmhining
equations (A) and (K), and which is of ver}' jfreat importance. Mi
If we suppcute that no nutlets exist in the Beid except in the^
form of electric circuits, the distinction which wc have hitherto
maintained between the magnetic force and the ma^E^^etic inductioa
vauiehea. because it is only in magnetized mattor that these quan
tities tlifier from each other.
According to Ampere's hypothesis, which will be esplained in
Art. 633, tbe properties of what we call magnetized mattor are dne
to motecular electric circaits, so that it is only when we regard th*
Hubstaiico in large masses that oar tiieory of ma^etixation ii
upplictibtc, and if our matlit'matical methods arc supposed capable
of taking account of what goes on within the individual molocolcK,
they will discover nothing but electric circuits, and wc ehntl find
the magnetic force and the magnetic induction everywhere identical.
In order, however, to be able to make uec of the olectrOKtatic or of
the elect romagnetiu system of meneurcmcnt at pleasure wc shall
retain the coolTtcicnt fi, remembering that its v«luc is unity in Uie
electromagnetic n'stom.
616.] The components of the ma^etic induction are by e^po*
taoDS^A), Art. 691,
The oomponents of tho electric current are by equations (E)
Art. 607, given by Jy 4^
Ja dy
m
dO
is'
' Hx
dP
dff
'di'
dx
dO
dF
Ji'
"^
ivm
4vv
dfi
*»" = ^^
da
If wo WTit«
I may write equation (1),
imilarl}'.
If we write
\\iien T '\% the ilJEtance of the given ptiint from the element (jJ,^,*)
tb« intcgratioiu arc to be extemktl over ull space, then
(ix
The qnnntitj x di«tp])onrs from the eqiiatioiis (A), anil it \» not
t\»iti to any phvsiotl phenomenon. If we Eiipjtojte it to he ten
rywhcre, .' will also bft wro cvetywhoro, an<i equatioDi (5),
litting the accmtit, will give the true ralnn of tho component*
■81.
' lb* nogitir* flifn I* vaplojtA )i«r# in nttlar t» duJ(o oiu cxpttaMMU cgmiBleat
*i libcw in vhicb (juatoniMH m vaifiojfL
617.] Wo may thcrorore adopt, as a dofinitioa of $(, ihat,
is the vectorpotential of the electric current, standing: in the sam^
relation to the eWtric current that the scalar potential staods ta
tJie matter of which it is the pot«Dtial, and obtained by a sinuhr
process of integration, which may be thus deecrJbcd: —
1Vom a given point let a vector be drawn, rcprcsontinj^ in nttg
nitiide and direction a given element of an electric current, dividrd
by the namerical value of the distance of tlie clement From t
^ven point. Let this b« done for «very olcmont of the i
current. Tlie rcsnltant of all the vectors thus found is the
tial of th« wliole current. Since the current is a Tcctor qnaotit
it* priteiittal is also n v<«tor. Sec Art. 122.
When the distribution of electric currants is given, there is one,
and only one, diiitribution of the values of 91, sncb Uiat !l iit ereiy
where finite and eontinuouf, and mtiitfieG the c<nattODs
V»91 = *s/*g, S.r9l = 0.
and vnniHhe» at on infinite distance from the electric system. Tbii
value i» that given by equations (5), which may be written
I
^ = p.jjj^dxdfdt.
QuatfrnioK Erpre«tloiu/or the EUciromagnetie EfHatioMt.
618.] In this tnatise we have endeavoured to avoid any procew
demanding from the reader a knowledge of the Calculus of Una
lernions. At the tame time we have not scrupled to introduce the
idea of a vector when it was noceteary to do so. When vre have
had occasion to denote a vector by a symbol, we have taetl a
German letter, the number of dilTerant vectois being so ^reat that
Hamilton's favourite synibots would liavc been cxhangted at ont^'.
Whenever theraforo, a Oermao letter is used it denotes a Ilamil
tonian vector, and indicates not only its magnitude hut its direction.
The constituents of a vector are denoted by Roman or Greek letters.
The principal vectors which we have to conntler are
Vcctw OcAiiitiiifntei
The radius vector of a point p * f *
The electromagnetic momentum at a pMnt M P G B
Tlic magnetic induction B air
The (total) electric ctirrent Q ■ v k
The electric displacement J) f f k
6 1 9.] oWtbhsi^^Spr^ioss^^^^^ 238
The cWtromotivo force II P Q R
The mectiaiiical forou % XTZ
Tbc v«lovitj' of a point ........................ & ov p x g i
The mu>nctie forci<! ......iii i... ^ a j3 y
The infousit^ of nmsnctiJiTttioo 3 ABC
The current of oomiuclKiii ft P J ^
Wv have also the foIluwiDg t>t:alur fundions ;
The electric {iot«nliiil +. ]
Tbv tnagTietic iioteiitinl (wbovc it exiet«) 11.
■Hie I'tcctric dciigity e.
The <]<!nsitf of mujfiielie 'mutter' m.
Beei<]cs these wc hnve the foIiowiD)> qiiftntities, indicating; physical
tropertiee of tho meditim at each point : —
C, the conductivity for electric currents.
K, the dielectric inductive capacity.
fi, the magnetic inductive capacity.
Tlieee quaotitieB are, in isotropic media, mere scalar functions
f p, but in general they are linear and vector operat«» on the
rector functions to which they are applied. A' and n are certainly
. always self>conjiif^te, and C ie probably bo also.
619.] The equaUons (A) of magnetic induction, of which the
~ dg dz
hfl written 9J = TvM,
Fwhero V \% the operator
.d .d , d
'T^^^dy'^dB'
V indi«at«s tliat the vector part of the result of this operation
' is to be taken.
Since 91 \* i^nbject to the condition SV9I = 0, V% is a pur«
TCetor, and the symbol V is unneccesary.
^TIw equations (B) of electromotive force, of which the first in
o . ,. dF d*
^u e = r«s«iv+.
no oquationa (C) of mechanical force, of which the first is
V . J* dil
ax dx
^become g = FfS^eV'i'mVa.
GEITERAL EQUATIONS.
Ifl Ions (D) of magnetization, of which the first is
a = a +AvA,
tecome S =^ + 4Tr3.
The equations (E) of electric currents, of which the first is
become 47iS=: W^.
The equation of the current of conduction is, by Ohm's Law,
St=Cik.
That of electric displaeemi
The equation of the total cnrrci , arising from the variatioD of
the electric dieplacement as well as rom conduction, is
When the magnetization arises frcm magnetic induction.
We have also, to determine the electric volumedensity,
To determine the magnetic volumedensity,
m = SV%
When the magnetic force can be derived from a potential
CHAPTER X.
DIMENSIONS OP ELECTRIC UNITS.
S20.] EvKnT clcetromajjnetic quantity may be ilelincd with
reference to the fundiimcDtul units of Length, Mass. nnd Tirav.
If we be^n with Uic Htfinition of (In? unit of electricity, « {jiven
in Art. 65, we tnny obtain definitions of tho units of every other
electromagnetic quantity, in virtue of the equations into wliich
they enter alon^ with quiintitie^ of olettncity. The syvtem of
onitB thns obtained i» culk0 the EK>ctrost»tic SyKtein.
If, on the other hand, we hei^:in with the definition of iJie unit
mn^etic pole, as given in Art. 374, we obtain a different syetcta
of units of the suue set of quantities, 'litis system of units is
not consistent with the former 8yBt«ni, and is called the Electro
I nweoetic System,
■ We shall bc^^in by stating those relations between the different
units which arc common to both systems, and we shull then form
■a table of the (limensione of the units according to each system.
^1.] We shall arrange the primary quantities which we have
to conxider in jiairs. In the first three pairs, the product of tlie
^Iwo quantities in each pair is a quantity of cncr^ or work. In
Bihe vecoiid thn'c pairs, the product of each pair ie a quautity of
ene^y referred to unit of volunw.
FUUT TURKB PaiB8.
Eleetrottalie Pair.
Symbol .
(1) Quantity of electricity ...... e
(2) Lineintegral of electromotive forco, or electric po
tential S
rouu. B
242
DINIKSIONS 0? tflfira
[623
(3) Quafitity or free nugoeU*!!), or strength or a pole . ■>
(4) Mkgnctic potential
EltrfrokiKctU Pair.
(5) Ei«otrokmet)c momeDtuot of a circuit .
(6) Electric current
a
f
c
StxoKD Tnwa Paibs.
EUftrOttatic Pair.
(7) EU^nc displacement (mvararcd by KurraceMleDsity). 2)
(8) Electromotive force at a point . . . Si
Maynttie Pair.
(9) Ma^etic indiietion ■ Q
(10) Magnetic forcx; , ^
ElectrokimHie Pair.
(11) Intensity of electric current at a point . . S
(12) Vector potential of electric currenU . . . fl
622.] The following relations exist betveea tbe«e qi
In Uie Gr^ place, since the dimension* of mtxgy mm ~J^ I , and
[M ~[
jnli *vfl Iuiv« the
[*i] =[.a] = [fC] = [^].
[i;s] = t«*] = [«a] = [^].
Secondly, since e, p and 9( arc the timeintegnk <^ C, £, and G
"^'"''' ra=[S]=[l]=tn («
nirOly, since i^ Q, and j» are the lin^^.il»tegralH of (J, ^, aod M
reepectively, rj?] rOi r^i
Fioally. since e, C, and m are tlte sorfaeeintegnb or 3>, Q, and
TRB TWO HTSTBHS OF UNITS.
243
6!!3.] These fift««D eqnationH are not independent, and in order
I to deduce the dimeDBions of the twelve nnita involved, we require
one additional equation. If, however, we take either ^ or >» as an
independent unit, we can deduce the dimenEions of tlie rest in
r terms of either of theee.
« ^1 =[f]=B]
(3) ™i (5) [,] = W = [^] = [«].
(9)
(10)
(IS)
m m{ir\
624.] The relations of the 6r3t ten of these qnantitiea may be
exhibited by means of the following arniDgement : —
<f, 3), ■§, C and a.
m and p, ^ d, £
Jf IS, ©, Mand;5.
C and li, ^, 21, c.
The quantitiex in the first line are dcriv<'d from e by the same
oiKTations aa the eorren ponding quniilitleM in the second line are
derived from m. It will be seen that the order o( the quantities
in the first line is exactly the reverse of the order in the second
line. The Grat four of each line have the first symbol in the
nnmerator. The second four in each line have it in the dciio
K minator.
H All tho relations given above are troe whatever system of unit«
Hw« adopt.
^^■025.] The only eyet^ma of any scientific value are the electro
^HRk40 and the elLetrontu^elic system. The clectrosbitic system is
H
244 IJIMESSIOSS OP DKlTa. [626.1
founded on the definition of tlw unit of electricity, Arts. 41, 49, 
and niAy bt doduoed Irani the equation
whieh enpresaes that the reBiiltant fopcu S «t any point, d«e to tlie
action of a quantity of electricity ^ at a dintance L, is found l>y
dividing e by L*. Suhittitutiii^ tho oqtnition* of dimension (I) Bii<I
(8), wcfind rUfi r e \ r m T r If]
in Uie elcctro^Utic system.
The electrom^netic system is founded on a preiriKcIy fiiniilir
definition of the unit of utrengih of a magnetic potc, Art. 371,
leading to tho equation ta
whence [^] = [^] [fr^J^M"
and [f] = [i'i/l]. W = [i*3f *r'],
in the electrama^ictic system. From these results we find the
dimcn!«i«ns of tlie other quaatittca.
626.J Ta&U ^ Dimatnoiu,
Diamuiant in
Quantity of electricity .... * [f*,!/'!!''] £Z.*ifl].
Quantity of magnHism j
Electrokinctic momentum [ . j"! [L^M^] [i^J/tr'J.
of a circuit ) "
Electric current > iC ) ,,i„i„., ,. ...,„..
Magnetic potential} U] [i'^'?"'] [i*.W*r'].
Electric dieplaceroentl ~ 7Ti,,>. ., .. ■ ..
SnrfaccHlcnity } * ' " ' ® [i"*^*?"'] t^«i/»l
Eleclroiootivo force at a point S [ALV*?"'] [Z*Jf*J^.
Magnetic induction 89 [J^''JW^'] [£t.l/*r'].
Magnelic force ^ [UMiT'] [i.*ir*r'].l
Strength of current at a poiat fi [^iJ/Jr'J [£"'jtf*7''l
Vector potential V [Z*Jtf*] [iiifir'].
6fl8.]
TABLE OP DIMENSIONS.
246
627.] We hftve strtttdf Mnsidored llic products of the pairs of
tbeae quantities in the order ia which tliey .it:iiid, Tlieir mtios are
I in certain cases of scientific importance. Thns
GlMbwUtio EteotTCma4(iiDtia
lj^ = cnpacity of an accumulator
' cocAiciont of eelfinduction
of a circuit, or oloctro
magnetic capacity
_ t spocific inductive capacit}* l
Fe ~ i of dielectric J ■
^=1
:(
e
= magnetic inductive capacity .
j^ = resistance of a conductor .
d _^ ( spccilic rGMstiinoe of a 1
Symbol,
■ ?
Bjitom.
.yj.
L
[LI
. K
[«]
• f*
to]
, R
■T
■L'
r
[7]
ttaic
...I,
' 63S.] If the units of length, mass, and time are the xanie in the
two systems, the number of ekctrogtatic units of electricity con
uDed in one clcctromafjnetic unit is numerically equal to a certain
'^velociiy, the alwoUilc value of which docs not depend on the
magnitude of Uio rundamental units employed. This velocity is
f important physical tjuaiitity, which wc shall denote by the
mbol r.
Nnmier of EUctntttatie Unitt in one Meetromagnetie Vnit.
¥tyte,C.Q;t>,^.^ p.
T9fm,p,E,^,f&,% .V
For electrostatic capacity, dielectric inductive capacity, and con
ductivity, H.
For electromagnetic capacity, magnetic inductive capacity, and
siftance, f
Several methods of determining the velocity v will be given in
768780.
In the electrostatie cyiiteni tlio KpeeiRc dielectric inductive capa
city of air IN amumcd e<]ual to unity. This quantity is therefore
ented by j in the eleotromagnetic system.
246
DIMENSIOSS OP WXIT9.
[629.
In th« elect romaguotio Ej'stem the spcdfic nuigtictio inductin
capacity of air is aseumoil equal to unity. Thin (iiianttty is tbcn^
fore represented by ^ in the electrostatic ttystem.
M
<
Practical Sj/atem of EUctrie UmU.
629. 1 or the two syxtems of units, tlie electromagnetic iti of the
greater use to those practical electrioiaus who are oooupied with
electromagnetic telegraphs. Jf, however, the units of length, time,
and ma&s are those commonly used in other scientific work, soch
■a the m^tre or the centimetre, the second, and the •^tamme, the
units of rcsietancc and of oWtromotire foitw will Iw eu small thut
to express the (jiiuntiticH occurrlu); in practice enormouii numluT*
must he used, and thct units of qimntity and cttiiacity will be W
large that only cxcwdinfily xmall fmetioiiii of them can ever occur
in practici?. Priictiail vleotrioiaiis have tlierefore tuloptetl « «ct of
eU^ctrical unit* deduced by the electromagnetic flynteni from u large
unit of length and a small unit of ma:is.
The unit of length used for this purpose is ten million of m^lnt^'
or approximately the length of a qoadrant of a meridian of
earth.
The unit of time is. as before, one second.
The unit of mass is 10'" gramme, or one hundred millionth
part of a millipnimmc.
Tlie elci'.trimi units derived Tioni these fundamental units hav
beCD named after eminent eltetricul discovemri. Thus the praetif
unit of renistance is called the Ohm, and is ivpmented by the
rcsintanoecoil issued by the British Association, and dencribed in
Art. 340. It is expressed in the electromBgDetic ayKtcm by a
velocity of 10,000,000 metres per second. fl
The practical miit of electromotive forc« is oalled the A'^pII, and
is not very diflcrvnt from that of a Daniell's odL Mr. Latimer.
Clark liaa recently invented a very constant cell, whose eU
motive force is almost exactly ].(57 Voltw.
The practical unit of capaeity is called the Famd. The qraintitj
of clcetrioity which flows through one Ohm under the electrorooti^e
force of line Volt during om; second, is equal to the charge producid
in » condenser whoee cajwoity is ono Farad by an cleetnamotive,
force of one Volt.
The us« of these names i^ f<mnd lo be more «oDTeBient in practiii
than the coDstant repetition of the words ' dMttonuguetic units,'
^4
atimer «
uintit^
629]
PBACTICAL UKITS.
247
with the ftdditional statement of the ptuticular fundamental unite
on which they ore founded.
When yery large quantitieH are to be measured, a large unit
U formed hy multiplying the original unit by one million, and
placing before its name the prefix mega.
In like manner by prefixing micro a small unit is formed, one
millionth of the original unit.
The following table gives the values of these practical units in
the different systems which have been at various times adopted.
FViniAIOU'TlL
UVITB.
FsAoncAL
Ststu.
B.A. RsroBT,
16SS.
Thohson.
Vises.
Ill
Eartk'i Quadruitt,
Stetmd,
10" OramiM.
Mart.
Setond,
Gmmme,
Cudinulrf,
Second,
Oramvie,
MiUimare,
Second,
MiUignimnie.
Bemitmnoe
Ckpacitj
Qumntlty
Ohm
Volt
Fankd
F»nd
(ohkrgedtokVtdt.}
10'
10'
lO*
lO"
10*
10'
10'
10"
10"
lO"
10
CHAPTER Xr.
ON BSB80V AH1> STRESS IN THE ELECTROIlAOSBriC FIBLD.
EleetroHatic Enetyy.
630.] The «acrgy of the ejrsteco mxy be divided into Uie Potential
Energy and the Kinetic Kncrgy.
The >ot(>ntial energy due to electrification haa been ilre&dy coBl
sidored in Art. 85. It may bo written
r=}s(«*). (I)
where e is the charge of electrieity at a p?aee where the clcctrio'
jKitentiiil is ^, and the Kiimmution '\t to bo oxteaded to evcij y\»et
where there is electrifi cation.
If y. S' ^ 1*^'' ^^'^ components of the elwilric di^lncemeDtj tbaj
quantity of electricity in the element of volnme dxdydi is
in
itnd
'^dxdj/dt,
where the integration is to be extended throughout all spaee.
631.] Integrating this expression by parts, and remembering
that when the distance, r, from a given point of a &aite eWtrified
)<y)itein becomes infinite, the potential '!> becomes an infinitely small
quantity of the order r"', and that/, </, i bwomc infinitely small
quantities of the older r^', the expreiwion in reiluccd to
where tbi^ integration is to be extended throughout all space.
If we now write P, Q, R for the oomponeuts of the clectromotiTe
i/4' ti"^
force, iastood of — ^ , —
<tt
'ty
and — i , we find
r = \fjf{P/+Qfi + RA)dt<fy'tt.
m
3]
240
[Hence, the electrostatic oncrgy of the whole 6eW will l>e the «ame
we Biippose thjit it reeiJcs in every part of the field where fiW
bncnl forci." nnd ck'ctrical (lis placement occur, instead of being
conliiictl 1o the places where free cicctricitj' is found.
Ttw! encrpry in unit of volume is half the product of the electro
motive feroi And the eUctric dieplaectncnt, multiplied by the cosine
of the un^lo which these vectors incltide.
tin Quaternion languag'u it is — 16'@£.
1
Magnetic Energy.
* 632.] We may treat the energy due to magnetization in a w»y
ilar to that pursued in the case of electrification. iVrt. 88. If
B, C an the components of maj^Detization and «, ft y the
componetits of ma^etic forc«, the potential eoei^y of the eyet«m
of magnets is then, by Art. 389,
I
~\Jff(Aa+Sff+Cy) d3>dyds.
(»}
iC integration being extended over the apace occupied by mag
netized niatt«r. This part of the energy, however, will be included
in the kinetic energy in the form in which we ahall presently
obtain it.
633.] We may transform this expression when there are no elec
tric currents by the following method.
We know that ia db de
Henoc, by Art.97, if
do. ^ da. da ...
»=.u' ^=df' ^=df ('>
if alwayB the case in magnetic phenomena where there are no
currents, rrr , ,
i jJJ{aa + 6^+cy)d^dydz = Q. (9)
the integral being extended tlirooghout all space, or
^ffJ{{a + i^A)a■^(8 + ivJi)fi + {y+4l,C)y)dxJyd^^^). (10)
Hence, the energy due to a magnetic aystem
^kJJfi^«+^P + CY)drdyd: = ~^JjJ(<,*i^^ + y^)djcdfds,
• Sm Appandix 1 •! tbe and of tbij Chnfltt.
j S60 SXBBOr AND STRESS. [634,
^K SlfCiroJkiiutie Snerpg. fl
F 6S4.] We have already, in Art. 578, expressed the kinetic eiMfgf
I of a sj'fitcin of mrrenta in the form
T^nipi), (12)
nlivrc p is the electromagnetic motnentnni of a circuit, and i ii
the strength of the curpcnt Jtoning rouD<l it, nod the BOmaiatiaa
extends to nil the circuits.
But w'l; have proved, in Art. 690, that p majr be expressed «
a lincintegral of the form
where F, G, U arc the componente of the elect ramof^etio mo
I meutum, 91, at the point {xt/i), and the int«^mti<in is to be ex
tended round the closed circuit *. We therefore find
I If H, t>, IP are the components of th« densily of the cnrrent at
any point of the conducting: circuit, and if S 'i» the tniti«Tcr«e
^^ Motion of the circuit, then we miy write
" 'S = "* '% = " '%=•"■ ('.)
and wc niny utso writ« the volume
Sdi = dxdgdz,
and we now find
r= iJJJ(FM+Gv+fftf)dxtlfd:. (16)
where the integration is to be extended to every part of apac*
where there are i>lecirio currents.
635.] Let us now substitute for u, r, v tiuAr values an given by
the equations of electric currents (E), Art. 007, in terms of the
compoueuts o, ^, y of the ms^ctic force. We then have
where the integration is extended over a portion of space including
all the currents.
If we integrate this by part«j and remcinber that, at * gnaL
distance r from the system, a, 0, and y are of the order of mug
nitude r~*, we find that when the inta'^ntiou is ext^ndid tbro ngby
oat all Bpoeci, the cxgiresston is rvdoocd to
BLBCTROKINETtC ESEROT.
251
I
I
I
By tlie equations (A), Art, &91, of mo^ctiv induction, we may
flubstitate for the quiintiticii in »mall bnickets tlic cotnponeots of
tna^etic induction a, b, c, so tliut tlic kinetic cni>r^y may be
written . ^,j
^'^ {'^jjj{«^'rb&\cy)d;cdsdt, (19)
wliero tbe inte^n^tioa is to be extended throuf^hout every part of
>pac« in which the magnetic furce and mu^netic intlnction have
va1u4H differing from zero.
'File quantity within bmckete in this «ipre:j!9ion is th« product of
the magnetic induction into the resolved part of the mn^rnvtic focct^
Id ita own diix'cUon.
In the laiiguagL' of qiiiitirnionM this may be written more simply,
where 35 is the magnetic induction, whoee components are ft, h, e,
and ^ is the magnclic force, whose components are o, ^. y.
63&] The eleotrokinetic energy of the sy&tem may therefore lie
•xpnssed either aa an inte^^l to he taken where there are electric
currents, or aa an integral to be taken over every part of the field
in which magnetic force exists. The first int^'gral, however, is the
natural expression of the theory which mipjiOBcs the currents to act
upon each other directly at a distnnecj while Ihe Bwy)nd is appro
priate to the theory which endeavours to explain the action between
(he Gurrente by means of some intermediate action in th« space
between them. As in this treatise wc have adopted the latter
method of investigation, we naturally adopt the second exprt^ion
as giving the most signihcant form to the kinetjc energy.
According to our hypothesis, we assume the kinetic energy to
exist wherever there is magnetic force, that is, in gt'nvral, in every
part of the field. The amount of this cnergj' per unit of volume
ig — — 5®$, and this energy exists in the form of some kind
of motion of the matter in every portion ufupace.
When we come to consider Fumday'it discovery of the cfTcet of
msgnetisnt on polarized light, we shall jioint onl reasons for be
lieving that wherever there are lines of magnetic force, there i»
n rotfttory motion of matter round those lines. Sec Art. 821.
Maffvffic and Kleclroiiitetie KHtrffj/ compared.
6S7.] We found in Art. 423 that the mutual potential enei^
m
ETTEBOT AKD STRESS.
[63».
<^ two mnguctic shells, <^ stivn^lhs ^ anJ ^', imd boiiDded hy tht
doeod curves « and »' wepcctively, is
**'//"
C08«
■forf/.
where < is the angle between the directions of dt and A", and r ,
IB the distance between them. fl
We alao found in Art. 521 that the mutual en«i^ of two cipcnit^^
« and /, in which current* i and i' flow, is
•■'//^
tbd/.
le vor>1
If i, I are equal to ^, 0' respeotivelj, the ni«haniea!
between the magnetic shells is equal to that between the
res)on<ling cleotrio circuits, and in the same direction. In the oue
of the magnetic eliells, the force t«nds to diminish their mutual
potential enerjjy. in the cfflsfl of the circuits it tends to increase i
mutual energ'y, because this energy is kinetic.
It is impossible, by any arrangement of mngnctized matter, to
produce a system corresponding iu a,\\ respects t« an electric circuit
for the potential of the magnetic system is single valitod at eve
point of space, whercus that of the eUetric s}'stcm is manyvalued.
But it is always ]>o8siblc, by n proper urningrrocnt of iutinitety
small electric circuits, to produce a syifteni corresponding in all
rcHi>cctH (o any magnetic sytitcm, provided the line of integration
which we follow in calculating the pot^^ntiat iit prevented fmrn^
passing through any of thet« amall circuits. This will be mc
fuliy explained in Art. 833.
The action of maguel* at a distance i* perfectly identical with?
that of electric currente. We tht^rcfore endeavour to trace both
to the same cause, and since we cannot explain electrto curreat
by means of magnets, we must adopt the other alternative, and '
explain magnets by means of molecular electric currents.
638.] In our investigation of magnetic phenomena, in Part III
of thia treatise, wo made no attempt to account for magnetic action
at a distance, but tnatcd this action as a fundamental fact of
experience. We tlicreforo a«5iimcd that the energy of a mngnelio ■
fVKtcni is >Dtential energy, and tluit thi« energy is dmin'uked whCB
the parts of the system yield to the magnetic forces which aot
on them.
If, however, we rt^rd magnets as deriving their propertiea froai*
eteetric currents circttlating within their moleooles, their
AUPRRirs TnitonY op maosets.
I kinetic, and the foitie between tliem is such that it tendii (o
move them in a direction such that it' the strengtha of the currcntii
^^were miiintained constant the kinetic energy would incrf.a.*e.
^R This mode of explainin;> mat^etism requires us also to ahandon
^Vtlio method followed in Purt III, in which we rejrarded the magnet
^ as a continuous tmd humogx;ncous Lody, the minut4.'et pact of which
has Diabetic projHTties of the eamu kind as the whole.
Wo mn»t now rctjiini a itmjrict as containing n Einit<^, tbon^h
^ very grent, number of electric eircnit«, so that it has essentially
HiB molecular, ax diiiting^uixluH] from a contiuiiuiis Gti'ucturc.
If we cuppose our mathematical machinery to he so coarw; that
Aur line of integration cannot thread a molecubr eirouit, and tliat
•n immense number of magnetic molecules are contained in our
element of volume, we shall still arrive at results simiUr to thoso
of Part HI, but if we suppose our machinery of a Gner order,
and capable of inveatigating all that goes on in the interior of the
molecules, we must give up the old theory of magnetism, and adopt
that of Ampfcre, which admits of no magnets except those which
coastst of electric currents.
H Wit must also regard both magnetic and electromagiie^c energy
Has kinetic enci^, and wo must attribute to it the proper sign,
Hm given in Art. G35.
^P In whAt follows, though we may occasionally, as in Art. 639, &c.,
■ttcmpt to carry out the old thcorj' of ma^notitim, wc kIiiiII find
•that we obtain a perfectly consist*ut ByMt<;m only when wo abandon
that theory and adopt Ampere's tJtcory of molecular current*, as in
Art 1544.
The energy of the field therefore consists of two parts only, the
eleotroetatic or potential energy
and the electromagnetic or kinetic energy
' = sV//<
4 + J ,8 + c y) rfx 1^ th.
I
OH TUB PORCeS Wllica ACT OS AK KI.F.MRNT QV A BOOT PIAOKD
IN THB ELBCTROMdGSCTIC HF.IJ*.
Forces actiitif oti a Magnetic Element.
*630.] Tte potential enfliify of th« element itxtlfd: of a lx>dy
ma^Detized with an intensity wliovc components arc A, B, C, and
■ Sm ApfwaiUx II kt tli« and of tlib CbipUr.
EVEBOT A>D STRZ8K.
[64a
plnc«(] in n field of msgnotio force whoae components arc a,
Honce, If the force urging the element to move withoat rotat
.in tJie direction off is Xjdjrdj/dz,
^■*Si + ^^ + ^^' (I)
and if the moment of the couple tending to turn the elemoot about
the axis of k from^ towards z is Ldjtiydt,
Z = SyC0. (!)
The forces and the momenta corresponding to the axes of f and
t may be written down by making the proper siihstitutions.
640.] If tlie magnefiBed body carries an elrctric current, of
which the components are u, v, ie, tixn, by etiuntions C, Art. 603,
there will be an additional electromagnetic force whovv components
are J^, I'„ /j, of which S^ is ^1
H<fi0^ the total force, X, arising from tlie nMfglMttiBlQ of tie
mol^ale, as well as the current passing through it, it
da ax dx
,.^
The quantities a, h, e are the componeDta of mogDOtic inductioii,
and are related to a, fi, y, the components of Boagnetic force, by
the ex^natioDs given in Art. 40O,
A = a + 4 nA,
b = p+ivB,\ (5)
c = y + inC.
The components of the current, m, r. k, can be eipreased in tcrmi
ofa, 0, Y hy tJic equations of Art. 007,
dy
M
Hence
da
at
^»
Aaie = f —
ax
di'
da
,] THEORY OF STRRS8. 265
By Art. m.  +  + g = o. (8)
IklnlUplyin^ tliijt liquation, (8), hy a, and dividing by 4 v, we may
Ittdd the result to (7), and we find
atso. by (2). I = ^((i^) y(cy)i3). (10)
= J_(8j,_.e), (11)
fbere X ia the force rrfen(.'d to unit of volume i» llie directioa of
p, iml L ill the moment of the I'uices iiliout this axix.
Om tkt Bxplanalion ofihete Foreea hi/ the Jlj/polkesU of a Medium
in a Stale of Stress.
64] .] Let us denote a stress of any kind rofciTed to unit nf area
[by a vymhol of th« form /*„, where the first suffix, ^, indicate)) that
tlie tionnnl to the enrfaec on which the stress is EUpj)i]se<I to act
is pwrsllcl to the axtit of k, and the second stiQix, ^, indicat«s that
the direction of the stress with which the jiart of the body od
the positive iiidt! of the surface acts on tlie part un the negative
nde is parallel to the flxta of k.
The directions of h and k may bo the same, in which case the
' stress ia a normal stress. Tliey may be oblique to eaeli other, in
which case the stress is au oblifjue stress, or they niay be perpen
dicalar to each other, in which case the stress ia a taiigx^ntial
■tress.
^ The condition that the stresses sliall not produce any tendency
^■to rotation in the etemcnbiry portions of the body in
t tl>c<
»:
f» = P».
In the case of a majjpietized body, however, there is such a
lency to rotation, and therefore this condition, which holds in
tl>c ordinar)' theory of stress, is not fulfilled.
Let OS consider the effect of the stre^see on the six sides of
the flementary portion of tlic body dxdgds, taking the ori^n of
eoor<iinati« at its centre of gravity.
On the positive face liyih, for which the value of .r is \ dx, the
ar»^
SlfEBQT AITD STftieS
[641
I'amllel to x,
Fiirellel to y,
Parallel to s,
<iP.
{P«+hA^'lx)d,ds = X,„'
4P
(la)
The forces acting on the opposit* side, — 1'_„ — J'_„ and —Z.,.
may be found from tbeii« by chiiiging \\k fign of dx. We may
exproat in the same way t!ie xyitt4'mit of thrre forces acting on e»di
of the other fnces of the filuincnt, the dirMtion of tlic force being
indicated by the capital Ictl^r, and the face on which it actx hy
the KuQtx.
If Xdxdgd: is the whole force parallel to x acting on the eleineat.
Xdxdgd^ = Jr,.+.v^,+A,.+x^+x^+aL,.
,dP„ , dP„ ^ dP„.. . .
wheuce J = ^i*«+^P^+^i>..
de
(13)
If Ldxdj/dx is tlio moment of the forces alouf the axie of a
tending to tnrn the element from y to ;,
Ldxdyd: = Kds{Z^,Z.,)\ds{T,,Y.,).
= {P„PJdxdyd^,
whcnee L = P,,~P^. (M)
Comparing the ruhies of X and L given by equations {fl) and
(11) with those given by (13) and (14), we find that, if we make
'* = o*"
the force arising from a Hviitem of stress of which these am the'
coiDpODeDts will be suticalty equivalent, in its eflbcts on
H2.]
MACNTTIC
251
Blcmcnt of tti« body, witb the f<irc«8 ariaiDg Trom tltc mngnctization
> doctric eumnte.
'MS.'] Thu nature of the streas of whieh theec arc the camporXTDts
i»y b(! rttMly found, by making the axia of j biiwct the angle
Iwtwoen the directions of the magnetic foreo and the miij^netic
intiiivtton, ftnd takin<; the axis a(y in the jthine of these dinictions,
and na>a«tirc<l towards the side of tlie magnetic force.
If we put J^ for the numerical value of thi? magnetic force, 3 for
(hat of ibe magnetic indactioD, and 2« for the angle hutweea thoir
directions,
a=.ftco«<, j3= ^sin*, y = 0, 1 , .
a = ©«»(, B =S9einf, e = 0;/ ^'^
P^ss — I&Jq cos ( sin e,
(17)
P = — — ©^ 009 e siu f.
4 IT
Hence, the state of rtress may he considered as compounded of —
(I) A prwsure equal in all directions =  ^'.
(3) A tension along the line bisecting the angle Wtwccn the
directions of the mi^netic force and the magnetic induction
L=l^«^cos^c.
(3) AprcMurcalong the line biitecting the exterior angle between'
in directions = — © fi sin' «.
r iv ^
(!) A couple tending to turn every element of the gubstanca in
the plane of the two directions /><?»• the direction of magoetio
indnctioD to Uie direction of magneUc force = —SQ^aa2t.
VThcn the magnetic induction is in the same direction as the
magnetic force, as it always is in Ihiidit and nonmagnetized solids,
then ( =: 0, and making the axis of « coincide with the direction of
IUic magnetic forcc^
TOt. II. S
p^
and the tan^ntial streesce disappear.
The stress in thia case is therefore a hydrostatic pressure — ^*,
I
combined with a longitudinal tension j © •& ^ong the linos of
force.
643.] When there is no magnetization, S = ^, and the stress is
itill further simplified, being a tension along the lines of force equal
to — ■ ^^, combined with a pretwi"" in all directions at right angles
8T7
to the lines of force, nume
poaents of stress in this impoiM
1
1
,al also to ^ 6*. The com
8 T
dse are
P„= ^{y'^^^^
P». = K = i^^y<
I
(19)
The 2! component of the force arising from these Btresses on an
element of the medium referred to unit of volume ia
I I da ^dfi dy) 1 ( d$ ^da) 1 ( rfy da)
I ,da (f/3 dy\ 1 ,da rfy\ 1 a/^^' ''"X
~4n'^^dx'^dy'^dz'ili^^dz~di^~'ilr^yd^~dy''
Now
da da dy
da dy
d}di='^'''
rfj8 da
dx
wh«re « is the density of austral magnetic matter referred to unit
dx dy
6451
TKKStOir ALONG LINES OP FOIICE.
259
of Tolume, and v and w are the componenU of electric curronfs
rcfi'iTcd to uoit of area perpendicular to y and z respectively. Hence,
X= aw+ t'y— W/S, 1
f (Equitloui of
Simuariy r= fita+iea~uyA Elcetrommnello (20)
P Z = ylW + KjS— CO. ) **"*
644.] If we adopt the theories of Amii^re and Weber aw to the
nntiirr of ma^etic and diamagnetic bodies, and assume that toAg
netic and diamafn^ctio polarity are due to molecular electric currents,
we get rill of imaginary ma^otJc matter, and find that everywhere
r
= 0. ""d rfa dS dy
J + J + J ■ ='0,
ax ay dc
(21)
(22)
so that the otjiiations of oWtromagnctic force become
^^ 7=«auy,\
^W 2=ii^va.'
P TIkm are the components of the mechanical force rtferrcd to unit
of volume of the substance. Tlie components of the mafjnctio force
■re o, 0, y, and those of tlie electric current arc u, r, w. Theae
ecjiiationit arc identical nith those alroudy eetabliifhed. (K({Uiitiona
(C). Art, 603.)
645.] In explaininfT the olectromagnHic force by means of a
state of stresM in a nicdiiira, we are only following out the con
ception of Faraday*, that the lines of magnetic force t«nd to
ehortcn themselves, and that tliey rcpttl ea<?li other when placed
side by side. All that we have done is to express the value of
the tension along the lines, and the pressure at right angles to
tbem, in mathematical language, and to prove that the state of
■tress thns assumed to exist in the medium will actually produce
tiui observed forces on the conductors which carry electric currents.
Wc have aseerted nothing as yet with rc8p«:t to the mode
in whiuh this state of atrees is originated and miiintuiu«d in the
m^am. Wc have merely shewn that it is poK^ihle to conceive
the mutuiil action of electric currents t" drpcud on a particular
kind of fIreiH! in the surrounding medium, instead ol' being a direct
aod immediate action at a dietaoce.
Any further explanation of the slate of stress, by means of the
potion of t;he medium or otherwise, muitt be regarded as a separate
fed indepeudeiit part of the theory, which may stand or fall without
aflecting our prcMMit position. See Art. B32.
• jffjy. tia.. !!8e. 8M7. 32$S.
m
EITEBOT AKD STBXSS.
Iq first part of this treatise, Art, 1 08, we shewed that lie
obeervGd ek'ctrostatic forces may be conceived us operating throagh
the intervention of a state of stri^ss in the surrounding' meiJium,
Wo have now done the same for the electromagnetic forces, ami
it remains to be seen whether the conception of a medium capable
of supporting these states of stress is consistent with other known
phenomena, or whether we Ghall have to put it aside as unfruitful,
In a field in which electrostatic as well as electromagnetic actioo
is taking place, we must suppose the electrostatic stress described
in Part I to he superposed on the electromagnetic stresa which we
have been considering.
646.J If we suppose the total terrestrial magnetic force to be
10 British units (grain, foot, second), as it is nearly in Britain, then
the tension along the linos of force is 0.1 2S grains weight per
square foot. The greatest magnetic tension produced by Jonlc*
by means of electromagnets was about HO pounds weight on
square inch.
^
• SttirgBoii'« AhiuiU of Slretrlcll;/. vd. v. p. 1B7 (1S4CI) ; or Philaiopkical Xagatlmt.
Dec. ISGL
APPENDIX I.
[The roUoving note, derived horn a latter written by PrafHnr Clark Maxwell to
I n«ft«Bar Clu^vla]. u iicpoiiuit in coneexlaii with Arta^ 3S9 and 891 : —
Id Art. 389 the cnci^ <Iue to the >rMi^nc«< of a in&f;itet whoM msg
DOtixntion cotnponcntu arc .f,, B,, C,, placed in a field whOM nUifiieUo
foroe compoiKnU are a,, ^,, j,, ii
//A^.<^ + S,^ + C,y^)<ix'Ivdz,
then the intefpvlioD is couSDed to the niugii^ in rirtue ot A^, B,, C,
^bdug MPO evtiywhcrc else.
Bnt ihc vholc pncrgv is of the form
Itiie intc^mtion extending; to every part of apaee wlirre tliero an mag
taftimd bodiee, and A,, B,, C, denotiuj;; the components of mgpMtiutiou
at AD}' point (otlorior to the mngiKit.
The whole energy tlius coniiistA of four part* ;—
i//AA,a, + &^.)<Lrdydz. (I)
vbicb IB constAnt if the nia^'netization of the inognet b rigid ;
~\///i.Ka, + &c.)<Ixd,jd^ (2)
wliieh is equal, by Qruen'M Tlicomn, tu
\///(A,a, + Ai:.)dxdi,d:, (8)
awl i///{A^a,+&c.)dxdyds: (4)
wliich lost we Dia; 8ui>pose to arise from rigid magnetisations and therm
ion cooRtrat.
BeiMc tlie rnrialilc part of Ute encrgj of tbc moveable magnet, a.s ri^dly
il, b tlie aura of the ezpreasions (2) and (3), tik..
Remembering tliHt the diaplaccmunt of the mi^ict aliens the value* of
S> i^i' Vk ^^^ "^^ IhoM of A^, B„ C„ we Cod for the component of llie
force on the magnet in aQy direction 0—
I
If]
(".t^^'t^'^'t""**
I
I
If instead of a rangnct wc hare a body maguetiiicd tiy induction, th*
exprtnion for ibe for« must be the same, vi*., writing A, = ka, Ac.,
we have fi'f , dn, , ,J,^, . dy,. , , ,
In Ihi* expression a ia pnt for n, + o,, Ac, but if either the magnet ixe<!
body be smnll or k bo small we may neglect a, in compariaoti with a,, and
the exprenian for the furee beeumea, as in Art. 440,
The worii done by the magnetic forces while a body of ainuU inductile
ca>ucily, magiietiied inductively, ia carried uff Ui infinity ic only half
of tliat for the aiinie body rigidly nia;^etieed to the wime uriginal
•Ireogtli, for n« tbe induced magnet ia carried off it loica iU atrength.]
APPENDIX n.
{OliJMitJuii hail liMU bikoii lu di« cxprenBion omittunnl io Art 639 ftr
the putetitiail energy per uiiH volume of the medium arising from nwE
ueUc forces, for the reason that in finding that vspressioo in Art. 389 ««
Assumed the force coinponcnta a, 9, y lo }>o di:riviibl« from a iiot^otid.
vhcreu in Art*. 639, 610 tliiH is not tlie case. Tfain objectiwD extemit
to th<! exjtrcti^Dn fur the force JT, which it the tipaoe rarititioii uf the
tntTgy. The puqtose of this note is to bring fonrsriJ gome CMtddernliom
tending to confirm the kccunic}' of th<^ t«xt.
We oonmder tlicu tb<> uleuitml dxdydz isolated twvm Uic rat of tht
medium with u current of electricity (u, t', te) flowing tliroiigli H. 13ia
fortna uoUn^ on the element ari»e fium two causes, vix., first, in rirtna of
tbe element being roagnntisird, next, in virtue of ib orrjring h curtent
The force dno to the hitter CflU»e is (X,, I',, Z^dasdt/dt, giren in (3),
Art. 610, viz. X, = re— wb, etc.
To entimate the other we oliacrve thai, if no currtiit wen; flowtn)(
tlirougU the eleuicDt. the tnngnetic force on th« eletnint would («
(X', 1", Z') dxdyds, where
ax dy efa
4
Jltit in widiiiou to this there will be forcea from the current of eleotriot)*
neting on tbi ntngncfism of tlie fjices of the element. Kuw it Is sheim in
Art. 653 that in pasting from the negative to the poaitire side of *
current sheet the conipoumt of th« magnetic UfK» nOTiBal to tbc shcel
experiences no discontinuit} ; neither doe* Uw component in th« diroctiuo
of the current, but the componeut paniliol to the tihcet aikd nornud to (br
current changes by 4itr, where i is the strength of the cuTreat per nnit
Uugtb. Let us then suppose that the faces of our eleraeut which ait
perpendicular to the axis of y in a right handed system of ftxee an
portions of an infinite current sheet in which there is flowing a cumnt
parallel to tbc axis of e equal to tcdAf. There will tlicn be a nmgnctie
force BAvtedxdyia over and itboye the force /I  Jxdffdx previotudy
reckoned in tbc estinmte of th« forcn* pnmllcl to dx and ndiog on the
dxda faces. In like m&uner we ihould find an MlditiMml (breo
—Civvdifdyd*
acting on the dxdy faces. Vr'e have thus to add to X' • term
X"= ii:B«>4iTCv.
and the total ^T'Componeiit of tlie magnetic force will 1m
My
''^*^i'+%
^dy
tlMttt,
tl wUl be observed tbot tlM force (X'
Talaeor{£,if,^).]
r'.Z'^iixttsolatlam
CHAPTER XII.
CURRENI^HEBTS.
I
Gi?.] A cVKBEirT'SQiiitT b an iofinitely thin stratum of con>
ductin;;: matter, bounded on both sides by iosulatinfr media, eo that
filectric currents mny 6ow in the ahcct, but CHnnot e6ctk[K> from it
«xccpt at (wrtnin points called Electrodes, where curreats are made
to «nt«T or to Icavo the sheets
In order to conduct n finite cleetric current, a no.] sheet must
have a Unite thickness, and oug'lit thurc^fore to be considered m
DOndador of throe dimensions. In many cases, however, it is
practically convvnicnl to deduce tlie electric properticn of a real
conducting tihcetj or of a thin layer of ooiled wire, from tliow! of
a ourrentshett it* dcltned above.
We may therefore r^'ganl u surface of any form a« a curreutfhcct.
Having selected one side of this surl'ncc aii the positive side, we
shall always eiipj>of!e any linvH drawn on the tiiirliico to be looked
at from the positive side of the surface. In the case of a closed
surface wo shall eoUKidrr the outttide as positive. See Art. 294,
where, however, the diroutiun of the current ie defined a« seen from
the negative aide of the sheet.
Th« CttrrtHtZuHclion.
618.] Ijet a fixed point A on the surface be chosen as origin, and
let a line hedrawn on the surface from A to another point P. Let
the quantity of electricity which in unit of time crosses this line
from left to right be <^, then i^ is calUd the Current function at
the point i*.
The currentfunction depends only on the ])osition «f Uie point P
and is the same for any two forma of tlte Hno AP, provided this
:
line MO be trnn^fonnod hy continiiouf motion from one (nna to tlii*
ollter witliout. passing: Uiroag'h an cleotroile. For the two forms of
tlic line uill enclose an arcu witliin wliJcli tliere in no electrode, utd
tbererore the Game quantity of electricity wliicli cntent Ike ares iicre«
one of the lines must issue aoroi^ the other.
If t denote the length of the line AP, the current acrow d» from
left to right will be ^ dt.
If is oonstsnt for any curve, there ia no current across it, Suob
a enrve is called a Currentline or a Strearaline.
649.] Let yp be the electric potential at any point of the sheet,
then the electromotive force along any ctom«nt tJ> of a curve will bo
provided no electromotive force exists except that which arises from
difTereDCeB of potential.
If ^ is constant for any curve, the curve is called an Eqni
potential Line.
660.] Wc may now Biippose that the position of a point on Iha
Rheot is defined by the values of ^ and ^j/ at th.it point. Let rf^ be
the length of the element of the ei]uipot«Dtiul line >j/ int«rc«pted
between the two current lines ^ and ^ + d^ and Kt dji^ he the
length of the clement of the current line ^ intercepted between the
two eqaipot«ntial linc« ^ nnd ip + ii^. Wc niay consider f/^, und</<,.
as the sides of the element dipiiyj/ of the slieet The eli>utromotive
force —dyfi in the direction of tt», produce* the current rf^ acroM f/i,..
Let thi.' Tcttistance of a portion of tlie sheet nhoae length is tU^,
and who«c breadth i* da^, be dt,
(fa,
where a m the spocific resistance of the sheet referred to unit <if
area, then , , di, , ^
1 ds, dL
whence 4 = "^
651.] If the slieet is of a substance which eonduots oqnally weli
in all directions, dtj ia perpendicular to d»^. In tlie ease of a she
of uniform rceistanoe a in constant, and if we make ^ « o*^, wc
sJiall have A, d^t
and the streamtines and cquipoteutial Uneawill ent tlu suriaaeiat
little t>4»aro(;
i
MAGNETIC POTEITTIAI,.
265
I) tit foUoirs from tUia tliat if 0, and t^i' arc conjii^nie funotinns
(Art. I$3) or and yfi', tho curvett <^ may be strxum'Iinoit in the
lebeet for which tlic curves ij'j'are the ooiivapondiii^ I'tiuipotontial
lines, One case, of coume, in that in which 0, = ^' and t^,'= —0.
' In tJiis case the «>ijui potential lines become curreiitliiie^ and the
kcuiTeDttines etjiii potential lines *.
If ne have obtained the solution of the distribution of electric
ctineDts in a nniform sheet of any form for any particular case, we
may deduce the distribution in any other case by a proper trans
I formation of the conjugate funotions, according to the method given
in Art. 100.
65a,] We have next to determine the magnetic action of a
cnrrcnt«hcet in which the current is entirely confined to the sheet,
thorn bring no electrodes to convey the current to or from the
sheet.
In this case tho currentfanction i^ has a determinate value at
evorj' point, and the streamlines are closed curves which do not
intfrrsect «Bcli other, though any one stxeamlioe may intersect
itwlf.
B Consider the annular portion of the eJieet between the stream
^■1inc« nnd i^+hif). This [lart of the sheet is ft conducting circuit
Hin which a current of strength &</> circulates in the positive direction
■.round that part of the sheirt i'or which tp is greater than the given
H Taluo, Tho nmgDetio elfoct of this circuit is the sumo as that of
H« magnetic shell of strength S^ at any point not included in tiko
B lubHtimoc of tho sbell. Let us suppose Utat the shell coincides with
that part of the currentsheet for which ^ has a greater value than
B'it has at tho given streamline.
H By drawing all the successive streamlines, beginning with that
^■for which has the greatest value, and ending with that for which
^■it« value is least, wo slmll divide the eurrenteheet into a Kcries
^^of circuits. Substituting for each circuit its eorr^eponding mag
^Bnetio Hboll. wo find that the magnetic ciTect of the currentxhoct
^■■t any point not included in the thickness of the sheet is the >^ne
^L aa that of B complex magnetic shell, whose strength at any (loint
^Bia C+<p, where Cis a constant.
If tlie currcntshoet is bounded, then we must make C+i}> =
at Ui« bounding cnrvo. If the sheet forms a clowd or an inlinite
Mirfaoc, there is nothing to determine the value of the constant C.
■ 8m Tlioaucin, Can^. and Dub. Math. /auni„ voL Hi. p. 280.
206
[653
053.] The magnetio potential at any poiot on either side of tbo
currentEhcct is given, aa in Art. 4 1 5, by the expression
"=//i
vra dJS,
wbere r is the distance of the g^ven point from the olemeni
sariaoe dS, and is the angle between the direction of r, and th:
of the normal drawn from tlie poeitive sido of if^.
This expreagion gives the magnetic potential for oil poinU
inelutled in the thickness of the cumentslMirt, and we know
for points within a conductor carrying a current there is no «ich
thing ax a magnetic potonfial.
The value of it is discontinuous at the cnrrentaheet, for if Qi
is its value at a point just within the currentshwl, and il^ ita
Tolue at a point close to the first but jost ootaido the curreot'sbeet,
where is the currentfaoction «t that point of the sh«et.
The value of the component of magnetic foroe normal to
sheet is continuous, being the same on bi>tli sides of the sheet
The component of the magnetic force parallel to the currentliMS
is also continuous, but the tangential coin{>onent perpendicular 1«
tlic currentlines is discontinuous at the slieet. If * is the length
of a curve drann on the sheet, the component of magnetic force
ot9
rfft,
and for the
in the direction of </« is, for the negative side, —  . , — .. — ^^
positive side. ^ =  ^4 ^ . ^
The component of the magnetic force on the positive ride tliere
fore exceeds that on the negative side by — 4 it 7 ■ At a given potut
this quantity will be a maximum when da b perpendicular to the
Gurrentlinea.
On He Indaedon of Blecirie Currenlt in a Shttt e/ Infiiiilt
Conduellrity,
654.] It was shewn in Art^ S79 that in any circuit
where E is the impniwe«l tlectrorootive force, p the e1«ctro1nB
momentum of Uie cirtriiit, li the reiustanec of the (lircuit, and i
current round it. If tltere i« no impreascd electromotive force 1
DO resistance, then ^ = 0, or pia ootutoai.
at
" 656.]  PLANE SHEET. 267
I Now />, ihe electrokinetic momentum of the circuit, wan dicwa
in Art, 388 to be measured by tlie surface integral of magnetic
induction through the circuit. Hence, in tlio cuite of a ounent
she«t of no rcaietance, the surfaceintegral of magnetic induction
through any closed curve drawn on the surfiice must be constant,
ami this implies that the normal component of magnetic induction
nimains oonstant at every point of the current •sheet.
1655.] If, therefore, by the motion of magneto or variations of
cunvnte in the neighbourhood, the magnetic field ia in any way
alt4.Te(I, electric currents will be set up in the currentsheet, such
that their magnetic effect, combined with that of the magnets or
cuTTcntx in the 6eld, will mointain the normal component of mag
■ nrtic induction at every point of the sheet unchanged. If at first
there is no magnetic action, and no currents in tlie sheet, then
the normal component of magnetic induction will always be zero
N at every point of the sheet,
B The sheet may therefore be regarded as impervious t« magnetic
induction, and the lines of magnetic induction will be deflected by
tlic sheet exactly in the same way as the lines of flow of an okctric
current in an infinite and uniform conducting mass would bo
ectfd by the introduction of a sheet of the same form made
if A suhetnnw of infinite reai^ance.
If the sheet forms a closed or an infinite surface, no magnetic
actioiiH which may take place on one aide of the sheet will produce
any magnetic efieot on the other side.
TAeor^ of a Plane Cirrfnl'sieei.
656.] We have seen that the external magnetic action of a
cuiTent>sheetia equivalent to that of a msignetic shell whose strength
at any point ix numerically e<iual to if>, the currentfunction. When
the sheet is a plane one.n'e may exproe* all the quantities required
for the determination of tiWtroiiiagiiclic effects in terms of a single
funHion, P, which \a the potential due to a sheet of imaginary
matter spread over the plane with a surfacedensity ^. The value
of P is of course rrA,
irbere r is the distance from tli« point (x, y, 2) for which P is
calculated, to the point («'',/, 0) in the plane of the sheet, at which
the element li^il^ is taken.
To find the magnetic potential, «c may regard the magnetic
I
k
at»
CntESin^SBEBtS.
[65.
Rbcll as oonKirtinf* of two enr&cee parallel to the plane of tf, tbe
firfit, whose equation is « = )«, having the Rurfstcc^Dritj ~, and
the Kcoond, whose eciitation in z =: — \e, having tbe sarfacesli
e
TIm) potentials doc to thcM turiaoes wiD be
J'H)
and P,
e
■(i)
e ,
respecttvelr, where the soffises indicate that t— a pat
in the first expression, and £ ■!■  Tor ^ in Uic i«cond. Expandio^
these expressions hy Tnvlor's TlKorein, addinff thtrm, and tbtn
making c infinitely small, nc obtain for the mitgaiAic potential do^^
to the sheet at any point external to it, ^
fl= —
dP
d:'
(2)
657.] The quantity P is synintctrical with respect to the plane of
tbe ahect, and is thcretorv the same when — ^ is substituted Tor £■
n, the magnetic potential, changes sign when —x is pnt for t.
At the positive surface of tbe sheet
" = ?="'♦• (=
At the negative snrfaoe of the sheet ^j
Within tbe sheet, if its mafrnctic effects arise from the magnet
ization of its substance, thi? maj;nctic potential raries ooDtinn
ously from 2t^ at tlie positive surface to — 2s^ at the negative
Burface.
If the sheet contains electric currents, the magnetic foiee
within it. does not satisfy the condition of having a potential
The magnetic force within the sheet is, however, perfectly deter
minate, ^i
Tbe normal component, ^M
da d'P ...
is the same on both ndea of the sheet and throaghont ita aafi
stance.
If a and p be the componentu of the magnetic force pantllel
VECTORrOTENTIAL.
I
and to y at tbe positive suriace, and a', ^ those on tlie negative
flarfaw, ^ „
  ' (6)
« = — $=
(?)
Tho iMjuutioiiti
^
ds
dx
iF
dU
da
it
dx ~
if
JG
dF
dil
(«)
Within th« she«t the components vary continuously from a and
,^ to a' and 3'.
^^^^^P da di/ ~ dz
which conncct the components F, 6, U of the vectorpotential due
Ito the curnntBlieot with the sualar potential it, are satisfied if
Itrv make Ap ,IP
F=~, G = ^, /f=0. .(9)
ay dx * '
We may also ebtnio these values by direct iDt'?^ratioii, thus for F^
Since tiie integration is to be estimated over the infinite plane
tflbeet^ and since the liriit term vitiiit^lics at infinity, the cxprescion \»
[reduced to the secund t«rm; and by substituting
d \ . rf 1
y~ for — T''
"S T ay r
\*aA remomberin^ that ^ dq)enda on of and /, and not on x, jf, e,
U obtain F=^fJ^^d^dy,
If Q' is the niBj>netic potcntiiil due to any mng'netic or electrie
sysbem €xt«raal to the sheet, we may write
'=~fsi'dz.
and wo sball tlien have
&=
dy
dx '
U'=0.
(10)
for the component* of the Toctor<potential dae to this system.
270
CUSREtfTSnEBTS.
[658.
658.] Lot ii» now determine tJie electromotive Toree at w»y pointfl
of the shfct, supposing the shwrt fixed. ~
4
, »uppoaing
Let X and f !« the compoiienU of the eiectronwtive force paralle!
to J and toy respectively, then, by Art 598, w« have
'iio.'nt
If the electric resistance of the sheet is uniTorm and equal to <r,
X=aM, r=<Tr, (H)
whore «i and c are the components of the current, and if ^ ia thai
current function, J^ ^^
But, by equation (3),
(15)
Tlencv, eqtiatioas (12)
itt the positive siirraco of the currentsheet,
and (13) may be written
2vdydz~ dydt^ ^ ' dx
2itda><h dxfU^ '*' ' off'
where the values of the expressions are those corresponding tu tbt
Iiositive surface of the sheet.
If we differentiate the first of these equations vrith rrspect to
and the second witli respect to y, ond add the rc«ultti, we obtain
(16)
The only value of ^ which satisliM this equation, and is finite
and continuous at every point of tlie plane, and vanisheA at si
inGnite dietftsce, is \b = 0. (I9l
Hence the induotion of electric currents in an infinite plane she
of uniform conductivity is not uecompanied with diBcrcnoc*
electric potential in different parts of the tsheet.
Substituting this value of <^, and integrating equations {iGJj
(17), weobuin „ dP dP dF ,, ,. .„„(
Since the values of the currents in Uie nheet are found
diffen^ntiutin^ with respect to * or y, the arliitrary runctinn ol
and / will disfl])i>c3ir. We shall iherufore Iciive it ont of accounk
6603
DBCAT OF CITRRESTS IK THE SHEET.
in
ir we dbo write for — , the single symWl ff, wbich repmenls
L oeitain rolocity, ttie equation bebwcca P and i^ becomea
!>.] Let o» first suppose that there is no external mAj^etio
actings on the current sUcot, We nrny therefore suppose
: 0. Tiie case then becomes that of u sy«teiQ of etei^tric ciirreuta
ih tlie sbeit left to thcmsulves, but nirting on one another by their
il induLtioD, and nt the suine time losing their energy on
aunt ol' the resistance of the sheet. The result is expressed
by the equKtJon ^rfP dP
the solution of which is P =/{*, y. (c + ^0) (23)
* llencej the value of P at uny point on the positive side of tha
rTfbsM coord inn te.« are x, y, x, and at a time /, is equal to
itwP tit the point *, y, (;+ Rt) at the iniitiint when ^ = 0.
If therefore ft system of ctirrent« is excited in a uniform plane
sheet of infinite exttnt and then lelV to itself, its magnetic effect
at any point on the pogittve wide of the sheet will he the same
aa if the syntem of currents had been maiutnineil constant in the
sheet, and the sheet moved in the direction of a normiil from ila
negative nido with the constant velocity I{. The diminutiun of
the electromagnetic forces, which arises from a decay uf the currents
in the real case, is accurately represented by the diminution of the
t force on account of tlie increasing distance in the imaginary case.
PiF
M
di.
(24)
If we suppose that at first P and P" are loth zero, and that a
magnet or electromagnet is suddenly magnetised or brought from
an infiuite distance, so as to <>liange the value of P" suddenly from
into to P', then, since the timeintegral in the second member of
(24) vanishes with the time, we must have at the finrt instant
P = —P" at the surface of the sheet.
* rnWBqiuU(au(lQ)Bnd(S31aF«' provHltabstrueoDljriit the surhavof tlteilucl
fcc whicib t — 0. lio csjiroMan (33) latinfln <S2) genLnlly, uid tberdoro hIio at
tha inrbM «f tka tbeet. It alia MiUin the iiUitv coiiditluiu uf iLo problem, uul U
IbmGirD » aalutlon. ' Any ctbtr tuluUun mud iltlTar (run Uite by ft i^«l«in of olacad
cimanlf, ilmndinii! ou tbe tnittkl tMa al the ilwel, dM iIu« to *i>7 aitanMl otUM
Mid wLich tWnlore niuit iIucbv rapidly, fienn (iiMv wa wtnim* ut vtunitr or put
tino tlin b i&c only wlutioo u( lb« problm.' Set Ftahmor CScric Muwell ■ Papct,
lUfiU ioe. PttK., XI. pp. I90lliS.l
272
CntHgaWBj^l^B,
Hence, the system of tarrent* excited in the sliect hy the suiUeii
introduction of the system to whieb P' is due is such that at tht
surface of the sheet it exactly nentmliw* Uie mngaetic effect of
this system.
At the surface of the sheet, Ihorcfore, and consequpntly at «II
paints on the ncf^tivc dde of it, thv initial fvxtem of cuTTnil*
produces SLti cfftxt exiictly equal and op[>o»itc tci thiU of Uie
magnetic system on the positive sid«. We itwy exprcM tht< by
saying that the efltct of the curront« \m equivalent to tliat of la
ivutge of the mofpietic system, coiueiding in position vritli thnt
system, hut opposite as rrgurds the direetion of its magtietiutJOD
and of it^ electric currents. Such an imi^ is called a ntffative image.
The clleot of the currents in the sheet at a {oint on the podtin
side of it is equivalent to that of a positive ima^ of the ma^iMtic
sysU'm on the npfjativc side of the «hcet, the lines joioiD^ cone
spondin^ points bein^ bisected at right angles hy the sheet.
The action at a point on cither Ride of the sheet, due to the
currents in thi; sheet, may ihertfore he reg^flfdcd as due to an
image of the iiiiignotic sj'stcni on tlie side of the sheet opposite
to the poiut, this image being a positive or a negative image
according as the point is on the poattive or the negative side offl
tlie sheet. ™
661.] If the sheet is of inRnite oonductivity, 2t = 0, and the
second term of (24) is zero, so that the image wiU represeat the
effect of the currents in the sihect at any time. ^
In the case of a real Nheet, the resistance H has some fiuite v»1u«. 
The image juat deaerihed will therefore represent the effect of tho
currents only during the first instant after the sudden introduction
of the magnetic system. The ourrents will inimediatfly begin to
decay, and the effect of this decay will he accurately represented if
we suppose the tno images to move from their original positions, in
the direction of normals drawn from tho sheet, nith the oonitant
velocity R.
662.] We are now prepared to investigate the system of Mirrrntw
mduced in the sheet by any system. J/, of mngneta or electro
nagnets on the positive »de of the sheet, the position and stnogth
of nhich vary in any manner.
Let P", a« before, he the function from which the direct actJoa^
of this system is to be dedneed hy the equations (3), (d), ftc.,
th«D nr it will be the function oomhtpottdiDg to the system
6640
UOVUfO TSA1L OP tUAGBS.
m
presented by jt S' Tliis quantity, ivhicli ia the increment of 3f
in the time 9^ nwy be regimlLil as ititelf rqimcoDUog n maguetio
system.
t If we supjioEC tbut at tlto time f » positive image of the system
ait
jfit ifi formed on tho nogatiro dde of the shoot, tho mo^Dttie
action at any point on the positive adi of the sheet due to this
ima^ ^vill be equivalent to that due to tho currents in tho sheet
excited by the change in At dnriog tho liret instant after the
ch&Dgo, and the imitgc >vill eontiiitie to bo equivalent to the
curreatfi in the sheet, if, lui soon as it is formed, it begins to move
in the negative dircetion of s with the coiistnnt velocity R.
I If we BHpposc that in every auecessive element of the time an
inuigc of this kind is formed, and that as snnn as it is formed
it Ijcgins to move away from the sheet with velocity E, we shall
obtain the conception of a trail of images, the last of which is
in process of formation, while all the rest are moving like • rigid
body away from the sheet with velocity li.
I 668.] If i^ di'jintes any function whatever arisiing from the
action of the magnetic system, we may find P, the corresponding
function arising from the currents in the sheet, by the following
nrooew, which is merely the symbolical expression for the theory
of the trail of images.
. Let /', denote the value of P (the function jurising from the
pnrrents in the sheet) at the point {a, y, i + Rt), and at the time
t—T, and let i^ denote the value of P" (the function arising from
jthe magnetic system) at the point («, ^, —{s+Bt]), and at the
(5.6)
iet^T. Then
dr 4ts (it '
d e<iuatlon (21) becomes
d7=sr' **''
id we obtain by integrating with reepoct to r from r=: tor = ec,
tbe value of the function P, whence we obtain all the propertiM
rtfafl eorrent sheet by ditferentiation, as in eciuations (3), (9), &c>
664i.] As an example of the proccta here indicated, let us take
e case of a single magnetic pole of strength unity, moving with
miform velocity in a straight tine,
VOL, II. T
m
m
CntHEKTSHEETS.
Lcl the coordinates of the pole at the time i be
f = u/, 1? = 0, C=: c + vol.
TIic coordinates of the image of the pole formed at the ti
t~T are
€ = M{er}. n = (», f=(ff+»(^r)+i!r),
and if r is the distanee of thia image from the point (x, g, t).
To ohtain the potential due to the Imil of image* we have
calenlate ^ r" dr
~dTJ, T"
If we write Q» = u» + (S— »)',
_^y = glog{Cr+u{*uO + (fiw)(«+*' + »01
+ a term infmitely f^rent which however will disBf^iear on diSi
entiation with regard to /, the value of r in thifl expreesion \m\
found by making r = in the expression for r fpven above.
DifreKntinting this expression with reepect to /, and poi
f K 0, ne obtain the magnatie potential due to the trail of i:
^^ n>f. + .)u. _^,_^,^^^
^ Q Qr+ii»+(i?»)(s+c)
By differentiating this expression with respect to 2 or
obtain the components parallel to ir or « leepcctively of the mag
netic force at any point, and by puttin<> r = 0, z = c, and r => 2r
in these expressions, wc obtain the following valne« of tho CODI^
poneats of the force acting on tho moving pole it«clf,
J = 
1
i'^Q+S
ks!'
Q
QiQ
J>! J,
^Mh
( + KW))
665.] In these expressions we must remember that the moSo
is aapposed to have been g<»ng on for an infinite time before
time coiuictered. Uenoe we must not take tv a positive qoantil;
for in that case the pole must hare passed through the sheet
within a 6nite time.
If we make u = 0, and tp negative, X =0, and
or the pole as it approaches the sheet is r«p«Ue<l from it
668.]
FORCE OS MOTINO POLS.
275
If we make » = 0. we fin^ Q^ = u' +^,
X = 
uR
and Z= .
The oomponent X repa'sent* n retarding force nctJD^ on the pole
in the dirccUon opposiUr to that of its own motion. For r given
valae of K. X is a masiinum when u = 1.27 Jt,
When the sheet is a noucouductor, R=ys and X = 0.
When the sh«et x* a perfect conductor, R = and A' = 0.
The component Z represents a reimlsion of the jmle from the
[ elMet. It increases as the velocity increases, and tdtiniat«ly becomes
—^ when the velocity is iotiuite. It has the Bame vnltie when
a is zero.
666.] When the magnetic pole moves in a curve parallel to the
sheet, the calculation becomes more complicated, hut it is easy to
see that the effect of the nearest portion of the trail of images
is to produce a force acting on the pole in the direction opposite
to that of its motion. The eScct of the jxirtion of the trail im
mediately behind this is of the t^mv kind as that of a magnet
with ita axis parallel to the direction of motion of the pol« at
I some time before. Since the nearest pole of this magnet is of the
' nun« name with the moving pole, the force will consist partly of
a repiiUion, and partly of a force parallel to the former direction
of motion, but backwards. This may be resolved inbi a retarding
force, and a force towards the concave side of the path of the
nioTing pole.
667.] Our investigation does not enable ua to solve the case
ID which the sjBtcm of currents cannot ho completely formed,
on aoeonnt of a discontinuity or boundary of the conducting
sheet.
It is easy to see, however, that if the polo is moving parallel
to tfae edge of the sheet, the currents on the side next the edge
will bo enfeebled. Hence the forces due to these currenl^t will
be lees, and there will not only l>e a smaller retarding force, but,
since the repulsive force is Icaxt on the side next the edge, the pole
will be attracted towards tbe edge.
Tieory <^ Araga't Sotating Pist.
Arago discovered * that a magnet placed near a rotating
metallic disk experiences a force tending to make it follow the
T 2
270
[66E
motion of the disk, altbongh when the diek is at rest there is
no nction between it and the ma^i^npt. j^m
This action of a rutatinf; disk was nttrihated to a new kin^^
of induced m^>netization, till Faraday ■* expluned it by means «f
thtt electric cinrcntii induced iu thv disk on account of ite motion
throogh the field of muf^netic f<irce.
To dctcitninc Uie diiitrlbution of these induced correntd, and
thiir ifTrat on the magnet, we might mske uw of the results nlready
found fur a conducting sh^t at rest aot^d oil hy a moving magnet,
Bvatling ourselvM of the method given tn Art. 600 for trcatin<^ th*
electromagnetic equatiouH when referred to mnring Gystcms of axes
Aa this case, however, has a special importance, wu shall treat it
in a direct manner, beginning by assuming that the [Kilos of the
magnet are so far from the ed^ of the disk that the effect of tb
limitation of the conducting sheet may be neglected.
Making use of the same notation as in Uio [ii«ceding article
(556667), we lind for the components of the electromotive
parallel to x and y respectively,
a« =
dx J<(f
where y in the reeolred part of the nugnetic force normal to tkc disk.]
If we now express « and c in terms of ^, the currentfunctioB,
aod if the disk is rotating about the axis of t with the angular
«lodty w, ^_„. ^
^=„,, ^ = «jr.
Substituting these values in e<]uation8 (1 ), we (ind
dd, d^
''ds=>'''di'
Moltipl^'ing (4) by e and (5) by jr, and adding, we obtsiii
Multiplying (4) by jr and (5) by —*, and adding, we obtain
d^ _ d^ diff il<fi
1
/ dib (lA. d<b d<li
df^ dy
• £r^ Ra, gl.
>68.]
ARAG0 3 DISK.
2rr
ar~ =
It we DOW express tbeae equatioQs ja tetats of r and 6, where
2! = r COB 9, ^ = r sin $, (6)
hey become a ^= ymr'—r ^, (9)
Eqaation (10) is sntisfied if we nseumo imy arbitrary function ^
br r and 9, and make j. __ ^X
^1
tibetitatin^ these values in eqimtion (9), it becomes
Dividing by vr', and restoring the coordinates « and y, tliit
>eoome8 ''"x . ''^X _ " /,^i
This is the fundamental equation of the theory, and expresses thu
elatjon between the function, x and the component, y, of the mag
letic force resolved normal to the disk.
Let Q be the potential, at any point on the positive ade of the
iflk, doe to imaginary matter distribut«d over the disk with the
ar&ce^ennty x>
At the positive stirfacc of the disk
f = 2.x. (15)
HflOOe tlic first m«mber of equation ( 1 4) becomes
di^^ df~ 2niiz War* df) ^"'
Bab Mice Q ssitisfieii Iia])Iace'a equatioa at all points external
o the disk, d^,'PQ__d^ ....
dx* ^ dg' ~ '^ ' ^"f
lud equation (H) become*
i/j*
2l^^=">
0«)
Again, eince Q i» the potential due to the distribution Xi the
otentiol due to the distribution A, or }~,w)Ube ^. From this
(19 off
obtain for the magnetic potential due to the currents in the disk,
ill
dedt'
m
ttf
. ..wortw force uoimal to tbc diik
(80)
■ M>(Aiti*' due to external inag^et«, and
.!*?"'■
tAP
*/>(*
:yv
(ai)
. oaMCtic force norina] t« the disk due to
00*'
_ j^^oHion (18), remembering that
y = Yi + yi'
iiri« '"''' "^T*^ *o '' *'*^ writinj J for — .
fto*"
L^ of /" a"^ Q ^"^ expr«scd in tcnns of r i
If '''^^^^jjwdiak, f and f two new vnriablca buc^
t^ J?
the dis
sueb tlwt
.f =.?.,
. j«it becotoMi '»y integration with reep«ct to f, ■
IV form of this cxprcwton taken in coi^unetion with lb«
^Tj nf Art fi^^ Bbrn'H fJiat Uic nmgnetio action of the currcnl^l
"^ ilUk )B ^oi'">'^'>^ to thai of a trail of images of the
" fid errteitt in the form of a helix.
i*"^. _ggneti« iTBtem connete of « single magnetic pole of
" . ynitr, '1"^ beWi nill lie on the cylinder whoso axis is
%f
bcli
tjve eofl» of tte helix, will I>e 2w — . The magnetic effect of
jl „iil be the Mune a« if thi» bclix had been magnetized
uhew '° '''^ direction of a tangent to the cylinder perpen
^ ' Ur to it" '^^^ ^'^^ "^ intensity such tltat the magnetic moment
'*■ j^ftll jmriion ia namerically equal to the length of its 
]^M,ontbedist.
 rttem in the form of a helix.
1
sti«'
tbat
""the disk. The dixtanoe, parallel to the axis between oon
. unitV '"'' uciii ntti iiv vu lUB uj iiuutT miHSJu aj.ta i
***** f the di*It, ""d which parses through the magnetic pole^
""^ ■■ ^ij] begin at the pMition of the optical tmaiire of tli^J
>7o.]
279
The ealeulntion of the efftct on the magnetic pole would be
jmplicateJ, but it is easy to eeo that it will consist of —
(1) A (inigging force, piirallcl to the direction of motioD of
the disk.
(2) A repulsive ffTce Bcting from the disk.
(3) A force towards the axis of the disk.
Wbon the pole is near the edge of the disk, the third of theae
jrvGS niny be overcome by the force towards the edge of the disk,
adicatcd in Art. 667.
All these forces were observed byArago, and described by him in
Annales ile CA'imk for 1826. See aUo Folici, io Tortolini's
fyna/ji, iv, p. 173 (1853), and v, p. 35; and E. Jochmann, in Cretlta
\jounial, Ixiii, pp. 158 and 320; and Pogg. Ann. cxiij, p. 2U
[1864). In the latter paper the enuations necessary for deter
mining the induction of the currents on themselves arc given, but
Itliis part of the action is omitted in tlie snhsequent calculution uf
suite. The method of images given here was published in the
^roceedinga of the Rogal Socictj/ for Feb. 15, 1872.
1^
»;
Sjiheritat Cun«ntSieet,
670.] Let ^ be the currentfunction at any point Q of n spherical
cnrrentsheet, and let P be the po
tential at a given point, due to a
sheet of imoginarj' matter distributed
over the sphere with BDrfaccdensity
^, it is required to find the magnetic
potential and the vcctoriiotontial of
the enrrcntshect in terms of P.
Let a denote the radius of the
sphere, r the distance of the given
int from the centre, and jn the
reciprocal of the distance of the given point from the point § on
the sphere at which the currentfunction is 0.
I The action of the currentsheet at any point not in its substance
is identical witli that of a magnetic shell whose strength at any
point is numerically equal to the currentfunction.
The mutual ])otential of tlie magnetic shell and a unit pole placed.
at the point P is, by Art 410,
Fig. S».
/M
as.
Since/' is a homogeneous function of the degree — 1 in r aad it,
da dp
dp id , .
Since r nod a are constant throughout the surfivceiDtc^nttion,
But if P is the potential du« to » sheet of imaginaiy mat
of surfacedemity A, _ rr
and £1, the mn<>nciie [)otcntia1 of the cnrrcntsheet, maj be <
in terms of i' in the form
671.] We may determine F, the ecomponont of the rectc
potential, from the expressioa given in Art. 116,
'=//♦('/,•!)'*
where f, •?. C are the coordinates of the element dS, and l,m,nue
the directioncoBines of the normal.
Since the aheet is a sphere, the direcUoa<oo«aes of tlie notmal an
'=!■
BI =
a
But
and
so that
~ adf a3t'
multiplying hy ^dS, and integrating over the aurfaoe of the sphere,
we find Fi^^y^^.
HELD OF USIPORM FORCE.
281
iimiliiHy
adx a dy
The vector SI, whose componentfi arc F, 0, U, ie cnJontly pcr
cular to the rntliue vector r, and to the vector whoso oom
jpoaents *^ j~ > r > ^i^ 7 ■ If we determine the lines of inter
[SectioDs of the sphericAl surraco whoso nidiuB is r, with tho strict of
[ equipotontiul surfuees corrvspondJDf* to Tnlues of P tu arithmcttoiil
tprof^rcssion, these lines will iaclicat'i by their diniction tlie direction
of 91, and by their proximity the mnjpiitudc of this vector.
Id the language of Quateniions,
81 = i FpVP.
672,] If wc assume as the value of /' within the ephere
\ where I'f i< a Bpherical harmoQic of degree i, then out«ide the sphere
"Die currenifuiiction ^ is
The magnetic potential nithia the sphere is
example, lot it be required to produce, by means of a wire
coiled into the form of a spherical shell, a uniform magnetic force
a within the shell. The magnetio potential within the shell is, in
Khia CMO, a solid harmonic of the first d^ree of the form
a=—Mrcm9,
, where Jf is the magnetic force. Ileooe A = (d'^, and
<f = — 3faooB9.
^0 currentfunction is therefore proportional to the distance
I from the equatorial pbtne of the sphere, and therefore the number
of windings of the wire between any two small circleii muHt be
. proxwrtional to the distance between the planes of these circles.
282
CUnBBSTSHEBTfl.
If jV is the wbolc Dumber of windings, and if y is the strei
of tbc current in each winding,
<f,= J.Vyooefl.
Hence t]ic magnetic force witliin the coil ia
3 a
678.] Let us next find the method of coiling the wire in older
to produce within the sphere a magnetic potential of the form of a
solid zonal liarmonic of the second degree,
Here * = ^(Sco8"tfJ).
If tho whole number of windings le iV^, the number between the
pole und the polar distance 6 is \Ns\a*$,
The windinirs aro closest at latitude 45°. At (he oqitator t!
direction of winding changes, and in tho other hemiiipbcre t!
windings arc io the contrary direction.
liet y hv the strength of the current io the wire, tlicn vn
the shell 4 « r3
Let us now consider a conductor in the form of a plane c1
curve placed anywhere within the shell with its plane perpendic:
to the axis. To determine its coefficient of induction we have
find the surfaceintegral of — t over the pbine bounded by the
curve, putting y=l.
Now ll=_iJi^f(^_i{^+y^),
and J = n " «•
as 6a'
Henoc, if ^ is tJte are* of the closed curve, its ooefScient of in
duetioo ia a.
If the current in this conductor is >', there will be, by jVrt 563,
a force H, urging it in tho direction of e, where
^=yy5^ = _A^5yy,
J
and, since this is independent of r, y, t, the force is the some ia
whatever part of the shell the circuit is placed.
G74.] The method givm by Poisson, and dMcribed in Art. H
UNSAS cmiREjrrpnircTiOir.
283
may bo applied io currentslieeta by substituting: for the body
supposed to be wDiformly magnetized in the direction of s with
intcnBily /, u cnrrcntsbeut having the fonu of its surface, and for
whioh the currfiiitfiinctiori is d> = Is. (1)
[ The current* in tlie whuct will bo in pianos parallel to that of jy,
and tbc strength of the current round a slice of thickness dx will be
Id2.
The mngnctic potential due to this cnrreotflbcct at any point
i outside it will be
,dr
At any point inside tbc sheet it will be
as
The components of the vectorpotential art!
dm
S=0.
(2)
(S)
(»)
These reeults can be applied to several cases occurring in pnotioft.
676.] (l) A plane electric circuit of any form.
Let r be the potential due to a plane sheet of any form of which
the surfacedensity is unity, then, if for this sbict we sulistitute
cither A magnetic sheli of strength / or an electric current of
strength / round its boundary, the values of il and of F, ff, II will
be tltotie given above.
(2) For a solid sphere of radius a.
~ 4 »r a* , , ,
^ = ^ — when r i» greater than a,
and F= 5 (3 a' — r*) when r is less than a.
(6)
Hence, if such a sphere is magitclized parallel to ; with intensity
I, the magnetic potential will he
(7)
(8)
4w
and
11 = ^I ~iS outside the sphere,
D = —  /? inside the sphire.
3
If, instead of being magnetized, the sphere is coiled with wire
in e^nidiotAnt circles, the total strength of current between two
small eirclci! whofc planes arc at unit distance being /, then outside
the sphere tbc value of U is as before, but within the siihere
fi=^/,. (9)
I This is the cn»e already discussed in Art. 6T2.
284
CFBRESTSIIEETS.
[6?
(3) The case of an ellipsoid noUormty ma^etiz«) parallel to
« giv«D lin« liiu been discussed in Art. 437.
If the ellipitftid is coiled tvitk wire in parallel and cqnidistul
planes, tbe magoetic force within the ellipsoid will bo antform.
(4) A C^linilrk Magnet ttr SoUaatd.
676.] If the body itt a cylinder hnrin^ any fonn of section ind
bounded by pliincs ]>erpen(liGular to ii« generating lines, a&d
if l\ ig the potential at the point («, ^, r) due to a plane area <J
euriaci.'Hleusity unity coinciding with the pontive end of tliv
soteuoid, and Y.^ the potuutial at the wanie point due to a plane am
of snTfacedertsity unity coinciding with the negative end, then, if
tbe cylinder is uniformly and longitudinally magnetized with in
tensity unity, the potential at Uie point (j, _jf, c) will bo
n = r,r.. (lo)
If the cylinder, instt^d of being a magnetized body, ts nniformly
lapped with wire, so that tbere are m windings of wire in unit
of length, and if a current, y, is ma<le to (low tlirougb ttus wire,
the magnetic potential outside the solenoid is as before,
ii = «y(r,r,), (II),
but witbinthe space houoded by the solenoid and its plane ends
ii = ny{4iir+r,r,). (la)
Tlie magnetic potential is discontinuons at tbe plane ends of t^Hi
solenoid, but the magnetic force is continuous.
If r,, r,, the distanoes of the centres of inertia of the positive
and negative plane end respectively from the point (*, y, r), are
very great compared with the tranavene dimensions of tbe solenoid,
we may wnte
" \ — — .
(13),
where A is the area of either section.
The magnetic force ontaude the solenoid is therefore vety small, I
and the force inside the solenoid approximate^ to a foroe parallel to '
the axis in the positive direction and e<iual to \xny.
If tbe section of the solenoid is a circle of radios a, the Talue* of I
r^ and r, may be expressed in the sertea of splierioal harmonics
given in Thomson and Tait's }!edural PkUoto^/tg, Art. 546, £x. 11^
677.]
SOIBNOID.
285
In Uiwe expTcsnoQs r » tlie distance of the point (x, y, z) fr&m
ae centre of one of ttie circular endn of the solenoid, and the uioul
irmonica, /*, , P^ , &c., are those corresponding to the an^le $ which
mftlccs frith the nxie of the cylinder.
The lint of these espressions is diseontinnous when = ^ bat
ve must rcmetober that within the Golenoid wo must add to the
oa^ctjo force deduced from this expression a longitudinal force
677.] Let us now consiiler a solenoid so long that in the part
'of space whicli we consider, the terms depending on the distance
^Jrom the ends may ho neglected.
H[ The magnetic induction through any closed curve drawn within
^Rhe solenoid is ■i unyA', where A' is the area of the projection of
^■Ihe curve on a plnne noruinl to the axis of the solenoid,
^1 If the closed curve is outside the solenoid, then, if it encloses the
^bolenoid, the magnetic induction through it is \TsiiyA, where A is
^M;he area of the section of the solenoid. If the closed curve does not
surround the solenoid, tJie magnetic induction through it is zero.
If a wire be wound n' times round the solenoid, the ooefficient of
ioductioQ between it and the solenoid is
M= \isnn'A. (!6)
By supposing these windings to coincide with n windings of the
we find that, the coefiicient of self induction of unit of
th of the soleuoid. taken at a sufQcieut distanoc from it« eix
emities, is L = 4ir»M. (17)
Near the ends of a solenoid we must take into account the t«rms
Jepending on the imaipnar^' dixtribution of magnetism on the plane
ends of the solenoid. The efFuct of these terms is to muke the co
efficient of induction between the i^olcnoid and a circuit which sur
rounds it le«>< than the value 4 n n J, which it has when t)ie circnit
sumnndw a very long solenoid at a great distance from eitlier end.
H Let as take the case of two circulur and coasal solenoids of the
^^Binie length /. Let the rudius of the outer solenoid Ite c^, and let
it be wound with wire k» a? to hnvc i] windings in unit of length,
let the radius of llic inner solenoid be r,, and let the number of
indings in unit of length be n^, then the coelTicient of induction
between the solenoids, if we neglect the eflcct of the ends, is
U=Gg, (18)
'wh«Te = lirn,, (I9)
(20)
y = nc//»j.
286
CTTBRESTSHEETS.
[67S.
'J
678.] To (Ictcniune th« oflect of the positnc «nd of the soleooiib
we mu»t caloulatu the cucfTicicnt of mduction on the outur sol«no«l
due to the oircular disk which forni* the wivd of Uie iuo»r solenoi
For thisi purpose ne take the tecond expression for F, as givei
in ^Illation (15), and diOerentiale it with reapect to r, tb'u ^
the magnetic forco in the dirootiou of the radius. We then muJtipljr
this Gxpressioa by ivr^dfi, and inti^rate it with reepeot to/i from
u = 1 to u =  , _ ■ — . Tins {fives th« coefllcieDt of induetioa
with T«spcct to a single winding of the oater eolenotd ut a distsncr
t from the positive end. We then multiply this by ^h acid
integn>t« with respect to t from i = I to « = 0. Rnally, we
multiply tho result by Wjij, and so find the elTect of one of the
«nd« in diminishinfj the coefficient of induction.
We thii!! find for tlie vidno of the coefficient of mutual indue
betwocu the two eylinder»,
c,+/— r_
.where a = 4
2.4 2.3 c,*' "t'^
1.S.5 1 c,*/ 1 „p,* . 6V\ 0. ISS
where r is put, for brevity, instead of •/t^^ c,*.
It appciars from this, that in calculating the mutual induction
two coaxal solenoids, wo must use in the expression (20) instead of
the true length I the corrected length ^— 2e,<i, in which a portion
eqtial to aOj is supposed to bo cut olT at each end. When the
solenoid is very long compared with it« external nulius,
n
«=i
t~ii<
(23
679.] When a solenoid consists of a number of layers of wire of
8uch a diameter that there are n layers in unit of length, thi
number of layers in the thiokoess Jr is ndr, and we have
P
0=4aJn''dr, and y = ■wtjn'r*4r.
, tii^
If the thickness of the wire is constant, and if the induction take
place between an external coil whose outer and inner radii are » and
V respectively, and an inner coil whose outer and tooor radii are
f and r, then, neglecting the edect of the ends,
(?, = j^/V«.'('f)U*'')
(Ml
68o.]
iKOtrcnoif ooiL.
287
^TI>at thi« may bo ft mtucimiini, x nod s I>eing given, and y
riabl«, , »
• = 04p (20)
This (qHation gives the beat rclitUon between the depths of the
pritnary and M>ci>n<Iury coil for SD induction machine without an
iron core.
If there is an iron core of mdios ;, llien Q remains oe before, but
,.„«(^
(27)
(28)
I
I
f ^ is 'giren, the value of t which gives the iDaximnis value of ^ is
Vhen, as in the case of iron, « U a hirge number, z = ij, nearly.
If we BOW make x constant, and y and : variable, we obtain the
maximum value of Gy whea
z : f : g : : i : 3 : 2. (30)
The coefficient of )>clfin<Iuction of a long solenoid wbottc outer
Rod inner radii arc « and y, and having a long iron core whoM
nkdins is f , is per nnii length
4ir/"'j«;/'«'(p« + 4nKa«)rf7+»//*V(^f4ff«f')rfrj«»rfp,
= §7i''/«*(jf)»(«" + 2«y + 3/ + 2<jr«.*). (31)
680.] We have hitherto supposed the wire to bo of nnirorm
thkkncxti. Wo shall now determine the law according to which
the thiekn»N muRt vary in the dintTent layers in order that, for
a given value of the resistance of the primary or the twcondar^' coil,
tlie value of tiie coefficient of mtitunl induction may be a maximum.
Let the leaistAnce of unit of lengtli ofa wire, ttuch that h windings
occupy unit of length of the solenoid, be p»'.
The reoEtoDce of tJie whole solenoid i*
R^%itlj«*rdr. (82)
Thecondition that^ with a given value of A, Onuiy boa maximum
dG
3?
iG ^dR , ^.
IS I = C T, where C is some constant.
This givM ■* proportional to , or the diameter of the wire of
the exterior coil must be proportional to tJio square root of tlw
radius.
IH
288
CDB C BSTsnEETS.
In order that, for a g'iven ralue of S, g may be a maxinaam
«. = C(r+i^'). (35,
Hence, ir there k no iron core, the diameter of the wire of th«
interior coil thould be inversely as ike sqnare root of tho ntdiu*,
but if ttiere is a core of iron having: a hig^h capacity for magnet^d
izalion, tho diameter of the wire should he more nearly dirvdlji
projwrtional to the square root of the radius of the layer.
An EndltM Soknoid.
681,] If a solid he generated by the revolution of « plane ana .
about an axis in ibs own phino, not cutting it, it will hare tlie fovm
of a Ting. If this rinj* be coiled with wiro, tio that tho winding
of thv coil are in planes jutesing: through the axiv of the ring^, then,
if B is the whole number of wLndings, the currentfunction of the
4
layer of wire is ^ = — nyd, where $ is the angle of aximuth about
the axis of the Hog.
If £1 is the magnetic potential inside the ring and ii' that ou
side, then £la'=iint>+C = —2nye + C.
Outside the Hng, A' must satisfy Laplace's equation, and must
vanish at an in6nite distance. From the nature of tlie problem
it must be a fanotion of only. Tlie only value of 12' which folGls
these couditioDS is xcro. Hence
C=0, U = 2ay^+C.
The magnetic force at any point within the ring i* perpendicular
to the plane passing through the axis, and is equal to 2 ay,
where r ie the distance from the axis. Outside the ring there la
DO magnetic force.
If the form of a closed curve be given by the coordinatee z, r,
and of its tracing point as functions of t, its length &ODt a fixed
point, the magnetic induction through the closed curve may be
found by integration round it of the vector potential, the com
ponents of which are
i
t
1
I
,<
We thus find
2Ny
f'sdr^
I ' J*
.'ft f as
taken round Uie curve, provided the curve is wholly inside the ring.
Vlg.m
EITDLESS BOONOID.
^■f the curve lies wholly without the ring', bat embraces it, the
HvDig7>eiio iDdnction through U is
^■rh«n
I' tt aini
^"•'11?%^=^'^''^
rherc the accpnted ooordinntes rofer not to the closed carve, but to
. singte wiiKling of the solenoid.
Tlie magnetic induction through tmy closed curve embrftcing the
ring is therefore the same, and equal to 2nya, where a is the linear
rif .' J/
qoantitj i ^ 7, A*. If the closed curve does not embrace Uie
ring, the magnetic induction through it is zero.
Let a second wire be coiled in any manner round the ring, not
necessarily in contact with itv so as to embrace it m' times. The
induction through this wire is 2n it'ya, and thereibre 3t. the
coefficient of induction of the one coil on the other, 10 ^=2Hn'a.
Since this is <U)te independent of the particular form or position
of the second wire, the wires, if traversed by electric curreDte, will
rienc« DO mechanical force acting between them. By making
fthc second wire coincide with the first, we obtjtia for the coefficient
Wfinduction of the ringcoil
Z = 2a*(i.
CHAPTER Xin.
PABALLSL CUBBBKI&
Cylindrkat Omduetori.
682.] In a very importimt cla&s of eltctriciil arrang«menta th«
current is conducted tliroU):fti round wires of nearly uniform aection,
Mild fillier straight, or mch that the radius of curvature of the axi<
of the n ire is ver\' ^real coiniarnd with the radius of the trstuveisA
Mwtion of the nire. In order to be prepared to deal mathematically
with such arrangements, we shall begin with the case in which the
circuit couBists of two very long parallel conductors, with two pioM*
joining their end^ and we slial) confine our attention to a part of
the circuit which ia so lar from the ends of tlie conductors that the
fiict of their not being infinitely long does not introduce any
sensible change in the distribution of force.
We shall take the axis of i parallel to the direction of the con
ductors, then, from the syrometr}' of the arraugements in the part
of the field oonitiderod, everything will depend on U, the comi>oaent
of the vector potential parallel to :.
The components of magnetic indactioo become, by equattona (A),
.4J. <■)
For tlie sake of generality we vhnT) f^ppose the co^cient t
mugnetie induction to be fi, so that o =^a, i = p,fi, where a and
nre the compoucnts of the magnetic force.
The equations (E) of de«tri« carrents. Art. 607, give
I
a
4
STRAIOnT vritB.
688.] irUie outTvnt ix a fiinotiua of*, the di»t»Doc from tho axU
^of ;, «ud if we wrilv
x = rcosO, and y^raaff, (4)
I And ;S for the magiietie force, iu the direction in wliicli $ is nica»urcd
1 pvqicudiculur to the plani; througt tLe axis of ;, we have
^.^. = ^^(^0.
(»)
If C is tho whole current flowing tlirougli a section bounded by
[s circle in the plunc xy, whose centre is tlic origin and whoso
Tiiditis is r, f
C= I 2Trrii)dr = \^r. (ii)
It appears, thcreforo, thut the magnetic force iit a given point
due lo a current nrnmged iu cyliiulrieid »trula, whose common axis
i» the axis of i, deiwnds only oii the tuUil slreiigtli of the uiirreut
flowing through the strata which lie between the given point and
the axia, and not on the diiitribution of the current among the
» different cyliudrical strata.
For instance, let the conductor le a uniform wire of radius a,
and let the total current through it be C, then, if the current is
Dnifortnly distributed tJtrough all parts of the section, w will be
oooBtant, and 6' = u len*. (")
llie current flowing through a circular section of radius r, r being
l(t8B than a, is C'= itior. Ilcnce at any point within the wire,
lOatside the wire
/5 = 2
(8)
(fl)
In the subetanc« of the wire there ie no magnetic potential, for
Ewttliin a conductor carrying an electric current tho magnetic force
not fulfil the condition of having a potential.
Outeido tho wire the magnetic potential is
il=~2C0. (10)
Let ns guppoBC that instead of a wire the conductor is a metal
tub« whose externa] and internal radii are tij and a^, then, if C is
Khe current through the tubular conductor,
%« magnetic force within the tube is zero,
tube, where r is between a, and ■>,,
(11)
In the metal of tli«
(12)
\ 1
29S
vxnkthsjr
BBESTS.
and oataide the tube,
^
("
the same as wheo the ciirrent flows thronffh n solid wire.
684.] The magnetic induction at any point i« ^ = n^, and siacv,
by equation (2), rf/f
Tb« Taloe of 11 onteide the tube is J
A~2^,C\ogr, (uy
nhcre ^ i^ the value of^L in tlif space outnide the tub4>, and Ai» t
constant, tho value of which dopeuds on the pontioD of the return
carwnt,
In Uic tubstanee of the tube,
Ib the space within the tube // i* constant, and
II=A2^Chgc,+^C{l+^^,\os^^ (IB)
685.] Let the circuit be completed by a return curront, flowing
in a tube or wire {nrallol to the first, the axes of the tno currents
b«ing at a dieitanei' i. To determine the kinetic cnci]gy of tht^
i>yst«m we have to catcuUU; the integral
T= \jjJH«dsdyd:. (19)
If we confine our attention to that pari of the system which lies
IwtwMUi two planes perpendicular to the axes of the conductors, and
distant / fron each other, the expression becomM ^
T=\lfJHKd^dg. (80)
If we diKtinj^ish by an accent the quantities belonging to the
return current, we may write this ^
^ =JJirti^d^dy+JJja'icdrjf + jj Bwd^dg+jJH'm'djr^/. (sip
Since the action of the current on any point outside the tabe is
the tame ait if Uie Kiinie current had been concentrated at th« axis
of the tube, the mwiii value of ff for the section of the return
current w A—'2iJyC\ogb, a»d the mi.«n valoe of H' for the sectioa
of the iKisitivt' current i« A'—2^„C\vgh.
LOSGITCJDINAL TEN8I0K.
293
llence, io the expression for T, the first two terniB may be writt«o
InteiiTatiii^ the two lattvr terms in the onliunry way. nn^l lulding
tlie resulte, remembfritijy that C+C'= 0, wo obuiii t!i« valuo of
the kiiiotic enei^gy T. Writing: this k^C^, whoru / u th« co
LfBvitnt of solfimluetion of the syBtem of two cmduuton, we find
lU Die value of £ for unit of letigtb of the eyctem
+ 1*^17^=^ +
2fl/*
iog?L.
1;
(22)
(23)
'1— "« «^<')^
If tbe eondtictors ftre solid wires, a.^ and ^/ arc zero, nod
r /a
j = 2^ol"8 7V' +i(M + f')
* a, a,
It is only in the case of iron wires that we need take account of
the monetae induction in calcuUtinpf thiir gclFinduction. In
other casce we may make ^, ft, nud (i nil oqual to unity. The
smaller the radii of the win^s, und tlic greater the diatttnce between
them, the greater is the selfinduction.
To/nd (ht Sepnl^oH, X, betioeen the Two Portions o/ Win.
08C.] By Art. S80 we obtain for the force tending to increase b,
= 2f*ojC», (24)
wbic^ Sf^rees with AmpireV formnta, when >/, = 1, as in air.
<J87.] If the ku^lh of the wires is great compared with the
distance between tbem, we may use the coefficient of selfinduction
to ilt.it<;rmi&c (h« tension of the wires arising from the action of the
current.
If ^ M tlii< tension,
,rf£
z=\~c^.
= C»j^.^^ + f
(25)
In one of Ampere's experiments the pAnillcl conductors coDsiBt
of two troughs of mercury connected with each other by a fioating
bridge of wire. When a current is made to enter at the extremity
of one of the troughs, to flow along it till it reaches one extremity
294
PABALLRI, CtTBRENTS.
[688.
of the floating wire, to pnat ioto the otb«r trou'^h through liic
floatincf bridge, ttod »o to return tioog tfae second trough, Ibc
flontiag bridge moves a\nng the troaghs so as to leogthfio tb« part
of th« mercoiy traversed by the cnrrent,
Profcasor Tmit tios simplified the electrical conditionft of thii
cxpcnmviit hy substituting for the vrirt a Hooting siptinn of gls»
filled wilh morcury, eo Uiat the current flows in mercury thron
otft ita course.
This experiment ie sometimes adduced to prove that two clemcntA
of A current in the same Btraight line rejicl one another, and Uidb
to shew that Ampere's formula, which indicates snob a repulrion
of colIincfirelemcDtd, is more correct lliunthat of Orasamann, which
gives no action between two eiemcnltt in the tame straight line
Art. 526.
Bnt it is manifo^ that since the formulae both of Anipire and of
Gmsbmann give the same results for closed cireuitii, and since n*
have in the experiment only a closed circuit, no result of tie
experiment can favour one more than the other of these theoriet.
In fact', both formulae lead to the very same value of the
repulnon as that alrouly given, in which it appears that i, tli«
diMtiince between the parallel conductors, U an imiwrtant element.
When the length of the conductors i« not very gnat com
with their divlance ajiart., the form of tlie valia of li beooi
tomewhat more complicated.
688.] As the distance between the conductors is diminished,
value of L diminisheis. The limit to this diminution is when the
wires are in contact, or when It » a,+a^'. In this case
4
J
6S<>0 SnOMPM SBLFIJIDPCnOX. 295
I fThia IB a minimum wh«n Hj = u,', and Uicn
K L=2l(\og4 + \).
^^^ =2/(1.8863),
HH 3.7726^. (27)
V^rtiiB is the smallest \a\ac of the ReiriDduclion of n round wire
Hdoubled on itself, the whole len^h of the wire bein^ 2 /.
f Since the two parts of tJie wire must he inxulal^d from each
other, the Gelfindnetiou can never actually reach thin limiting
value. Ry using hrood tkt strips of metal instejul of round ww»
the selfinduction msty be diminished inde&nitely.
Oh tie EUeiromoiive Force required to produee a Current <f Faryiny
Intfiui^ ahiig a Cylindrkal Condactor.
68D.] When the current in a wire is of varying intensity, the
ive force arifiing from the induction of the current on
' is different in different part« of the section of the wire, being'
general a function of the distance from the axis of tlie wire
I as well as of the time. If vre suppose the cylindrical conductor
Lto consist of a bundle of wire« all forming part of the same circuit,
ISO tliat Uic current it compelled to be of uniform strength in every
part of the si^cticn of the bundle, the method of calculation which
_ we have hitherto ui>cd would be strictly applicable. If, howcwr,
f we consider the cylindricnl conductor as a solid ma«i« in which
electric currents arc free U> flow in obetlience to electromotive force,
tlie mtcnaity of the current will not be the same wt different
didtances from the axis of the cylinder, and the clectrom<itivo fonrCK
tlicmselvos will dt^pend on the diittribution of the current in the
different cyiiodric strata of the wire.
The vectorpotential //, the density of the current », and the
electromotive force at any point, must be considered as fiinctions of
the time and of the distance from the axis of the wire.
n Tlie total current, C, through the section of the wire, and the total
BiJcctromotive force, E, acting round the circuit, are to be regarded
Vm the variables, the relation between which we have to find.
^^^Irtt US assume as the value of //,
■K n=$+T,+ T,r' + kc. + T,r'', (I)
'^^Sere S, T^, T„ &c. are functions of tb6 time.
Then, Ihmi the equation
d*H IHH , ,„,
_. + j^=_4,«. (2)
we find UK = 7', + &c»'r.H'. (31
Bwef
296
P^SJILLKL CUneEKTS.
If p deDotw tlie specifio rciii»taDoo of the 8uh«tatM!e per mit of
volume, the elect roniotive for«e at any jo^int in p t, und this quj Ik
ex)>re«iH>d in U'rmij of the «l«otric poteiitiul oiid Uie vector pot«a^
7/ bv cquutioDs (B), Art. 598,
or
(*)
Comparing the cocffiine&ta of like powers of r in eqnatioos
{5} and (5),
it ^
P
* as dT„.
^^ P dt '
^" pn' dl
Hence wc mnj' write
<f*
^^'pdt'
dz
7.=
I ^r
(9)
(ID)
p{n\f3F'
600.] To find the toUt current (7, we muet integrate to over tbe ^
twclion of the wire whose radius is a, 1
Jo
Krdr,
(II)
Suhstitnting th« v&lue of v lo front eqnation {9}, we obtain
C = (7',a<+&o. + »7',fl"*). (12)
1'^ie value of H lit any point outside iJie wire dcpendtt only oB
the total currpnt C, nnd not on the mode in which it is Uislrihuted
within the wire. lU'Dce we may assume that the vxliie of //at the
nurlace of the wire is AC, where A ia ^ constant to be det«nniDed
liy calculation from the general form of the circuit. Vatimgtt = AC
when r = B, wc obt^ti ^
AC=Si■T^ + T^a* + &c. + T,a^^. (13^
If we now write = «, a is Uie value of tlie conductivity of_
anit of length of the wire, and we have
dT 2b* rf>7"
«o" rfT
+&«.),
ACS=T^J^^^,^
d'T . a' dT , .
M
('«)
VARIABLE CCBBE8T.
297
To eliminato T from thcae cqiuitioue wc mast Gnt reverse the
^Beriee (14). Wo thus find
fit*'
d*C
+ rfi«'»'^fVA«'"^f&c.
4t ^'"rf/ '" ^/»
W« have bJbo rroio (tl] and (15)
Prom the la«t two eqiiationB we find
.dC dS
dC
^) + '^+*°S?tV'^
dt dt
dl*
■^^ ■ ^'^^^..^O.^
+ AV*»''' rfT*
■ If/ is the wliolu length of the circuit, R its resiiitance, and E the
H electromotive force due to other cnuscs tliaa the induction of the
KxorreDt on itself, dS E I
F
«=g'
(17)
The 6r9t term, RC, of the righthand member of this eqaation
expresees tlie electromotive force rei^uired to overcome the reeist
I aace according to Ohm's law.
The second term, l{A + i) ^, expresses the electromotive force
tit
[ which nrould he employed in increasing the electrokinetae momentum
the circuit, on the hypothestK that the current is of uniform
th at every iKiint of the section of the wire.
Tlie remaining terms express the correction of this value, arising
lUw f»ct thut the current is not of uniform strength at diiferetit
from tlic axis of the wire. The actual B3«tcm of current*
\h»a a greater degree of freedom than the hypothetical syrtem,
in which the current is constrained to he of uniform streugth
throughout the section. Hence the electromotive force required
Bto produce a rapid change in tlie strength of the current is some*
H what less than it would he on thix hypntlieHia.
H The relation between the timeintegral of tlie electromotive force
H«nd the timeintegral of the current is
fEdl = njcdi + / ( J + i) c A ^ ^ + &c. (1 9)
IT th« cnneot before the beginiung of the time has a constant
298 PARALLBL CtTRREKTS. [fi^I.
vaiuc C^, and ir daring the tiiufi it rises to Hic vnliic C, , and ny
nmiDB coDNtnnt nt Ihnt vuluc, then the t«nDfi involving tite dtfler
ential coeflicivittn of C vanUh at both limits, And
fEdi = R fcJl+l{A + J) {CiCt), («J
tlie cuat value of the eleLtromotirc impolse u if the curreDt had
been uniform Ihroujjhout th« wire.
0« Ue Geometrifal MeaH Dislaxee y Tko Ttgarea in a Plaiu*
fiftl.] Til oalciilaling the plectromnfrn'^tic action of n current
flowing; in a ctriiijrlit conductor of any gircn soction on the current
in ft purallel conductor vrhoKc locution is also given, wv bavi; to find
the intejrral CCiT
where Jxiiy in an ot«nient of tlie area of tbe first sectioD, i*'^
uo element of the second section, and r tbe distance between
thctie eI<!Ricnt«, the integration being extended first over every
element of tbe fint Hcctioii, and then over ever^ element of tbe
second.
If we now determine a ttne U, sneh that tliis integral is equal to
A^ A^ log R.
wbere Ax and A,^ nrx^ the areas of tbe two Reetions, tbe length of fi
will be the «arnc tvbat'evcr unit of length we adopt, and whatever
sytttcin of logiirithme wo use. If wo supiiOKC the Mcttomt divided
into elements of equal nze, then tlic Ingaritlim of B, muIUptied
hy the number of pairs of cbmenls, will be ei]ual to tlie mm
of the logarithms of the diittaiK'eif of all tbe pairs of elements.
Here R mny be considemd a» the geometrieal idcad of all tJie
distances hetwcca pnirK of clement*. It is evident that the valoe
of R must be intermediate between tbe gteateet and the least
values of r.
If R4 and Hg are tbe geometric mean distances of two fig1lre^
A and B, from a third, C, and if Rji*a is that of tbo sum of tbe twe
figures from C, then
{A+B) log Ra^b = J log «^ tSlogR".
Hf means of tbis ntlalicm we can determine R for a ofHnpoond
figure wheo we know R for tbe parts of tbe figure.
• Trau*. S. B. RU*., l$Tl9.
69i.]
OBOMKTBIC JIBAN DISTANCB.
299
692.]
IBXAUPLBS.
(1) LH R be thi ni<>an diKtanoc from the point' to Hic lino
JS. Lut OP be iwriniiJioiiliir lo AB, tbcn
JJi{\ogIlH)=AP\og04 + PBhs0S+0PA'0£.
Fig. 41.
(2) For two lines(Pig. 42) of longtlis a nnd (> dran'o perpendicu
lar to the fixtnrmities of ii line of leiij^li c and on the same eide of it.
ai{2\ogR + S) = (c»— (<!—«)*) log v'c* + («4p + c» logc
— *>(« — iltan' +act«n"'  + fctiui' ■
^ ' c c e
Fig «.
(3) For two lines, PQ and ^5 (Fig, 43), wbo»c directions inter
ect at 0.
'Q.AS(2lo»/f +3) = \ogPR{20P.ORBia'0PS*<!o»0)
+ \ogQS{20<i.OSmn'0~QS*eo90)
 log PS (zOP.OSsin^ 0~PS»coaO)
\ogqR(20Q.OR^Q*0QR'cosO)
noo [OJ^.^soQ^s^Rt 0R\ p'^qos'.Ps'q}.
Fig. IS.
300
IlLEL CUBRENTS.
(4) For a point and » twctongle ABCD (Fig. ii). hoi
OQ, OR, OSf be pcrpciidicularg on the aides, then
AB.dJ){2U>gB+3)=2.0P.OQ\ogOJ+2.0Q.OItlogOB
+ 2. OR. OS l(^ OCi 2. OS. OP log OD
+ OI».D'd'J + OQKjd'B
+ OR'.Bd'C + OS'.C'd^.
Fig.it
(5) It is not necessary tbut tlie Iwo figures should be dilTerent, for 
we may find the geomotric mean of the distanoes between every [air i
of poiotfi io the soma figtire. Thus, for a ctratght line of Ivngtli a, 
\oeR = loga—l,
or R — aa~t,
S = 0.22313 a.
(6) For • reutangle whose ddc>e are a aad 6,
logV? = lDg>/^?T6W^' log/Y^l + ^i^loff/^r+Jj
.J>^*+$>n'j«.
^hen tlic rectangle is a square, whose side i* s,
log J? = Iog«+l Iog2+ ^H.
(7) The geonuttric nicnn distancA of a point from a circular '
ie equal to the greater of the two quantities, its tlistance from '
centre of the circle, and the radiua of the circle.
(8) Hence the Reometrio mean distance of any figure frol
ling l>nund<d by two concentric circles is equal to its g«ometria
mean distance from the centre if it is entirely outside the ring, l)ul
if it is entirety within tJic ring
6931
SELPlKDUmOlT OF A COIL.
where a, and a, ar« the outer and inner nulii of tLe ring. R is
in this case independent of tlie form of tb« figtire within the
ing.
(9) The geometric roesn distance of ftll pairs of points in tlw
ig is found from the equation
1 n 1 "a* 1 "i ,3a.'— a'
For a circular area of radius a, this becomes
]ogi? = logffli,
or R = a«"*,
R= 0.77«ea.
For A circulnr line it becomes
R = o.
693.] In calculating the coefficient of EelfindoctioD of a coll of
liform M!Ct4on, the radiiiN of curvature being grtiit c^tmparod with
dimeiwiottt of the tnitiBTer«e section, wc first determine the
trie mean of the distnuccs of every pair of points of the
section by tlic method already de8eril>ed, and thcji wc cakulste the
eoeffietent of mutual induction betwrn'm two linear oonducton of
tlte given forn), phK'ed at thif distance apart.
B This will Imi the coeffldent of Belfinduction nben the total cur*
rent in the coil is unity, and the current is uniform at all points of
^^.he section.
^1 But if tlicre arc n windings in the coU we must multiply the
^coefficient already obtained by »", and thus we slinll obtain the
'floeffldent of self'induction on the supposition that the windings of
Uie conducting wire fill the whole section of the coil.
^K But tlie wire is cylindric, and is covered with insulating material,
'•o that the current, instead of being uniformly distributed over the
section, is concentrated in certain parts of its and this iooreaaes the
ooeffioieDt of selfinduction. Be«ides tliis, the currents in the
neigfabooring wires have not the same action on the current in a
g iyea wire as a uniformly distributed current.
^B The oorrectiona arising from these considerations may be de
^Termined by the melhod of the geometric mean distance. They
are proportional to the length of the whole wire of the coil, and
may be expressed as numerical quantities, by which we must
multiply the length of the wire in order to obtaio the corroctioa
' coefficient of selfiuduotion.
^rf^
802
PABALLEL CtTIlREKTS.
[693
Let the diameter of the wina lie d. It is covered with iaffnlatinj;
EDat«ri»l, ftnd wound ioto a coil. We shftll suppose that tbt
eectioDs of the wires are in square order, as in Fi^. 4S, and that
the distance between the axis of each wire and that of the next
is i>, whether in the direction of the breadth or tiio depth of thr
coil. J) ie evidently greater than d.
Wc have first to dc^tcrmiiie the excess of eelfiDductioo of unit of
length of a cjlindric iviro of (liamotcr d over that of unit of len^li
of o fquarc^ wire of side D, or
. fl for tho square
^ Jt for the circle
= 2(logT + 0.138060fi).
The inductive action of the eight oea
round tvires on the wire under consideration
if loss than tlint of the corrcepondiog eight
equarc wires on the equaro wire in the miilil
by 2x(0197l).
The corrections for the wires at « greatv
distance may bo neglected, and the total
correction may be written
2 (log, T + O.I 1835).
The fionl value of the selfinduction is tlierefore
i= ii»if+2/(log,^ + 0.Il835).
! II ID the number of windings, and i the length of the
3f tlie mutual induction of two circuits of the fonn of the mean
wire of the coil placed at a dintancv R from each other, wbure £ i»
the mean geometno distance between {lairt of pointa of the •eetion.
J) n the distance between consecuUre wirea, and d the diameter
of the wire.
o
o
o
o
o
o
o
o
o
III
all
Fig. 16.
CHAPTER XIV.
CIBCCLAR Cl'BKENTS.
Magnetic Poienlial due to a Circular Current.
■J] The tnagnelic j>oUntinI nt u given point, due to a circuit
fing a unit current, t.i iiuincrirally e<junl tn th<. soliil aogU* «ul>
led hy the circuit at tliat point; see Arts. 409, 485.
When the circuit is circular, the solid angle is that of a cone
of the second degree, which, when the given point is on the axis
of the circle, becomes a right cone. When the point is not oa
the axie, the oone is an elliptic cone, and its solid angle is
numerically equal to the area of the spherical ellipse which it traces
on a Bphcre whose radius is unity.
This area can be expressed in 6nit« terms bj' means of elliptic
integrals of the third kind. Vfe shall find it more convenient to
eapand it in the form of an infinite scries of ^hcrical harmonics, for
the tacility with which mathematical operations may be performed
on the ^ncral term of such a series
more than counterbalances the trouble
of calculating a nunibcr of terms guITi
^ont to on«urc practical accuracy.
■ For the (take of generality wc shnll
B^nie the origin at. any point on the
sxif of the circle, that is to sny, on
tlie line throogh the centre perpen
dicular to the plane of the circle.
Let (Fig. ii) be the centre of the
circle, C the point on the axis which
wc assume as origin, II a point on the
circle.
I)«!*crib« a sphere with C as centre,
and C/l m radius. The circle will lie
Fig. IS.
on thia 8ph<rrc. and will form a mnall circle of the sphere of
M^hur radiuri c.
[[Mgumr rau
304
CIBCULAB CUBBEXTS.
[6
UA CH=e,
OC = fi = ««»tt,
Olf = a = enaa.
Let A be the pole of the sphere, and Z xay point oa the axis, aaj 
let CH = ..
Let li be any point in epace, and letCS = r, and ACJi = 9.
Let P be the point where CR cute the eph«re. 
The magnetic polt'ntial due to the circnlar corrent i» equal to j
that due to a magnetic Dhtll of strength unity bounded bj the I
current. As the form of th« surfacn of tlie shell is indifibnal,!
providid it i« bouuded by tlic circle^ vns may auppOM it to coincide j
with thi; 8urijivc of the «plicr«.
Wc have sliewn in Art. 670 that if F ia the potential dne t« a '
etratum of inalt«r of «iirfuce density unity, sprwd over the sar&n 1
of the sphere within tlie small cirelA, the potential duv to a nu^ 
Dctic shell of utrcngfth unity and bounded by the same circle is
Wf liave in the first place, therefore, to find F.
Lt't the given point be on the axis of the circle at J?, then tbe]
jiurt of the potential at Z due to an element dS of tlio spheric*!'
Hurlace at /* is JS
FP'
Thia may be expanded in one of tlie two serioa of spberictl huA
monius, AS f a ^ i
or :^7>„+P,f +&c. + i»,^+&c.j,
the first scries being convergent when 2 is leas than e, and tlie
second when x is greater than «,
Writing dS = — c> dp.d<^,
and inlcgrating with respect to ift between the limits and 2ff,
and with respect to li bettreeo the limits cosa and I, we find
or
'"=2»~jy*i>„rfM4&o. + p/'i*<rf»i + otc..
By the cliaracLcristic eqtiation of 7>^,
595]
SOLID AjrOLE SeBTSNDKD BT A ClftCUE.
icnce
£p,,.
eipression £aUs when i = 0, but since Pq = 1,
(2)
/
Ax tbe function ~j~^ occurs in every part of tUis invcstigntioD iv«
l)«U denote it by the nbbrcviatcd symbol i*,'. The values of /*,'
oorrespondiDg to eevnal values of i are given in Art. 698.
We are now able to writ* down the value of F for nuy point S,
whether on the axis or not, by substituting r for :, and mnltiplying
each term by the zoniil harmonic of of the game order. For
/' must be capable of expansion id a series of zonal harmonics of 9
wiih [Hoper cocfBcicnts. When $ = each of the zonal harmonies
btfoomcs equal to unity, uud the jioint It lies un the axis. Hence
the cocffictint« are Ui« terms of the expansion of f for a point oa
th« axis. We thus obtain the two series
or r=2,^lp+&o. + i^^7r(«)7;(fl)j. (4)
695.] We may now find m, the inagnetic potential of the circuit,
Ll>y the method of Art. 670, from tlie equation
We thue obtun the two aeries
« = 2«looa« + &c.+ ^jJ»/(«)i',(tf)+&c.jt (6)
 «2xain»«{ l5i>,'(a)/', («) + &c.+ ^^>/f«)iJ(tf}} . (6')
The series (6) is converf^ent for all values of r less than c, and the
series (6') is convergent for all \a1ues of r greater than c. Al th«
eurfaoe of the sphere, where r = e, the two series give t)i<* Muiie
for tt when $ is greater than a, that is, for points not
by the miignctic shell, but when is Jess th»D a, tliat is,
I on the muj^ctic shell,
o.'=M+4ir. (7)
If we assume 0, the centre of the circle, as the origin of co
ordinatea, we must put a = ~ , and the eerier become
VOL. II. X
^B VOL.
306 CTBCtrtAR CCRRSSTS. (696.
 = 2» j • + ;fiw+&o.+()'^y^^;'>ffp,..,,(tf)}. it)
where the orders of all tlic hnmiunics are odd *.
On lie Pofentiat Bnersg o/two GnuUtr Otrrenit.
096.] Let uii begin by «]p]HMnng tlic two miij^etic dicllx wirieb
nruoqiiivaleiit In tJie enrrents to be portions of two concentric spbem
their radii being e^ and c^, of which e, in the greater (VSg. 47).
Lei us also suppoKe tlmt the axes of tlw two shells coincide, and
that O] is the nng'le subtended ij
the radius of the firat shell, and n,
the angle subtended by the radius
of tbe second shell at the centre C.
Let w, be the potential due to tbe
first shell at any point tvilhtn it, thai
the work rcquindlooury thosocood
shell to an infinite distance is tiw
value of the surfnceintegn]
HcncB
4«*sLn»«,v5rPi'(''i)/'^('')'^'%+*«+^^(''i)Aw«'**,{'
or, snbMtiluting the viilne of the mt«ignils from equation (2), Art. 694,
J/=4«»m»a.sinS^,'ji^^A'(«.)Pi'^)+&e. + jj^^P/(aJP,'{«jj
* Tlia TtloA of ttie nlid tagia aubttadot by • dnlo nujr b« obtained ia a nan*
Fl«.47.
extended over the second sholl.
tuba
tWUMfallMIA—
• MBd
•agjlo nibtaadad by tha ebola ai tbe point 2 in tb* kxb b «Mlly 1
. I. I — CDMB,
bpwding tbi< tgcii«>ian Id ipliivlt^ buinanla, <t« ted
for th* «ii«uiniu or w fiir jwlnu no tbo iixl* for wbMi > ■« 1«» lli«n " or fn*— 1
tliui e tttftetiidj. TboB naalM eui sad/ b« ahewn to oounciil* witb Ibiao if
toik
i^B
POTBSTIAt OP TWO CTOCLBS.
M7
= 47l'illl*0
697.] Let Ofi next 8Uppc«e that tb« axis of on« of the sbelb itt
jburned about C as a centre, so tliat it now makefi an an^e 6 witb
be axis of the other shell (Fig;. 48). We have only to introduce
the zonal harmonics off into this expression for M, and we find foi
[the more ^aeral value of M,
This ix the value of the potential energy due to the mutual
Rction of two cireulur currents of unit strciig^.h, pluoi^d so that
'th« normalf through th« eontrvu of the i>ircU« meet in it point C
in «& angle 0, the di¥t«&ci» of the oircuinferencex of th« circle* from
the pciiil C heing Cj and e.^, of which f, ia the greater.
If any displacement (/xaitf'r* the value
^of M, then the force acting in the direo
H, „ rfjf
I
tion of the displacement is X =
tlx
For instance, if the axis of one of the
shells is froe to l»m about the point C,
m> as to cawHc tit vary, then the moment
jf the force t«n<liog to increase 6 is 0,
thtre dM
Performing the differentiation, and remembering that
rhcre P/ has the same signifioation as io the former equations,
I =  4ir" sin' o, sin* o, sin <i(* { J ^i>,'(ai,) ^/(oj) Pi'C^) + fee.
fi98.] As the values of P' occur frequently in these ealnilationn
'the following ttthlc of values of the first nix degrees may bt u>»ful.
In this table h stands for cos 0, and v for sin $,
/>,'= 3;*,
X a
CtBCDLAE CTOKENTS.
699.] It is BomeUtuea convenient to expreaa Uie series (or Jl ia
terms of linear quantities as follows :—
Let a be the radios of tbe smaller circuit, b Ute dittaoco of i
plane from the origin, and a = /a* + A*.
Let A, S, and V be the corres^wnding quantities for thv
circuit.
Tbe series for M may then be written,
Jf'
jitji
+ 2.3.w»~«»{(coe««isin*fl)
+ S.4.H' ^^^~*^^ a'{^ia')(cos*0Biii'g<
;^{»§— ^^'H
ir we cuke $=0, the two circles become pAiallel otui on
mae axis. To determine the attraction between tlieuj we ma/
diiJ^Tentiate M with respect to i. Wo thns find
700.] In calculating' the effect of a coil of rectangular aecti
WB have to inteffiate the expressions already found with res
to A, the radius of tHe coil, and It, the distance of its plane from
the origin, and to cxtond the integration over the breadth and
depth of the coil. ^B
Id some cases direct integration is the most convenient, boP^
there are others in which llie following method of approximation
leads to more useful results.
Let P be any function of « and y, and let it be roqmred to fi
the value of P where
In this expre«»ton F is the mean value otP witliin the limits
integration.
Let P^ be the value of P when x = and 5 = 0, tien, exiiandic
P by Taylor's IVorem,
Integrating this expreasioD between the limits, and dividing 1
result by xy, we obtain as the vatne of i^.
701.]
COIt OF BECTANOrXAH SECTIOS.
309
'Po+it{
.rf'n
+j,
• ^
)
tie' ■' d/
In the c«ec of the coil, liit the outer and inner radii he A + \ f ,
land ^— if respectively , and let the distance of the planes of the
I vrindings frum the ori^ii lie between Ji+\jj and j9— J >], then the
! IjTCfldth of th« coil is ij, nnd its depth (, these quantities being
[small compared with A or C.
In order to culculntc the magnetic eSVct of such a ooil we may
write the succcsnve terms of the scries as follows : — *
_B
e,= ^^{l+^.
".— g(A(A.4K...i^,.)'
/. , /2 25 35 JS , .
imsA'^
c*
')■
G, = 4^^^^^"^,^^^ + ^^{C*(9S'12A')+35A'£'(6A^4B')}
(P
24 C"
'ate.
24 C
+ ^l_.pj»C*(6a«44^) + 63J«£'(4£«J«)},
8ec.,&c.
The quantities <?„, (?,, (7^, &c. belong to the large coil. The
ralue of CM at points for which r is Icvs than C is
»=2ir+2C„CjrP,((J)C,^PB(fl)&c
The quautititw ffy^ffi' ^°* l>c]ong to the small coil. The value of
at' ut point* for which r \a greater than c is
Tlie potential of the one coil with respect to the other when Oie
total current through the section of each coil is unity ig
M = 0,y, P, W+ G.^a /•,{«)+&<!■
To find M iy EU'iptic IntegraU.
'01.} When the distance of the circumferences of the two circles
I is moderate as compared with the lodii of tlic unwller, the series
310
ClBCin.AB CCRREST3.
already ^ven do not omverge rapidly. In every case, boncri;r,l
may lind Uie value of 3f for two pwsUel cin:l«a by elliptic in1«^
For let 6 be tlie length of the line joining the ocntres of the 4
and let tliis line bo perpendiovW to the jiknee of the two
and let A and a be tho radii of the circles, tlien
the integration bein^ extended round both curves.
In this cuBo,
Aacm (^— *') ii<t> r/<fr'
where
_ 2V^
and /"and E are oomp1et« elliptic inte^nls to modultts r.
From thit, remembering that
and that c is a function of &, vre End
If r, and r^ denote the greatest and least valoes of r,
r,* = (J + «)* + «*, r," = (^«)« + i*.
and if an angle y be taken such tliat cos y = 1 1
where /"y and fy denote tho complete dliptic integnds of the ft
and second kind whose modulus i» sin y.
ltA = a, w>t> = ",and
^=2.eosyf2/;(l+s«»y>^).
The quantity ^ represent* the attraetion lielncen two panUlcl
ciraalar drcnita, the current in «acfa being utity.
ro..]
LISE9 OF MAQXETIC POfiCE.
On account of the importance of tlie qiuntity M in eteiHr»
la^Dttio calculatioDE tlio values of log(jI//lTrv'irt). which is n
ftmctioQ of c and tJiereforc of y only, have bwn tfthiilutH for
interralB o( 6' in th« vnliio of tho tn\g\e y between 60 and 90
fdegices. The tabi* will Ir' found in un appendix to this ehnpttr.
SicoHd Expratiion /or M.
An expreesion for 3f, which is Honictimce more convenient, is got
bv makiofr c, = J — ^ , in which case
To droK tie Lines of Magnetic Force for a Circular Current.
702,] Tho linM of mag^nctic force are evidently in planes passing
brough tbe axis of tho circle, and in each of these lines the value
'of M \» constant'.
Okleulato tho vaUio of £^ = t^ p — 7= from Lueendre'e
ablee for a saffioient number of values of Q.
Dmw rectangular axes of x and ; on the paper, and, witb centre
'at the point X = Ja (sin tf+ooaecfl), draw a circle with radius
\a {c<iiec$—(aQ(i). For all points of tJiis circle the value of Cj will
I be sin $. Uenoe, for all points of this circle,
I .,^.„ ATT I ._.. ._ 1 ^'i.
M =1 8iti/Aa ■
and J =
Now A is the value of x for which the value of M was found.
lenee, if we draw a line for which x = A,\X, will cut the drelo
two points having the given value of M. *
Giving M a, series of values in arithmetical pnjgression, the
TaltMB of A will be as a series of squares. ]>rawing thorefore n
series of lines parallt'l to !, fitr which x bus the values found for A,
the points where these lines cut the circle will be the points where
tbe corresponding lines of force cut the circle.
f * (Tla weatoA exprcaion fat J/ iDiy b« ilvduatd from tb« fimt by mMU of Uw
MuwIbk 'iHiiiP'Ptm*'""* In EU^dc IntegnU* : —
U
Vie*.
W'.
or e 
l*^
J'W(i*(i)^(<\>.
*w
l+«»
j(f,)aA>''(0.]
312
CIRCITLAB CCUBSKTS.
£;*>>
If we put m = 6sa, Jind ii = aw, then
^ = « = «»ir#a.
We may call n the index of tLe line of TorM.
The forme of thcw> lines nro (fiven in Fig. X^^II at Ihc end t^
this volume, Thiy arc copied from a dnning given bj Sir W.
Thomson in hi* iiajier on ' Vortex Motion *,'
708.] If the patition of a circle having a given axis is r^ardtil
a« defined by i, the distance of iU centre from a fixed point od
the axiii, and a, the radius of the circle, then M. the coeflSdm
of induction of the cirde with respect to any ^stein whaterer
of magneta or cnrrents, is subject to the following equation,
da^ "*" (16* " a da ~ ■ ^**
To piore this, let us consider the immbor of liooa of maguBlac
force cut by the circle when a or ft is made to Tsiy.
(I) Let a become a + la, i remaining constant. During this
variation the circle, in exjiariding, sweeps over an unnulur sorfiwe
in its own plane whose broiuUh it< ha.
If r is the magnetic potential at any point, and if the axis of f
he parallel to that of the circle, then the mognctic force perpen
dieular to the plane of the ring is ^ •
To find the mngoetic induction throngh tbc annular sarfaoe we
have to integrate p' , t?r .
Jt dy '
where ff is the angular position of a point on the ring.
Hut this quantity repi'csvnts the variation of M doe to the
variation of a, or ;— ia. Henoe
da .'(, djf
m
(2) Let h beoonu i+ti, while a remains constant. During thi«j
variation the cittl* BWWpf over a cylindrio sorface of ladius a andi
length it'
The magnetic force perpendicular to this snrGuK at any point iB<i
J— , where r i* tbc distance from the axis. Uenoe
DifTcrentiMinp equation (2) mtli reapcct to «, and ($) with
respect to b, we get
• JViM. &. S. Biiii. y<H. xxT. p, sir (1889).
704]
TWO PABALLEL CTRCLES.
da* J„ dy .'a
[WcntH
da*
d0.
rf'Jf
Jrify
313
(*)
(5)
(6)
Coegiwnf of Induction i^Two Parallel Cirelfi icien the J)i»l,iii£i itf
tm^en He Arcs ia tmall compared with the Uadiim </ eithtr Circle.
1704] We might deduce tlic value of M in this ciise from the
ex))anBion nf the elliptic integrals already given when their modulus
ia nearly unity. The following method, however, is a more direct
ap{)1icat.ion of electrical principles.
IFirtt Approximation.
Let a and af it be the radii of the circles and l the distance
between their plane):, then the shortest distence between tbeir
circumfcreuces is given by
p r = Vc" +i«.
"We have to find the magnetic induction through the one circle due
to a unit current in the other,
^ft We shall begin by eu]>[)OMing the two circles to be in one plane,
H^Coneider a small element hi of the circle whose radius is a + c. At
B a point in the plane of the cirole, <listAnt /> from the centre of ht,
B measured in u direction making an angle $ with the dircotiou of
it, the magnetic force i» perpendicular to the plane and equal to
1
I
.mn0U.
d$dp,
To c*lcwl*tc the snrfaee int(^ral of tht» force over the dpoee
which lies witliin the circle of radius a we must find the value
of the integral /•*• j^i sin
where r^, r^ are the roots of the equation
H2(a + f)Binflr+c» + 2a<?= 0,
tiz. »i «■ (ii+c)8infl+\/((i + c)*sin'^— tf*— 2a<',
^1= («+c)Bind— N/(fl + f)' sm0—<^—2ac,
. ,„ «^ + 2ac
CTIfCrLAB CUKBB5TS.
'When e n smalt compued to a wc may pot
r, — 2a 811) 9,
Intc^atiDg with regard bi p wc liavc
28*/ log( — sia'0J.an6d6= 2i«(lo(f— — 2), rnjarlj.
We thus find for the whol« induction
3/„ = 4«(log^2).
Since the magnetic force at any point, the fliBlantM of wliicb
from a curved wire iei small compared with the radius of curvature,
is Dearly the »ame as if the wire had been etmight, ne can calculate
iha dilferencc between the induction throng'k the circle vrhote
nuliufi ta a~e and the circle A by the formula
^f«AM^= Jirff {logclo^rj.
Hence we lind th« value of the induction between A and a to be 1
Mao = ina {log 8a— log r— 2)
approximately, provided r is small compared with a.
70£i.] Since tbo mutual induction between Iwo windings of till
same <,oil is a very important (juantity in the calculatinn of (.xl
porimental resuIU, I vhall now describe a method by which the
approximation to the value of M for this caae can be carried to any
required dCjErree of accum<^.
We shall assume that the value of M is of the form
M=iAA\osy+B\y
where A =
*•. ,'S^
,.«?•
and
+ &C.
5 = 2J + J?,«+ J?,"^ +J»j'^ + 5,^ +i?3'^ + 8te..
where a and a\as are the radii of the circles, and y the distance
between their phinca.
We hare to di*termino the value* of the ooeffioients A and li.
It is manifest that only even powers of ji ean occur in these quan
thiee, because, if the ngn of g n reveraed, the value of M nia»l
ntnain the same.
Wo get another *et of conditioru fVom the reciprocal property
of tlwj ooefEcient of induction, which remains the same wliichi'vir
circle we take as the primary circuit. The value of hi mufit then
1
705.] ISDirCTIOS DUE TO A CIBCDI.AK CURRBNT. 315
[fore renaiD the same wh«n ve sabstitute a+x for a, and — « for a
in the above npreedoii.
We thus fiml the following conditionx of rociprocity by cqiialing
[the (MMjfficwnU of similar coinbinatioDs of jd and y,
J, =— ^ —jf, 5, = i— i^,+^ — 5g — £,,
lrA,^A,+i«~3)A,+ ^'*^l^^'h , + &e. + A.,
From the ^neral cjiiatioQ of il/, Art. 703,
ne obtain another Bet of conditions,
iA^ + 2A'f + 6A, + 2A\=2A,i;
{«l){«_2K, + «(«l)J'„,+ 2.3^. + 2.3^',^, = (II2M/,
tc.;
4^,+ .1, = 2^3 +25', B, = 4^3,
6^,+ 3^= 2^,+ 65af2/rg= eJ'a + SJ'j.
= «(i.2)£,+(»+l)«7?,^,+ I.2^.+ I.2i?',,,.
Solving ilieso equations and eubstitating the values of the co
efBei«ntc, the s«rieB for .V becomes
k=^V'i»:^^^M
■iiva
m Sj?— y «*— Gayi*
4B(i^
• Sic.j.
" [Tliii molt may be obuboil dlrtctly bj the method rnggorttd in Art. 7M,
*{l. by tha apuudo'n* of tlii] »t1I>tk InlCKrali in tha eiiiiMubui fot Jf fuauri in
fcAit. 70t. SMO»yU7'a£'WiJ(ic/'uiwf(o>u, All, 75]
316
CIRCmAR CFRRENT8.
[706.
Jb find tie form of a coil for Kiiei ti« coefficient <^ «j^»
duetiott 14 a naximum, tie total Ict^ih and IHeknen ^ ikt
teire being given.
706.] Omitting' the corrcctionB of Art. 705, we find by Art. 67S
where n is the number of windings of the wire, a is the mcBn
radius of the eoil, and A is the geometrical mean distance of the
transverse section of the coil from itself. See Art, 690. If Uiii'
section is always similar to itself, R is proportional to its linear
dimensioits, and n varies as S^,
Siooe the total length of the wire is Zvan, a varies in'
as n. Hence
dn dR
i:=^R'
and we find the condition that L may ho a maxioiiun
If the transverse section of the coit is circular, of radins e, tlwD,
by Art, 692, R
and log — = V»
, da dR
and — = — 2 = >
a R
whuace ii = 3.22e;
or, the mi>an radiuii of the coil should be 3,22 times the radint of
the traDsvente section of the coil in order tluit HUch a coil may Iiave
the greatest coefficient of selfinduction. This nsult was found by
Gauss*.
If the cliunncl in which the coil ia wound has a square tnuisrerse
section, the mcim diameter of the coil should be 3.7 times the side
of the square »ectioD.
Wttht, QMtiiigMi MUtion, lee?, vol T. p, tn.
1
APPENDIX I.
^
HV Taiie iff the vaXutt ^
^l:^ ^'
_ fkt* Vftl \
^tn^Aa ' '■
60" <y
4»v5a
L« * .
Log " 
^*~VJ^
T 490 4 7 83
63" ac
T5963782
67' 0'
1.6927081
6'
1.5022651
86'
1599132!!
6'
1.6954642
ir
1.SOSOS05
42*
1.6018871
12'
1.6982209
18*
1.6078345
48'
16046408
18"
1.7009782
24'
1.5106173
54'
1.6073942
24'
1.7037362
SO*
1.5133989
64' 0'
1.6101472
30'
1.7064949
36'
1.5I6170I
C
1.6128998
36'
1.7092544
ir
T.5I80582
12'
1.6156522
42'
I.7120M6
4B'
T.52173G!
18'
16184042
48'
1.7147756
81'
T.5243128
24'
1G211560
64'
17175375
61" </
T.5272883
30'
1.6239076
68° 0'
1.7203003
ff
1.5300628
36'
1.6266589
6'
f.7230640
12'
1.5328361
42'
I.629410I
12'
1.7258286
le
T.5S56084
48'
1.6321612
18'
17285942
24'
T.S383796
54'
1G349121
24'
T.73 13609
30'
15111198
65" 0'
1.6376629
30'
1.7341287
^^H
36'
1.5130190
6'
1.6404137
36'
1.7368976
^^H
42'
T.516G8r2
12'
1.6431648
42'
17396C75
■
48'
1.5494545
18'
1.6459153
48'
17424387
■
84'
1.5S22209
24'
t. 6486660
54'
T7462II1
■
62' 0'
1.5549864
30*
1.6514169
69° 0'
17479848
C
1.8577510
36'
16541678
6*
17507597
^^1
12'
I.5G03I47
42'
T.6569189
12'
175353GI
H
18'
1.563277G
48*
1.6596701
18'
17563138
24'
15060398
54'
16624215
21'
1.7690929
^^1
30'
1.668801 1
66" 0'
1.6651732
30'
17618735
^^H
36'
f.57156t8
6'
1.6679250
36'
1.7646536
^^H
42'
15743217
12*
f. 67067 72
42'
T.7674392
^^1
48'
1.5770809
18'
16734296
48'
17702246
^^1
54'
15708394
24'
i6761824
54'
17730114
^^H
63= 0'
I582fl973
30'
1.6789356
70" 0'
1.7758000
^^1
6'
i.58.'i3546
36'
16816891
C
T7T85903
^^H
12'
15881113
42'
16844431
12'
T.7813823
^^1
18'
1.5908678
48'
i'687ig76
18'
i. 7841762
^^1
24'
1.5936231
64'
1.6899526
24'
17669720
a
J
1 318 ^^^H
APFBKDIX 1.
^
■
tl
Leg * .
L» *
I^B 7^^=^
^iwVH
TO" Si/
T7897696
75° C
19185141
79" SO*
■0576I3S
36'
17925692
6'
i9214613
36'
.MO9037
42'
1 7953709
12'
19244135
42'
0642054
48'
17981 745
18'
19273707
48'
■0675187
54'
18009803
21'
19303330
54'
•0708441
71" 0'
T803T883
30'
1!I333005
80' (/
•0741816
«'
18065983
36'
1.9362733
6'
•0775318
12'
T8094107
42'
19392515
\^
0808944
IS*
18122253
48'
19422352
W
0842702
84'
I 8 150423
54'
i 9452246
24'
■0876592
30'
18178017
76° 0'
19482196
SC
■0910619
36'
i 8206836
6'
19512205
36'
0944784
42'
1823A0S0
12'
19512272
43'
■0979091
48'
18263349
18'
i9572400
48'
■1013542
54'
18291645
24'
19602590
54'
1048142
72' 0'
18319967
30'
19632841
81' 0'
1082893
6'
18348316
36'
19663157
6'
■1117799
12'
T ■8370093
42'
19693537
12*
■1152863
18'
1810501HI
48'
19723983
18'
■1188089
24'
1<(1M3534
54'
19754497
24'
■1223(81
30'
18461998
77° 0'
19785079
30'
■1259043
3G'
18490493
6'
19815731
36'
■1294 778
42'
185190I8
12'
19846454
42'
■1330691
48'
1.85»7575
18'
198772*9
48*
■136C786
54'
I85761G4
21'
19908118
54'
■1403067
73= O'
18604785
30'
i9939062
82' 0'
1439539
6'
r'8633440
36'
19970082
6*
■1476207
12'
18662129
42'
•0001181
12'
■1513075
18'
18690852
43'
•0032359
18'
•1550149
24'
18719611
54'
•0063618
24'
■1587434
30'
18748406
78' 0'
0094959
30'
1624935
36'
18777237
6'
0126385
36*
■1663658
42'
18806106
12'
■0157896
42'
•1700609
48'
18835013
18'
•0189494
48
•1738794
54'
18863958
24'
0221181
64'
•1777219
74" 0'
18892943
80'
•0SG2959
83' 0*
•1815690
e'
18921939
36'
0284830
6'
•18S48I8
12'
t8951036
42'
0316794
12'
•1804001
18'
18980144
48'
0348835
18'
•1933455
24'
1 ■9009298
54'
0381014
24'
•1973184
30'
19038489
79* 0'
■0413273
30'
■2013197
36'
19067728
6'
0445633
36*
■2053502
42'
19097012
12'
•04 78098
42'
■3094108
48'
T9I2634I
18'
■0510668
48'
2135026
54'
19155717
24'
•0543347
54'
•21762S9
^^^^^1
^^^^^H
^^^^^H
H^^^HI^^^I
APPEKDIX I.
319
Lo. ^
1 ju.
Ug —
UVAa
"W ;=^
"°S r
84" 0'
•2217823
86" or
•313t(097
88° 0'
■4 38 S 4 20
6'
■2259728
6'
•3191093
6'
■4405341
12'
•2301083
12'
■324384S
12'
■4548004
18"
■2344600
18'
3297387
18'
4633880
24'
2387591
24'
■33S1762
24'
■472312T
SO'
■2430970
SC
•8407012
30'
■4816206
36'
•2474748
SC
■3463184
36'
4913595
42'
•2518940
4y
■3520327
42'
■G01S870
48'
■2S63fiei
48'
•3578495
48'
■6123738
64'
■2608626
54'
■3637749
54'
■5238079
85' 0'
•2654152
87" 0'
3698153
89° 0'
5360007
6'
•2700156
6'
•3759777
6'
5490969
12'
•274(1635
12'
•3822700
12'
•5632886
18'
■2783670
18'
•3887006
18'
•5788406
24'
■2841221
24'
•3362792
24'
5961320
30'
2889329
30'
4020162
SO'
•6157370
36'
■2938018
36'
•4089234
36'
■6385ft07
42'
•29873I2
42' 4160138
42'
•6663883
48'
■3037238
48' 4233022
48'
•7027765
54'
3087823
54' 4308053
54'
7586941
[APPENDIX ir.
In the very imiMrtant case of two circular coaxal coils Lord fiayleigli
has BUggeeted ia tlie uso of the foregoing tables a very coavenietit
fommla of appro simatiou. The formula, applicable to auj number of
vonables, occurs in ^Ir. Kerrificld's Report on Quadratures and Inter
polation to the British Association, 1880, aud is attributed to tlie lal«
Mr. H. J. Purkiss. lu the present instance the number of variables is
four.
Let n, n' be the uoinber of windings in tbe ooUb, ~
a, a' the radii of their oentzal windings.
b the distance between their oeutree.
2A, 2h' the radiftl breadths of the ooils.
2k, 2U the axial breadths.
Also let /(a, o', h) be the coefficient of mutual indnction for the central
windings. Then the coefficient of mutual induction of the two coils is
^f{a + h,a',b)\f{a~k,a',h)
+/(«, »'+A', h){f{a, a'^h", b)
+f{a,a',b + k)+f{a,a',bk)
+ f{a,a',b+k')+f(a,a',b^k')
J'
[APPENDIX ni.
Selfinduction of a eircutar coil of rectangutar aecHon,
If a <1(innt« the menti rmlius of u coil of n windiujts vrhnnc nxud
brnulth i« t> and radial brcndth is o, then the tclfintluctlon, m calculated
by meam of the eeiies of Art. 705, may be thrown into the form
L = lwn*{Xa + ~ + &c.).
where
K = log.8«2  ^ i\og,C ^)^ (log.ft~)
32' ' 96" '*"*' 96" ' 32' 96 A*
:H«dy+^<")iog.8''^6*+^'^4S{iog.ca
12'
i(3ft»+e^(lt*«^+^t«a^)
. * «'yi '37. 1 fi* , . 137.
41)0"
GO
120 P'
1 «*„ 117. 16',, . H7.
+ 240 ft»^'^" "60">2l0 ? tl^* 60>
2^0 (^•^'' + ^'^'4*> ('"« ^'^^~ W>
*24 6 c*12 ^ « 4' 120 « b
]
TOL. U.
CHAPTER XV.
BLECTROMAONETIO INSTRUMENTS.
6alvanomt;lert.
707.] A Galvasombter U ua instrument by mesiaH of nhich nn i
electric currt^nt is indiottvil or moiieureti by its muj>nctic nvtion.
When thv iustruni«nt U int«D(U<l to indic«tv the nctMUiieu of a
fix'tiU* current, it ix called a Sensitive Galvanometer.
When it !» intended to measure a current witli llie greatest
aecuracy in terms of standard units, it is called a Standard Galva
nometer,
All galvanometers are founded on the principle of Sdiweigger'a
}ktulliplier, in which the current is made to pass thmuf^ a wire,
which is ooiled so aa t« pass many times round an open space,
within wliicli a magnet is suspended, so as to produce within thi»
space an electromagnetic force, the intensity of which ie indicated
by the magnet.
In sensitive galvanometers the coil is bo arranged that
windings occupy the poritionB to which their infliwuce on
magnet is grcat(«t. They are therefore packed closely togvtli
in order to be near the magoot.
Standard galvanonictcre are constructed so that the dinH!R»ions
and relative positions of oil their fixed port* may be aocuratily
known, and that any small uncertainty about lliv podlion of the
moveable part« may introduce the nnalluit possible error into the
calculations. fl
In conntructing a sensitive galvanometer we aim at malring tho^
field or electrooiagiietic force in which the magnet is suspended oa
intense as possible. In designing a standard galvanometer wc
wish to make the field of elect romagnctie farc« near the magnefr^
as unifonn as {Kigsililo, and to know its exact tntaosity in tvr
of the Btrength of Uic current.
J
ated
..I
tJiafl
tlwrl
MBASUREMEKT OF TH2 COIU
323
I
On Stanttard GalvaHometers.
708.] In a standai'd giilvauonieter tlie etrenglh of th« current
1)Ss to be determinod from the force which it coerls oa the boa
led ma^et. Now the diHriljutioii of the mn^edsm within
Ae magnet, and the position of its centre when suspended, are not
capable of being determined with any great degree of accuracy.
Hence it is necessary that t!ie coil should be arranged so as to
produce a field of force which is very nearly uniform throughout
the whole space occupied by the magnet during iU possible motion.
The dimensions of the ooil muat therefore in general be much larger
than those of the magnet.
By a proper arrangement of several coils the field of force within
them may be made much more uniform than when one coil only
i» used, and the dimensions of the instrument may be thus reduced
and it« sensibility increased. The errors of the linear measurements,
however, introduce greater uncertainties into the values of tlie
electrical cotistants for small inatnimcnts than for large ones. It
is tbereforo bust to dot.erraine tlie dectricftl constants of smull
instrmnenta, not by direct mr'Osuremcnt of their dimensionM, but
by an electrical coinpnrisun vrith a large standard instrument, of
which the dimensions are more uci^urately known; see Art. 752.
In all standard galvanometers the coils are circular. The channel
in which the coil ia to be wound is carefully turned. Its breadth
tig. a.
is made equal to some multiple, n, of the diameter of the covered
WLie. A hole is bored in the aide of the channel where the wiro ia
T a
to enter, ood one tni of the corcrod wire is pnabed oat throngb
tbit iiole to form the inner oonnexioo of the coil. Tbe channel U
plaoedon s lathe, sad a wooden axis is listened to it ; see I^g. Of,
The end of a long ttring is nailed to the wooden axis at the mma
part of the circumfcrmoc as the entrance of the wire. The whole
is then turned round, and the wire it emoothljr and regularly laid
on tiw bottom of the channel till it is completely covered by m
windings. Doring ibis procen the siring has been woaad m time*
roond the wooden axis, and a nail id drivm into the string at the
Nth turn. Tlie wtudingit oT the Mtring xliould be kept exposed
so that tbey can miily l>e counted. Tlie ext^mul circumference
of the first layer of windings is then measured und a now layer
is begun, and so on till the proper nnmber of layers has been
wound on. 'Flie use of (be string is to count the number of
windingK. If for any reason we hare to unwind part of the coiL
the string is also unwound, so that we do not lose oar reckoning
of the octnal number of wiDdtngs of tlie ooiL The nails serve
to dialinguieh the nnmber of windings in each layer.
llie measure of the circumference of each layer furnishes a t««t
of the regularity of the winding, and ennblee ns to caleulate the
electrical constants of the coil. For if we take the arithmetic mean
of the circumforencee of the channel and of the outer layer, and
then add to this the circumferences of all the intermediate layeTS,
and divide the sum by the nuin1>er of layers, we shall obtain the
moan circumference, and from this we con deduce the mean radins
of the coil. The oircumference of each layer may be measured by
means of n steel tape, or twtter by means of a graduated wheel
which roils on the coil as the coil revolve* in the process of
winding. The value of the divisaonc of the tape or wheel must
be a«ccrtainod by comparison with a straight ttcale.
709.] The moment of the force with which a unit current in
the coil ftct« upon the suspended apparatus may be expreasod in
**» «"" ff,^, sia S+ G,9t sine P.'(0) +&c,
where the ooefBcients refer to the coil, and the coefficients g to
the suspended apparatus, B beiiig the nngle between the axis of
the coil and that of the suspended appaiatue ; see Aii. 700.
Wlien tlic suspended apparatus is a thin uniformly and longi
tudinally mognetixed bur nuignot of length 3 /and strength onity,
saspended by its middle,
TANOEST OALVANOMETEK.
325
The Tiiluos of the coefficient for a ma^et of length 2/ magnetized
b^in aoy other way arc smaller than when it is magnetized uni
pormly.
710.] When the apparatus is usod ae a tangent galvanometer,
Ithe cot) is fixed with its plane vertical and parallel to the direction
[of the earth's maffnetic force. The equation of equilibrium of the
^magnet is in this eaMC
My,ffcoBfl = my8in^{(?,^i+ffjj?,;i','{()) + &c.},
l^rhflTo wj7, is the magnetic moment of the magnet, Jl the horizontal
i>mpoaent of the terrestrial magnetic force, and y the strength
^of the current in the coil. When the length of the magnet is
■mall oompurcd with the radius of the coil the terms after the first
^bn G and g may he neglected, and wc find
y = 7T cot 0.
I
The angle usually measured is the deflexion, 8, of the magnet
which is the complement of 9, so that cot 6 = tan h.
The current is thus proportional to the tangent of the deviation,
and the Jnittninient is therefore called a Tangent Galviinometi'r.
Another mctliod is to make the whole apparatus moveable about
a vertical aii^ and to tum it till the magnet is in equilibrium with
its axis imnillcl to the plane of the coil. If the angle between the
,e of the coil and the miLgnctic meridian is 6, the equation of
brium i«
iBy,ffsin* = wy {G,y,f C,^,+&C.},
U . ,
^m Since the current is mea&ured by the sine of the deviation, the
^Binxtrument when used in this way is called a Sine Galvanometer.
^M The method of sines can be applied only when the current is
Bbo steady that we can regard it as constant daring tbo time of
adjusting tJie instrument and bringing the magnet to equi
librium.
^P 711.] We have next to consider the armngement of the coils
of a standard gal vn no meter.
The simplcjit form is that in which there is m single coil, and
lio magnet ia suspended at its centre
Let A \h> the menu radius of the ooil, ( its depth, ij its breadtll,
n the number of windiogn, the values of the coetEcients ar«
ELBCTROMAOyRTIC INSTRCllENTS.
<7.=
2w»
{..A^»5i'
becomes 1 — 3^=.
A'
0, = 0, &c.
Th« principal correction is that ansiaff from G^. The aeri«a_
The factor of correction nill differ most from unitrf when th«
magnet is imifonnlj magnetized and Trben d = 0. In this cose it
It vant»he!i when tail 0=2, or whvn tho de^
flexion 18 lan'i, or 20*34'. Some obstfrvers, therefore, arrange
their experiments bo as to make the ohwrvtHl deflexion as near
this an^le as jioasiblo. The hest method, however, is to use a
magnet so short compared with the radios of tho coil tlist
comction may be nlUigether neglected.
The suspended msg^net is carefully adjnsttd «o tliat its cent
shall coincide as nearly ss posiiilile with the centre of the ooil.
however, this adjustment is not perfect, and if the coordinates o!
1 ho centre of the ma^et ivlative to the centre of tlte ooil are x, y, i,
t being measured parallel to the axis of the coil, Uie factor of
correction is ( I + s '^ ) ■
"When the radiiiti of the coil is large, and the a<Ijui^tmcnt of tlii
magnet carel'iilly made, we may assume that thi» correction is
insemnble.
Gatiyain's Arrangement,
712.] In order to ^t lid of the correction depending on O^
Gangain con»tnictcd a galvanometer in which this term was ren
dered 7^ro by «u«iiending the miigmet', not at the centre of tlie
coil, but at a point on the axis at a disbinoc from the centre equal
to half the radius of the coil. Tlie form of G^ is
and, HJnce in this amngement S = iJ, G, = 0.
Tliis arrangement would W an improvement on the first form
if we could be sure that the centre of the suspended magovt ii
>1
I
s
I
■
I
GALVAKOHETER OP TWO COILS.
327
exactly at the point thus dotinotl. The position of the centre of the
ma^et, however, is alwa};! uncertain, aud this uncortuinty iHtro
duces a factor of correction of unknown amount dqtondin^ on G.^ and
of the form (l — J j), where z is the unknown excess of distance
I of the centre of the msgnet from the plane of the ooiL This
COTTcctioD depends on the tiret power of j . Hence Gaugain's coil
with ecccntriciitly EUgpcnded ningnct \% subject to for greater un
l^cerUinty than the old form.
HelinAoUz'a Arrangement.
713.] Helmboltz converted Gaugain's galvanometer into a trust*
vorthy instrument by placing a ^cond coil, equal to the first, at
an equal distance on the other side of the magnet.
By placing the coils symmetrically on both aides of the magnet
we get rid at once of all terms of even order.
Let A be the mean radius of either coil, the distance bctwcpn
their mean planes is made equal to A, and the ma^^nut is sui^penilud
at the middle point of their common axis. The cocfficicnte are
16,r«l, eS
G = 0.0612
ITJJ
3s/5d[«
(31 f ^36 A
G.= 0,
Cj= — 0.73728
where n denotes the number of windings in hoth coils together.
It appears from these results that if the section of the coils be
rectangular, the depth being f and the breadth rf, the value of
&,, as corrected for the 6nite size of the section, will be email, and
will vanish, if (^ ie to t}' as 36 to 31.
It is therefore quite unnecessary to attempt to wind the coila
! upon a conical surlitce, as has bovn done by some inetrumunt. makers,
for the eonditioDH may 1>e satisfied by coils of rectansuUr Mction,
which can he con«t.Tucted with far greater accuracy than ooiU
wound upon an obtuse cone.
The arrangement of the coila in Uelmhottz's double galvanometer
repreaented in l^g. 53, Art. 723.
328
BtBCTROMAONCTIO INSPrRCMRllTO.
C714.
The field of force due to th« double ooil i* irprcsonted in Bection
in Fig. XIX nt tlio Liid of tltis volume.
Galvanometer tf Four CoiU.
714.] By combining four coiU we mfty get rid of the co«fBd«nt»
0^, &,, and 6'^. For by any symmetricAl combinations
C, G.
s> "ai
we get rid of the coefficients of even orders. Let the four coils
be parallel cirtles belonging to the Esme sphere, iMrresponding'
to angles 6, 4>i ""^j ^^d it— ^.
Let the nnmber of windings on tbe first and fourth coil be «, H
and the number on the second and third pn. Then the condition
that Cj = for the combination gives
aein*^P,'(fl)+ji«an»0i's'(*) =0, (I)
and the condition that G^ = o gives ■
«sin>flf;(^)+;.f.8in**P/(^)= 0. (2)
Tutting sin'd = jf and ain*ip=f, (3)
and expressing /*' and P^' (Art. 698) in terms of these quantities,
the e<^uatious (l) and (2) become
8» 28«' + 21 a:^ + 8/.y28/)y + 21/^ = 0.
Taking twice (4) from (5), and dividing by 3, we get
Hence, from (4) and (t>),
x5xi ar'7x6
w
J> =
and we obtain
y»67y
P =
32 7*— 6
Both X and y are the squares of the stnea of angles and most
tbcrefbrc lie between and 1 . Henoe, either * is between and f ,
in which com y is between f and I, and p between 00 sod f,
or else x is between 3 and 1, iu which case y ia between and
f, and p betwoon and J{.
Galvanometer (f Tkrae CoUt.
715.] The most convenient arrangenHnt is that in which x =s
Two of the coils then coincide and form a great circle of the ^here
whose radius is C. The number of windings in this eoinpound
coil is 64. The other two coils form small eiroles of the sphere.
The radius of each of them is s/}C, The distance of either ul
4
OALTAirOSraTER OP THREE COIIS.
820
' tbem From the plane of the first is </• C. The number of wmdiuge
^OD each of these coils is i<i.
120
Tho Tftlue of ff, is — ^ .
c
This ammgemeat of coils is represented in Fig. 51.
Fig. GO.
Since in this threecoiled ^Ivanometer the first term nfler G,
[■which has a finite value is G, , a lar^ portion of tho sphere on
^whoso surface the coils lie forms a field of force sensibly uniform.
If we could wind the wire over the whole of a Hphcrioal surface,
^M dwcrihed in Art, 672, we should obtain a field of perfectly
tunifonn force. It is praetically impossible, however, to distribute
rtlie windiDgi on a spherical surface with sufficient accuracy, even
if saeh ai coil were not liable to the objection that it forms a closed
[ aurfacc, RO thnt its interior is inaccessible.
By puttinfr the middle coil out of the circuit, and making the
eurT?nt How in oppOEit« directions through the two side coils, we
obtain a faId of force whioh exerts a nearly uniform action in
tthe direction of the nxis on a mngnet or coil suspended within it,
with its axis coinciding with that of the coils ; see Art. 673. For
in tbie can all the ooeffieienU' of odd orders disappear, and since
Hence the exprcftnon for the majpietJc potential near the centre
of the coil becomes
I
BMCTROMAOSETIC nraTKHMENTS.
[71*
On He Proper Tkieknw of lli* Wire ^ a GalvanomeUr, tke Ezftmal
Besulanct being ffwm.
716.] Let the fonn of the channel in whioh the ^IvanometerJ
coil is to bo wound Le fiven, and let it be required to determiiM,
whittivT it ought to be lillcd with a long thin wire or with a sbor
tbi<rk wiro.
Let I bo the len^lli of the wire, y its radius, ^4^ the radius
of the wire when covcrod, p its speciBc resistance, g the vnloo of
G for unit of leugUi of thi, win', nnd r the part of the resistoooe
whioh is in(le]iendent of the gat van o meter.
The resistance of the gftlvunomctdr wire is
p I
It =
xy'
The volume of the coil is
The elect romagnetio force is y 0, where y is the strength of the
current and 6 = gl. ■
If £ is the eleotrotnotiTe force acting in the circuit whose
resistance k R + r, E = y{R+r).
The electromagnetic force due to this electromotive force is
^/
which we have to make a maximum by the variation of; aad /.
Inverting the fraction, we find that
is to be mode a minimum. Hence
If the volume of the coil remains coostani
= 0.
or
Eliminating dl and t/y, we obtain
P y + i '
9
T
Hence the thickness of the wire of the gs]Tanomot«r should be
such that the external resistance is to the misUuiw of the gal
viitiomcter coil as the diameter of the oovcrod wire to the diameter
of the wiio itMir.
QALVAKOUETBKS.
331
On Sensitive Galvanomeffrs,
717.] In Uie construction of a eensitive galvanometer tlie aim
of erery part of the amngeinent is to produce tho grftatest pnesible
deflexion of tho magiiet by means of a given siiiall electromotive
I force actiuff between the electrodes of the coil.
The current through the wire produces the greatest eflect when
it is placed as near as possible to tho auspended magnet. The
magnet, however, must be left free to iiscillnle, and therefore there
is a cerlaiu spnc<' which must be left empty tntliin the coil. This
dffinifs the tut«nial boundary of Uie coil.
^m Outside of this space Oiioh wiudiug must be placi'd so as to have
^^tlie greateHt possible effi'ct on tlve magnet. As the number of
^■windings incniises, the most advantageous positdoiiB become tilled
^^□p, so that at lust the increusiil resistance of it uew winding
diminishes tine effect of the curnnt iu the former windings more
[than the new winding itself adds to it. By making the onter
pindingd of thicker wire than the inner ones wo obtain the greateixt
letio eiTeet from a given electromotive foree.
718.] We shall euppoue that the windings of the galvanometer
are circles, the axis of the galvanometer jiassing through the centres
of these circles at right angles to their planes.
Let raiaO bo the radiuK of one of these circles, and rcosO the
distance of its centre from the centre of the galvanometerj (hen,
if / ia the length of a portion of wire coinciding with this circle,
y the current which flows in it, the
foree at the centre of the gal
"'Wn(MBet«r resolved in the direction of
the axis is gjn &
yt
If we write
H = *» sin 0,
0)
I
this expression becomes y ^ ■
Heuec, if a surface bo constructed
umtlor to those represented in section
is Fig. 51, whoso polur equation is
/^ = ii'naO, (2)
wberc r, is any constant, a given length
of W)r«? bint into the form of a circular
Kit 51.
arc wdl produce a greater magnetic
effect when it lies within this sar&oe than wh«a it lies oiit«vi)K \^.
332
ELECTEOMAOKETIC IHSTBCICBNTS.
It fallows (rom this that the outer surface of any la^r of wire
ought to have a conBtant vainc of ar, for if « is greater at one place
thitit another a portion of wire might be traosferml from the Stst
place to thu second, so as to increase the force at the centre of the
galranomctor.
The whole force due to the coil is yCr, where
6
fdl
(3J
the int(>gTation being extended oTer the whole length of the wire,
X being considered as a function of /.
719.] Let 1/ be the radius of the wire, its tniDsrerso »ecttoQ will
be isj^. Let p be the specillc resistance of the material of which
the wire is made referred to unit of volume, then tlie reustance ofa
length / is ~4r • an^ the whole resistance of the coil in
n
p fdl
w
wliere f is considered a function of i.
Let r* be the area of the (luadrilateral whose angW arc tlui'
BectJonfl of the axes of four neighbouring wires of the ooil by a
plane through th<^ axis, then T^l is the volume occupied in the ooil
by a length / of wire together with its insulating covering, and
including any vacant ajtace necessarUy left between the windingt
of the coil. Hence the whole volume of the coil is 
r=fT^di, (5)
where J' is considered a function of/.
But sinee the coil i« a iigure of revolution
r= 2it f J r^aaedrde, (6)
or, expressing r in terms of x, by equation (2),
Now 2ii / {aiifffidd ia a numerical quantity, call it^, tlwn
where V^ is the volame of tlie interior apace lefl for
magnet.
Let us now eonaidcr a layer of tho coil contained between
surfaces x and x^dx.
SEysmVB OALVATTOinCTEES.
383
The volume of this layer ia
[tvliore dl is the length of wire in this layer.
Thi^ giv<s us dl in tonus of dz. Substituting: this in equations
f(3) anil (4), we fmd ^^ ^^dn
dG^N^^,
(")
rhere dO and dR represent the portions of the mlucs of G and of
' due to this Uyer of the coil.
Now if M be the given electromotive forcCi
rliere r is the reeistanco of the external part of the oircsit, in*
ei)eDdent of the giLlvunometer, niid the force at the ceotie is
We have therefore to make p — at nuudmum, by properly ad
Justinf* the section of the wire in each layer. This also neccfisarily
tvolvee a variation of Y because ¥ depends on g.
Lot Go and li„ be the vidues of G and of fl + r when the given
flayer is excluded from thu calculation. Wc have then
g _ gp + rfg
Ri.r~ Ra + dR*
to make thia a maximum by the variation of the value of j' for
I given layer we mual have
(12)
I..4G
^.dR
Q
RTr
(IS)
g..
Since da is very small and ultimately vanishes, ■— will be
iibly, and ultimately exactly, the same whichever layer is ex
Eoludcd, and wo may therefore regard it as constant. We have
hereforc, by(10)and(ll),
(14)
constant
If the method of covering the wire and of winding it ti Mch
that the proportioD between the space occupied by the metal of
i
the wire boars the same proportion to the apoice between the wires
whclLer the wire is thick or thin, then
y d^ _.
and we must make botli jr and Y proportional to x, that is to nr,
the diameter of the wire io anjr lajrer must be proportional to the
linear dimension of that layer.
If the thiclcncss of the insulating oovenng is constant and equal
to {>, and if the wires are arraDg«d in si]uare order,
Y=2(jf + i>). (15)
and the condition is
^to:i) = co«rta»t (16)
Tn this case the diameter of the wire increases tnth the diameter
of the layer of which it ibrms part, bnt not in »o hiph a ratio.
If we adopt the first of these two li^^iothcaee, which will be nearly
true if the wire itwelf nearly lills up the whole epnve, tfaOD we may
put n = ax. y = ffy,
where a and p are constant numerical qnantities, and
lt=N^ '
tsaa the sise and form of th« frcti^
where a is a constant depending upon the Aae and form of the free
space left inside the coil.
Honco, if we malce the Ihidtness of the wire %'at7 in the nune
ratio as x, we oblain very little advantage by increa»in^ tlie ex
tenuil size of the coil after the external diDieusiona have beooow
a larffc multiple of tlie internal dimensiona,
720.] If inen««o of resKtanoe i> nnt roganlcd aa a defect,
when the external reaiatancie i* far gri>ater than that of the gaW
vanometer, or when our only object ia to produce a field of intense
force, we may make >/ and Y constant. We have then
= ;^1(«— ),
N
I
ise
where a ia a oonrtant depending on the vacant upace inside the^
coil. In this case the value of Q increases uniformly as th^f
dimensions of the coil are incnnued, to Uial tJiere is no limit t^^
the value of G except the labour and expense of making the coiL
«». Wj
0ALTAN0METRE9.
Oa Suspended CoiU.
721.] In ih« ordinary gnlvannmeter a suspended magnet v» Mted
on by a ftsed coil. Hut tithe ooil can be tuupended with sufilcieDt
delicacj, we may determine the action of the magnet, or of another
coil OD the suspended coil, by its deflexion from the position of
eqailibrium.
We cannot, however, introduce the electric current into the coil
unless there is metallic connexion between the electrodes of the
battery and those of the wire of the coil. This connexion may he
made in two different ways, by the Bidlar Suspension, and by wires
in q>posite directions.
T!Tie bifilar saspension has alresdy been described in Art. 459
as applied to magnets. The arrangement of the upper part of the
snspcnsiott is shewn in Fig. 54. When applied to coils, the two
fibreti are no longer of silk hut of metal, and since the torsion of
a metal wire capable of supporting the coil and tmnsmittioi; the
current is much greater than that of a silk fibre, it must be taken
Bpecially into iiccouat. This suspension has been brought to great
perfection in the iiiHtrumL'nts constnieted by M. Weber.
The other method of suspension is by means of a single wire
which is coiim'cttil to ouc extremity of the coil. The other cs
treniity of the coil is Cfinnectcd to another win: which is made
to bang down, in the sanie vi^rtical straight line with tbc first wire,
into a cup of mercury, as is ^hewn in ¥\g. 36, Art. 720. In certain
cases it. is convenient to fasten the extremities of the two win.'S to
piece* by which they may he tightly strtitchcd, care being taken
tliat the line ()f these tvires passes
Hitough the centre of gravify of the
coil. The apparatuii in this forcn
may be use<l when the axis is not
vertical ; see Pig. 52,
72^.] The suspended coil may be
used as an exceedingly sensitive gal
vanometer, for, by increasing the in
tensity of the miiirnetic force in the
field in which it hangs, the force due
to a feeble current in the coil may
be greatly increased without adding
[ to the mau of the coil. The mag^
nctio foroe for tliis purpose may be
produced by means of permanent magnets, or by electromagneta
Fy. &2.
S8(
ELECTROWAOITETTC IltSTBlTMESTO.
[7»;
excited by an auxiluMT cuircat, and it may be powerfully^ coDcen
trated on the stispeiulod coil bj meAiM of soft iion ansatures. Thus,
in Sir W. Thonnfon's ntcordiof^ nppamtuii, Fi^, B2, th« cnil is sus
pended betwotn the opposite polos of the cloctromagnvte A' mi S,
and in order to conccntr»t« the lineH of mngnctie fono on the ver
tioal sides of thi> coil, n piece of Bofl iron, D, t» IiximI brtnreen the
poles of the magnets. This iron becoming inii<^etizc<(l by iodao
tion, prodnccs » verj powerfiil fiold of force, in the inlerrals between
it and the two magnettt, throug^i which the vertical sides of tJie
coil are free to move, so tliat the coil, even when the current
through it is vety feeble, is acted on by n eonsidenible force
tending to turn it about its rertioal aicia.
723,] Another application of the suspended coil is to dctcnnine,
by comparison with a tangent galvanometer, (he horizontal coi
poni>nt of terrestrial magnetism.
The eoil is fiuepcaded eo that it is in stable eqailihriiim whi
its plane is parallel to the magnetic meridiaD. A current y is
posted through the coil and causes it to he deflected into a neir
porition of equilibrium, making an angle with the magnetic
meridian. If the suspension is bifitar, the moment of tho couple
which produces this deflexion is FeinO, and this must lie vqual to
HygKOSiO, where //is the horizontal component of terreKtrial ma^
Detism, y is the current in the coil, and g is the sum of the areas of
ati the windings of the coil. Hence
/7y = ^tantf.
3
If A is the moment of inertia of the coil about ita axis of sua
pension, and T the time of a single vibration.
J
and we obtain
^y = 5^tantf.
If the same current passes through the coil of a tangent gal'
meter, and deflects the magnet through an angle 4^
where CistheprincijMdcoDstantof the tangentgalTanom6tcr,JVrt.710,
From these two equations we obtain
as of
4
I
B^
tV
AGtaii0
V /a tan tan ^
=tV — 0^ —
f tan^
This method wbs given by P. Kobbauscb *.
E1.ECTR0DTNAM0METER.
ddt
t,] Sir William Thomiton has conrtmclixl a singIt instrnm^nt
hy meaQS wf wliieh (In; oliKtrviiUon* roqiiiretl t* (khrmiiie // and y
he taadc ftiinulhtnioimly hy th« wm# observer,
ooil is jiuii]H;iKlo(t BO ni! to W in <^iiilibrium with its [>iiin«
the ntagnetjc mcriiliMn, and is dcfl«ct«d from this position
the current IIowb tlirougli it. A wry nmttll mngiRt is sup*
at the c«ntrv of the wil, and in deflated by the cnrrent in
the direction ftppoaite t« tliat of the dcHexion of thu coil. Let th«
deflt^xion or tho eoil be 0, and that of the toni^net 0, then tlie
jnnablt! jiiirt of the energy of tho system is
//yysinfl + my GBin(tf— ^)— i/mco8i^— /'costf,
Biflereutiating with respect to and ip, we obtain the equa
Wona of eqniUbnum of the ooil and of the muf^et respectively,
Uy$eoit$ + myGco8(0—<f)+FanO = 0,
—myG <:oe($—4>) + Ilm sia^ = 0.
From th(«e equations we find, by eliminating ZT or y. a quadretie
^uatioD &om which y or II mny be found. If m, the ma^ctie
noment of the suspendod magnet, is very unall, we obtain the
followinff appTOsimate values.
y = YWt
^eosUsin^
— .ifeiu^ein^
,i
9
M Rin
coa0
I"
(i^ COB fl cos (9—0) ' g ciwtf
In these expressions G and g are thi> principal elcctiic contitJintM
of the coil. A its moment of inertia, T its time of vibration, m the
mBgnetiG moment of the magnet, // the intensity of the horizontal
ma^etic force, y the streugftli of the etirrcnt, B the dellcxion of ihe
coil, and ^ that of the mn^net.
Since the deflexion of the coil is in the opposite direction Ui the
deflesion of the magnet, thcBc values of II and y will uUvays be
I.
Weber's SieefroitytaiKometfr.
7S5 ] In this instmroent a small coil is anspended by two wires
within a krger coil which i» fixed. ^Vhen a cnrrent is made to
flow ihrmigh both coils, the suspended coil tends to place itwlf
parallel to tlie fixed coil. This tendency ia counteracted by the
mumcnt of the forces itrittin); from the bifilar suspension, and it, is
also atfocted by the nelion of terreBtriul ma^etism on the «u«<
^iwnded coil.
VOL. II. Z
BLECTSOMAOSETIC I^reTRtTMKSTS.
[73S.
In the ordiasry uso of the invtrarnvnt tiic i>Ianos of the two oolh
aro nrarl)' nt right angles to vnch other, so tliat tlic madial action
of the currents in the voiU may bo ns grvat ii< puwfibU, aod thr
pldoe of th« (UEpendod coil is ncnrl)* »t right nnglm to tlic nui<picUc
Di«ridiui, so ttiut the action of tcrroxlriid tua^ctinn may be t»
small as jKJBsible.
Lbt the magnetic oximutli of the piano of the fixed coil ba a,
and let the angle nhicli the axie of tfao snspoDcled coil makes with
the plane of the fixcul cuil be 0+fi, where is the value of thi
anijiio whon the coil is in ivjuililirium and no current ie flonnnj
and is the deflexion due to the current. Tlic <^()^4ltioa of «q
Ubrium is
''yyty!<'oe(tf+^)//?Vii«in{tf+j9+ft)/'*intf = 0.
Let us suppose that the ioHtrumcnt is adjusted so that a uui
are both very small, and that Jf^fi is utaail compared with
Wn have in this case, approximately,
If the deflexions when the >«igns of y, and y, are changvd
as follows, 0^ when y, is + and y, + ,
"i " ^ It ^1
^» .. + » — »
*4 " — t» +•
then we find
f
If it is the same current which flows through both coita we may put
y, y, = y'. »nd thus obtain the value of y.
When the currents are not very constant it is best to adopt thb
method, which is called the Method of Tangents.
If the eurrmt* arc so constant that we can adjust ff, the utg]
of the torsioDticad of the instroment, we may get rid of thi
correction for terrestrial magnetism at once by the method of siimb.
In thiit method /3 is adjusted till the deflexion is zero, so tlist
If the signs of y, and y^ are indicated by the suffixes of fi
be fort!,
/"sin^, m ~F»ia^ = — Gffy^y^^ 11 gy^ sin «,
i*sinj3, = — /'«n^4» — Gy)r\yj,— //jry^sino.
Jf
and ri/t = j(i^(Binj3,+BinftBin ftsin ft).
1
■i*ias^'
Fk. f.8
7. a
340
KSCVBOMAOyETIO IHSTRtTUBSTS.
Thia is th<; method nduptcd by Mr. Ij»tiinrr Ckrk in his nw
of the iustniiiiisnt constructed by the El«»trioal Committal! tif
British As»<M.ititi»n. We nre )D<lebLed to Mr. Clark for the drawin
of th« rlecti'odynamftmottr in Figure 53, ia which II«Imbultx
nrrxngvmcDt uf two coiU is a(lo[)te(l both for the fixed and for U:
Kitspcndt'd inil* The torsion^hnad of the iuBtrumetit, by which
thu bifilur «u»ik'iision is atljueted, ia represented in Fi^. S4. T1i»
Fig.6».
iHfiiality of thfi tension of the tiispension wires ts ensured by their
beang atlndied to the extremities of n sitk thread which fwmx ov«r
a wheel, and tht'ir distitnco is reguliiti4 by two guide>whecls. which
con be sot At the proper distsnco. The sti^x^nded coil can Ix! oiovitl
Tertically by means of n screw iieting on the suspensionwhtcl,
and horizontiklly in two directiona by the sliding pic<vs sliewn
the bottom of t'ig. S4. It is adjusted in azimuth by menns of III
torsion screw, which t»ma the oniionh»id round a vertical nxi
(see Art. 159). Tlie aximuth of the suspended coil i* aseertaimil
by oboervinj^ the reflexion of a scale in Uie mirror, Hlietvn just
beneath the axis of the siiiipindcd coll.
■ Ib ttw K^tnl inalniinmt, th« virM Aumr'ng Ihr carnal to ■nil fhini tka i
an not ^nad oat t* di>[<l»t4 in the Hfure. bol arr kipi n* cIme logniiet «■ ;
liUe. ■> M to utatntloo «m1i uUior'* ckulroBupictH: kctina.
CI,
i
I
CITBBBNTWEIOHIE.
341
The intrtninwnt originntly cinwtTOctfd by Weber is described in
his Eteitrodjfmimivri'' MtiftheithnmuufW' It wne intended for the
mcMurcnwnt of smaU oiirrent«, and Uienforv both the fixi<d and
the suspended ooiIk i:'oneixt4Hl of many windings, and the enspended
coil occupid] II UrgiT juirt of the ^ystv*: within the fiscd coil than in
xXk intttniniint nt \\\f IlritiNh Assoeiut.i(>D, w)iich wiis primarily in
tcndt^l a* a Mtandiird inslnimeiit, with which mnrv ecinisilive inxtru
metitM might be coniparE>d. 'Ilie exjierim«nt8 which he mode with
it rumixh the most coniplcf* exxirimpntal proof of the acoiirafy of
Ainp^re'n formulu ks^ a>{>It<d to cltiacd currents, and form an im
portant part of th« rettearehea by which We)>er has raiitixl the
iiimit?riciJ determination of electrical quantities to a very high rank
as regards precision.
Weber's form of the electrodynamometer, in whioh one coil i*
suspended within another, and is acted on by a couple tending
to turn it almut a vertical axis, is probably the best tttted for
absolute measurements. A method of calculating the constants of
sach an arrangement ia giien in Art. 6'J7,
726.] If, however, we wish, by means of ■ feeble current, to
produce a ooosideiable electromagnetic force, it is better to placo
the snsjiended coil parallel to the fixed ooil, and to make it cnpabtc
of motion to or from it.
The siisiended coil in I>r. Joule's
current weigher. Fig. S.'S, is horizontal,
and eapabto of vertical motion, »nd the
force between it and the fixed coil is
estimated by the weight which must
be added to or removed from the coil
in order to bring it to ibe snmc ri'lative
position nith rBsM>ct to the fixed coil
that it has when no current jiiuescs.
The snspendod coil may alxo Im
to tJie extremity of the hori
Sontol arm of a torsionbalance, and
nuiy W pUoed Wtwcin two fixed coils, one of which nttiacls it,
while the other re[ieU it, as in Fig. 3tf.
By nrraiiging the wnla as described in Art. 729, the force acting
on tlie K\ii>t>ended coil may be made nearly uniform within a small
distance of the position of equilihrium.
Another coil may he fixed to the other extremity of the arm
of the toTsioDbslancG and placeil between two fixed coils. If tbtt
Flt. S6,
ELECTSOVAQirBTtC ISSTItl'MBSW.
two fa«peiM)e<l coils are BuniUr, but wilU the oorrent flovriti^ in
oppofite directioiu, the effect of temxtrial raa^etum on the
FifrM.
pocition of tlw ami of th« lomionbalance will be coin]>letelj'
sliminaliHl,
727.] If tho »u»pcnd«d cml b in tlie shape of a lonff solenoid,
and IK uipiilile vf nioring jnrallf^l to its axia, so as to jiniis into
tlie interior of a lai^r lixed solenoid having: tl>^ same asiit, then,
if the cun*nt is in tiie same direction in l)oth eolenoid*, Uie ww.
[tended solenoid will be siickeil into the Bxrd one by a force nhicli
will be nearly uniform as long as none of the extremititw of the
Bolenoide are near one another.
728.] To produce a uniform lor^ritwlinal force on a Hmall coil
placed between two equal ooiU of much lurgor dimcnsioua, we
should mnko the mtio of the diameter of the 1ar^ coil» to Oie dis
tauuG between their plantti that of 2 to V^3. If no Kcud the eame
currmt througih these coJU in oppoeito directions, then, in the ex
prcwion for m, the terms involving odd powers of r dis^pear, and
«incD kin' a = 4 and eon' a = i, the temi involving r* disappnrt
nlw, and we have n* the variable ]>art of u
which indicateH n nearly uniform force on a email siifpeuded coil
The armnf^Rient of the coils in this case is that uf Ihtt two outer
coils in the gulviinometor with three coils, described at Art. Tlfi.
See l'1{f. at}.
729]
SUSPENDED COILS.
343
7S9.] If we wish to BUspend a coil between two coiU pla<^
6o near it tlmt llie distance between the mutuatly xcting u ires U
» email ooTnjiiired with the radius of the ooils, the moat unirorru ionn
is obtained by tnukiii^ the radius of either of the outer coil.i exceed
;hat of the middle one by = of the distance between the planes
v3
of the middle and oot«r coils. This followg from the expression
(proved in Art. 704 for the mutual induction bctwvcn two circular
I currents.
So mnny of the m«i6nromont« of ol«!tri«iI qtuntitin '
ilepend on obeiTvationi) of t)ii! motiou of a vibrntin^ body IJiat we ,
shall devote some atk'Dtion to the nature of this tDOtion, and tb*^ j
best nn'tliods of obeorviny it.
The small oscillationH of a body iibout a position of irtable «qm
Hbriiiiu are, iu gcnrml, Himilar to those of a jioint act«d on by
II forc^e varying dincily lu tb<^ ditttaooe from a fixed point. In
the «iu« of tbo ribrating' Iiodies id our experiments tli<re iR alao
a reitiHtnoce to tlie motion, depending on a variety of causes, suoh
a» the viccotiity of the air, and that of the sufpcnaion fibre. In
many electriejil intitriimeutH there is another eause of reatstance,
namely, the refiex action of eurrents induced in condueting oircuita
placed near vibrating magnets. These currents are indoced by ttie
motion of the magnet, and their action on the magnet is, by the
law of Lenz, invariably opposed to its motioD. This is in many
eases tho principal part of the resistance.
A metallic circuit, called a Damper, is sometimes phiecd dc
a magnet for the eipress purpose of damping or deadening it
vibrations. Wo tthall theroforu speak of this kind of resis
as Damping,
In the COM of alow ribrnttOD*, such as can bo ca»ily obMrvedJ
the whole reiiistaace, from whatever causes it may nriec, appears]
to be proportional to the velocity. It is only when the velocity]
is much greater than in the ordinary vibrations of clectronuigneti'
instruments that wo have evidence of a rcsiHtance pro[>ortional U
the square of the velocity,
W« have therefore to investigate the motion of a body tmhjr
to an attraction varjing as thi! dixtanee, and to a tvuslance nryiiij
as the velocity.
I
MOnOS IK A LOOAniTHMlC SPIBAU
845
731.] Tbe following application, by IVifes^or Tnit •, of th«
principle of the Ilodojrrapii, eoalilps us to invcati^t« thi* Iciud
of motion in a very simple manner by means of the e<juiangular
spiral.
Let it be required to End tbe acceleration of a particle which
ilcEcribce » logarithmic or equiangular spiral with uniform angular
Telocity u about tJie jKile.
The property of this spiral is, that the tangent PT makes with
the mdiug vector PS a constant angle a.
If V IS the Teloci^ at the point P, then
t> . sin a = 09 . SP.
Henoe, if we draw SP' parnllcl to PT ani oqual to SP, Hie velocity
at P will be given both in magnitude and direction by
v=
aina
SP'.
^Hence P* will be a point in the bodograph. But Sf is SP turno)
tbrongh a constant angle n — a, so Unit the hodograph described
[Ijf P" is the same as the original spiral turned about its pole through
[an angle t— a.
The acceleration of P is represented in magnitude and direction
by the v«loeity of P" multiplied by the name funior, . — •
UcDCC, if we porfonn on SP" the Bome <^cration of turning it
• Pfoc S. 8. ^tiL, Da«. Ifl, 1SB7.
846
ELECTKOJIAOSFTIO OBSERTATIOSS.
[733^
\
Uirou^li an vmgh «— a into tbc poffltmn SP^, the aoceleraiioo otP
will hn c<uul ID inuj;nitud« itii<l (HrvctiuQ to
Bin* a
SP^.
where SP" IB etjuttl to SP turned tbroof^h nn wngle 2ir— 2(1.
If wo dnw PF cqtui nnd parallel to SP", the occvlerntion will W
w
i'/ whi«h we may n^olve into
Sin' a
PS and —. PK.
siu'a
The Tint of tbcM coinpon«nt8 is a central force towards 8 ptv
portionnl to t)ic iliitlanoo.
The KCvoiii] 16 in a diroction opponte to the velocity, and einoe
sin a GOB a
u
PA" = 2oOBaP'S=— 2
this force may \tv wriH«n
Hina
The acceleration of tlit particle is lliorefote compounded of tw
partf, the tirnl uf ivhicli J* iin uttniclive force (ir, directed towards ^
and proH>rtionul Iri the distance, and the cceond is —2kv, a resist*
anee to the motion proportionul to the velocity, where
ui* , , co*«
11 = . . , and * = to r^ •
Bm*a BIO a
If in these cxjiressionB we make = 41 the othit heoomes a ciicls^
and we have ^ = u^, and k = <i.
fienoe, if the law of attraction renuuns the same, ft e ja,, and
U ^ Wq SID o,
or the angular velocity in diflerent spiraU with the anme law of
nllntction ie proportional to the sine of the angle of the spiral.
732.] If wc now consider the mutioa of a point which la Uw
projection of the moving point P on the horiaonia) line XY, we]
itliiill find that its distance from f and its velocity are the hnriKODtel]
compunentB of those of P. Hence the acceleration of this point is
ttlxo an attraction tonnrds S, equal to n timee its distance ttma $A
lo^tboT with a retardation equal to 2k times iti< vdiicity.
We have therefore a complete construction for the recttline
motion of n point, snhject to an attrnetion praportjona] to
dittance from a hiced point, and to a re«i«taneo proportional l>i
the velocity. The motion of such a point is tnmply thu horixont
7551
SCALE BEAniKCS.
347
]inrt of the ]noti<Mi of another point which moves with unifonn
angular velocity in a logarithmic 8]>iral.
»733.] The filiation of tbe spiral ia
To <I«t«r[nin<; the horizontal motion, wc ]>nt
ft ^ ss wf, a: = a + rem^,
where a is the value of a? for the point of equilibrium.
If we draw BSJ) making an nnfjlc a with the Vertical, then the
tangents SX, 3)T, GZ, &c. will be vertical, and X, Y, Z, &c. will
\>e the extremities of suceet»ive oEcillationfi,
1734.] The observations which are made on vibrutin;; bodicM are —
(I) Tlie neat e reading at tlio stationary points. Tlieee arc culled
Elongations.
(2) The time of passinf* a dcRnito division of the ecnli! in the
positive or ncgjitive dircetion.
(3j The scaiereitding at certain definite timeB. Observationa of
this kind are not often made except inthe case of vibratious
of long period *.
The quantities which we have to determine are —
(1) The scalereading at the position of erpiilihrium.
(2) The logarithmic decrement of the ^ibrations.
(3) The time of vibration.
»To deiemine tie Seadiiig af Ike Potillon of E^itiiiirium frvm
Three Comecuthe Elongal'wnK.
735.] Let it,,Z2, *■, bo the observed scale readings, eonvqiODdin^
to the elongations X. }', Z, iind let a be the reading at the ponition
,gf equilibrium, S, and let r, be the raluc of S£,
^k jTj— IT = r, sina,
HProm tl
"when .
thcce values nc find
whenee
a = J'l'^II^A
'When X, does not ditTer ranch from f, we may ase as an ap
Bioximatv formula
• Se« Gmh, SemlUUt du MagnetltehtM Ytrttai, 1830. II.
ELECTBOMAOITETie OBSBltVATIOSS.
7'o ileferniiu lAe Logtirilkmic SttrtmviU.
736.] Tlic lognritlini of the rgtio of the amplitude of .a vibiatiwi ]
to tliut of tli« nixt follottin> is calloJ th« Logarithmic DecremeDt.
If we write p for tliis rutio,
P =
*i«t
^ = %»P. A = log, p.
Z ie called the common lo^rithniic docremeat, And A the Napienu
lo^ritbmic decrement. It is innnifcal that
\ = L log, ] = n cot a.
Hencfl
a = oot~'  >
wliicli <lc1ermin»i the angle of the lo^rtthmic spiral.
In mnkin^; a spodiil dote rmi nation of A wc allow the body to
perform a consideralilc niimlHT of vibrations. If c, is the amplitodp
of the first, and '■„ that of the «"■ vibration,
^'.i^oy
If vre suppose the accuracy of observation to be tlic sunic li^ '
small vibrations as for large ones, then, to obtain the best value
of A, we should allow the vibrations to subside till lh« ratio of c^toj
e, becomes tnost Dearly equal to i, the base of ibc Napi«
logarithms. Tlii.i gives n the ncarent whole number to  t I .
Since, however, in most csks time ik vatoabV, it is bi^irt. to take
the second Hit of obncrvalions bufore the iliiiiinutiou of amplitud^j
bo* proceeded so far. H
737.] In ocrluiii ni^es wo may havo to determine the poettion'
of eqnilibriiim from two eoii«evutive elotigatioiis, the logarithmic
decrement being known from « special e>Li>criment. We have tJicn
l^mf of FU/ralion,
738l] Having detcrminMl the ecaloreadiog of the point of cqui
librium, n coni^iicitous mark ix pbiec<l at that point of the suli
or as near it as powiblc, and thv limrN of the piuuigt* of llua
are noted for several successive vibrations.
Let na suppoae that the mark in at an iitikuown bat very
distance f on the tiaiitive n itli of tJie point of equilibrium, aai
Of TrBBATIOM;
1^ JN the obwrvcd tJin« or the first tnneit of the mark in tb« positive
I direction, ami f., t^. Sic. the time;; of the followiufr tniDsita.
If T be the time "f ribrotton, iin<l /'(, P.^, P^, &c. the times of
I transit of the true point of rquilibrinm.
ai
»here r,, t',, &o, are the successive velocities of transit, which we
ay suppose uniform for tho very small distance x.
If p is the ratio of th« amplitude of a vibration to th« next in
sticccseion.
»u
I
^— r,
]•
and
^B Tlie liiue of thv eccund ])aes3^> of the true point of equilibTium ia
^^ Three transits are suSicient to determine these three qimnlitieei,
b«t •ny preat«r niiniber may bo combined by the mcthoil of least
(Kjuares. Tlius, for five transits,
Tlie lime of the third transit is,
739.] The anme method may be extended to a series of any
number of vibrations. If the vibrations are so rapid that the time
of every transit cannot be recorded, we may record the time of
every third or every fifth transit, taking care that the directions
of suceetisiye trantdte are opposite. If tho vibrations continue
re;tutar for a lougr time, we need not observe during the whole
time. We may beofin by observing a snllicicnt number of tran»it.t
to determine approximately the period of vibntion. T, and the time
of llie middle transit, /', noting whether this traiinit ih iu tlie
positive or the n^^attve direction. Wo may then either go on
countinir the vibrations without recording the 1itn<'* of transit,
or we may leave the ap]>arat«s uiinittelicd. \\c iheii oliser^'e a
If three transitd are observed at tiroes fj, ^, /,, we find
The period of vibration is therefoie
je oft
^ = lV2'» + '4'.2MA('i2^ + 2',2'.+'J~(2fjTs)
^'
S30
UCROHJlGSTnC OBBETlTTtmL
[74a
mtnaA tarn* at ttamti, sad dedoee the Hate of ribTatioB T ai
Uw ttBe gf anddk tna«t l". Botu^ the iinetion of this tnneil.
If T sad V, the pgriodi of vibtatioa aa dedncid fnim (lie l«i>
■eU of obamBtinfM, are nckriy e^oal, «e m^ proceed to n nun
acmralc drlcrmtBadoD of the petiod b^ cMabining' the two sebn
of obMmtioiuL
Piriding P'—P bjr f . the qnotint oc^t to be vetj ataiij
ut iDt«ger, even or odd according as the tnuxita P and P' are
in tb«! tuae or in opponte dirrctMAS. If thi« i* not tbe case, tbe
iieneii of olwervationa is wortUesi, bat if tbe niatilt ia very aearij
a whole narober «, we divide f—P hj a, and thai Gnd the wko,
\alae of T for lh« nholt? time of swinging.
740.] Th« time of vibration T thna fimnd is the actual m«aB
time of vibration, and is sabject to oorreetionii if Wf wish to dMlore
from it the time of vibration id infinitet; amaU arcs and withnit
damiin;jf.
To reduce the obaerred time to the time in infinitclj email am,
we observe that the time of a vibration of lunplitude c is in gti>cnl
of the form r=y,(l+«c*),
where « is a ooofficient, which, in tbe ciseof the <»n]inaiT pendolu
is ^i Now the amplitudes of the Ruocessive vibimtioDS are rJ
Sfi
~\
cp;.
.cf^', so that the whole time t^jn vibratioDs is
where Ti* the Umt* dedaoisl from the obMrvations.
Hf^nvc, to (ind the time Ti io infinitely small ores, we
approximately,
To lind the time T„ when there is no damping, we have
7; mT,ana
741.] TliP pciiation of the rectilinear motion of a body, attrncle^H
to a fixed jioint and risiftted by a force varying u tho velocity, ^^
wbrrp T is till coordinatf of tUp body at tlwj time /, and a u thq
eoonliuute ul' the point of etiiilibriuni.
74»J
nBTLEXIOir OP THE OALVANOMETini.
I
I
To solve this i<quation, leb
w—a = tf*'y;
the solution of wbich is
^then
(2)
(3)
y = CcfM{ •/at' ~ifl+a), wheni is less than w; (i)
y = A^ £/, wlicn i is ecjunl lo w ; (5)
and J = C'ci)s^(s/^— fci*/ + o'), when £ is greater than u. (6)
The vmluc of « nwy lie ohtoinetl from that of ^ by equation (2).
Whcji k i$ li'!W than to, the motion consists of an infinite series of
o^Ilutionit, of oonstunt [H^riodic time, hut of continually dwreasin^
am[ilitudit. As A increases, the iwriodic time tK'comes longer, and
the diminution of amplitude becom«it more rapid.
When i (half the coefficient of resiNtaiiie) hwomes equal to or
jireater than u, (the square root of the acoeknttiou at unit ilistftnco
from the point of equilibrium,) the motion ceases to be oscillatory,
and during the whole motion the body can only once pass through
the point of equilibrium, after >vhi<;h it nachetf apoxitioii of greut^wt
donation, and then returns towards the point of c(uilibntim, con>
tinoally »pproacbin^, but never rea^^hin^ it.
Galvanometers in vhieh the resistance it) so groat that the motion
IB of this kind arc called deatt deaf gatvanometerH. They are useful
tin many experiments, but especially in telegraphic signalling, in
which the exiatcnco of free vibrations would quite disguij^e the
movements which are meant to be observed.
H Whatever be the values of i and w, the value of a, the scale
' reading at the point of equilibrium, may be deduced from five scale
readings, j>, q, r, t, t, taken at equal intervals of time, by the formula
fc q(r'i—qf) + ripl—r')+»(qr~fi>)
■ net>
■ to t
H eurr
Port
Ok tie OiservalioH of the Galntnomelir.
742.] To mi°a»ure a constant current with the tangent galvano
meter, the instrument is adjusted with the plane of ita coils parallel
to the magii.tic meridian, and the zero reading is taken. The
eurreot is then made to puss through the coils, and the deflexion
of the magnet corresponding to it« new portion of equilibrium ib
observed. Let this be denoted by ^.
llien, if y/ is the hoiizontal mii^nctic foreo, G the coclficieut ol
Iht! fpilvjDometer, and y the stronffth of the current,
//
tan f ,
(I)
jnRcmoMAosino ob8bhvatioit». p'45.
If tbe coofficient of torsion of the suspeasion fibre is TMJI{tee
Art^ 452), w<! must use the corrected formala
y = ^(tan* + r^«OC*). W
Setf falve of the Ikfit^on.
743.] In some gulvniiorneteri the number of windings of Uie
coil tlirouf^h which the current flows can be alterwl at plearank
In others u known fraction of the current can he dirertecl front thr
galvanoRietvr by a conductor called a Shunts. Id either case t)ie
value of O, the eifect of a onit>outTent on the magnet, '\» maik
to vary.
Let UK determine the value of Q, for which a given error in the
otwervation of the deflexion correapond^ to Uie smaUeei error of the
deduced value of the Rtreiigth of the currenL
Diirerviiliating equation (1), we Bad
I
A
Eliminating G, J~ ~ n~ **" ■*^' (*^
This ia a maximum for a given value of y when the deflexion it
45". The value of should thprofore he ndjtwtwl till (Jy is a*
nearly e<Uid to ^ bk in pottifihlo ; wi Ihat for strong currents it ii ]
better not to ukc too sensitive a gatvaiiomi'ter.
On tie Bat Method of applying the Current.
744.] When the ohMtver ig able, by ntoatis of a key. to make or
break the conncxioss of the circuit at any instant, it is advisable io\
crate with the key in snch a nnr as to make the magnet arrive
~ftt itK i>OKition of r<iailihrittm with the least possible vcloci^. Thfta
lulluuiug metJiod was devised hy Gauss for this purpose. f
SuppuM that the nutf^et is in its position of rqailibrium, and thai
there is no current. The observer now makes contact for a sliorbj
time, so that tliv mugnct is set in motion towards its new posit
of ecjuilibrium. Ho th«'n breaks outjict. Tho force is now towa
the origitml position of icjiiitibrium, and tho motion \* retarded,
this is so managed that the magnet comiH to rei>t exactly at thi>
new position of equilibrium, and if tJie observer again makes* con
tact at thai inslant nnd maintains the contact, tlie magnet wilt
remain at ivst in its uew jio^ition.
anatnai
a sboft^
positiofM
toward^l
ded. ifl
^ 745.J MEASUMKBUT 0? A CTTBREITL 808
B If we neglect tb« effect of the resistance* nnd also the iD«qn»lity
^bT the total force acting in the new and the old positions, thon,
^^nnce we wish the new force to generate as much kiiit^tic enei^
daripg the time of ita first action as the original forci^ dcBtroys
while the circuit is broken, we must prolong tlie Sr^t action of the
^rfnrrent till the magnet has moved over half tJte digtanoe from the
^prst position to the aeoond. Then if the original force acts while
the magnet moves over the other half of its course, it will exactly
Btop it. Now the time required to pass from a point of greatest
elongation to a point half way to the position of equilibrium is
onesixth of a complete period, or onethird of a single vibiatioo.
The operator, therefore, having previously ascertained the time
of a single vibration, makes contact for onethird of that time,
breaks contact for another third of the same time, and then makes
contact again during the continuance of the experiment.. The
magnet is then either at rest, or its vibrationB ar« eo small that
observations may be taken at oni?e, without waiting for the motioD
to die awny. For this purpose a metronome may be adjusted so a*
to beat three times for each single vibration of the magnet,
^1 ^e rule is somewhat more complicated when the resistance is of
sufficient magnitude to be token into account, but in thitt case the
Tibrations die away so fast that it is unnecessary to apply any
corrections to the rule.
When the magnet is to be restored to its original position, the
jit is broken for onethird of a vibration, made again for an
time, and finally broken. This leaves the magnet at rest 10
former position.
If the reversed reading is tobetuken imme<liatelyaiUr the direct
le, the circuit is broken for the time of a single vibration and
then reversed. This brings the magnet to rest in the reversed
rrition.
Meatnrement by tie Firtt Swing.
745.] TVhcn there i« no time to make more than one obscrvatioa,
the ciirrcnt may he measured by the extreme elongation observed
in the first swing of the magnet. If there is no resistance, the
permanent deflexion ^ is half the extreme elongation. If the re
nstonce is such that the ratio of one vibration to the next is p, and
if fly is the zero reading, and ff, the extreme elongation in the first
swing, the deflexion, 0, corresponding to the point of equilibrium is
^ =
Aa
354
ELBCTBOMAOSSTIC OBSERTATIOITS.
la this way tbe deQcsion may be calculated wiUiont wuting f«r
the magnet tn come to rest in its positioa of cquilibriom.
Ih mah a Strie* pf Oiaervathus.
746.] The best way of making a coDeidemble namber of meaniitf
of a conaUint eurrent is by obaenring three elongations while tJuj
current is in the positive direction, then breaking contact for about
the time of a Ginglo vibration, so as to let the magnet swing
into tbe position of Degutive <l<.>flcxion, then revcr«in> the coTreat
and observing three succesEivv elongations on the negntive oAv,
thou breuking conUct for the time of a nnglo vibnition and iv
pcflting the observations on the pottitivA Hide, and m on till a soffi
cient number of observutionB have been obtained. In this way tbe
errors which may arise from a change in the direction of the earth'*
magnetic force during tbe time of observation are eliminAtod. The
operator, by carefolly timing the making and breaking of contact,
can easily regulate tho extent of the vibrations, .w a^ to make Uiem
snfBviently small without being indistinct^ Tho motion of the
magnet is graphically represented in Fig. 59, where the nliirii
rpproscnts the time, and the ordJnato the deflexion of the maglMt.
If flj ... (1( be the observed elongations, the deflexion is given by the
equation 8^ = tf, + 2tfg+fl,tf42tfj— tf,.
I
Kg. ss.
Melkoil 0/ MnUipticatiwt.
747.] In certain cases, in which the deflexion of the galvanometer
magnet is very small, it may bo advisable to increase Uio visible
efi^ by reversing the eurrent at proper intervals, so as to set
np a Ewinging motion of the magnet. For tliia purpose,
ascertaining tbe time, T, of a single vibration of tlie magntt, tb
corrent is sent in tbe poative direction for a time T, then in tfaj
reversed direction for an equal time, and so on. When the motiod
of tlio magnet has become visible, we may make the reversal <ii \
current at the observed times of greatest elongation.
Tjct the magnet Ixt at the positive elongation fl,,, and let
current hn wnt through tbe coil in tbe negative direction. 1
748.]
TfiANSIEKT CCBBENTS.
355
Ppo
I
r
lint of equilibrium ib then — ^, and the magnet tvill swing to a
ne^tive doagmtion 0^, such that
Similarly, if the current is now made positive while the magnet
swings to 0j, p^i =— l'i + 0>+ 1}^.
or />*fl, = fl« + (p+l)'*;
>nd if the cnrrent is reversed n times io sucoceeion, we Gnd
rhenoe we may find ^ in the form
* = («/>""^o)
p1
p+l 1p
If M is a number so great that p'" may be iK^lectcd, tlie ox
^reeaion becomes p^i
= ff„
I
P+l
The application of this method to exact measurement requires an
accamtc knowlrdgo of p, the ratio of one vibration of the magnet
to the next under the influeticc of the resistances which it expe
riencex. The unceTtainties ariEiiig from thu iliniculty of avoiding
irregiiUritien in the value of p generally outweigh (he advantagoa
of the large angular elongation. It is only whitre ^ve witili to
establish the existence of a very omall eiirreut by oauKing it to
(produce a visible movement of the needle that this method in really
valoable.
On the Meaturement of Tramienl CurrenU.
748.] When a cnrrent tasta only during a very small fraction of
the time of vibration of the galvanometermagnet, the whole <]uaa
tity of eJeotticity transmitted by the current may be measured by
the angalar velocity communicated to the magnet during the
pas»ige of the current, and this may be determined from the
elongation of the first vibration of the magnet.
^H If we neglect the resistance which damps the vibrations of tbo
^■magnet, the investigation bfcomos very simplo.
Let y bo the int«nsity of the current nt any instant) and Q tliw
quantity of electricity which it tranvniitv, then
q^jydt.
0)
A a z
ss»
BLECTBOIUOKBTIO 0ISBBTATT0K8.
[74»
Let 3i be the magnetic momettt, and A the moment of inerta at
the uu^et and suspended sppamtos,
A^ + MII»ae = ,Vffyoo«*. (J)
If the time of the passage of the current ia very small, wc na;
JDtegnite with respect to / during this short time withoat r^ionlii^
the change of S, and we find
A^=MG<ioa0oJyJl + C = 3fGqoM9^ + C. (3)'
Thig sheirs that the passage of the quantity Q prodnoee an aagolar
momentum StGQ cios 0^ in the magnet, where $„ is the valoe of t
at tti« intitant of lujsago of the current. If the magnet is initiallj
io ei]uilibrium, we mnj make 0^ = 0.
The magnet then swings freely and reaches an elongation tf, . If
there is no reoistance, the work done against the magnetic Ebrce ,
during this swing is Jfi7'(l— costf,).
The energy otanmttnicated to the magnet by the current is
iA
Equating these qnantides, we 6ad
d$
= 2^(lC0Bfl,),
whence
i'V
HE
sin \0^
= ^Qby(8).
A
Bat if 7 bo the time of a single ribration of the magnet,
and wc find
J_
It T
m
in
where B is the horizontal magnetic force, G the coefficient of the
galvanometer, 7* the time of a single nbmtion, and 0, the firrt
elongation of the magnet.
749.] In many actual experimente the elongation is a
Ka^V; and it \s then ensy to take into acoonnt the effect of restst
anoe, for wc may treat the equation of motion as a linear equation.
Let the magnet be at rest at its position of equilibnnm, let
angular velocity c be communicated to it instantaneooaly, and
its first elongation be $x.
\ firrt J
amsnl
■Mist"
mm
MBTHOD OF KBCOIU 357
!I1ie equtttioD of moUon U
^ = C»,Bec^flt'»"flco8(«,(;+^>. (9)
■When t = (i,$ = 0,MA.~ = Cu^ = p.
When «,(+/9=,
tf = CrfH'"''cos^ = (l,. (10)
Now ^— = <u' = co,'sec*A (12)
tanja = . w,= ^, (13)
•=^«. (14)
H«oo .. = f:^.>^^
(15)
[whkh giYo the first elongation in terms of the quantity of elec
pcatj in tho trnnEieot current, nod convtrrsely, wliero jT, is the
. tioM! of a eingle vibration as affecteil by the actual resigt
BDM of damping. When K is sniiiU we may use the approximate
fonnula rr t
Q = ^lil + h>^)9,. (17)
Method of RteoiL
750.] The meUiod gntfa above EnpposeH the magnet to be at
TMt in its position of Cfiuilibrium when tlio transient cuircnt it
paaaed through Uie coil. If we wish to repeat the experiment
we must wait fill the magnet is again at reat. In certain caeea,
however, in which we are able to produce transient currcnte of
equal intensity, and to do so at any desired instant, the following
metbod, described by Weber *, is the most convenient for making
.tinned aeries of observatjons.
* JlavUtUt iri MajiuiUAm Vm{Ht, ISS8, p. W.
35?
ELBCTROMAOinrnC OBSBRTATtOSS.
Suppose that wc set the mnfpiet Biviagin^ hy means of a trmsiait
current whose valao is Q„. If, for brovity, wii writ«
6 ./¥+l^ _.i,^'.
e •
= jc;
(18)
M r,
then the first elongstion
fl, = JfQ„ = fli(My). (13)
The Totocity instantaneously coinnimiicat«d to the mugoei il
Btartinff is MG . ,^„,
When it returns through the point of c<]uilibri(im in a negaiin
direction its velocity will be
», = — or*. (21)]
The next negative elongation will be
0, =«,** = 5,. (SJ)
Wlien the magnet returns to tJie point of eqnilibriom, its veloci^
will be „^j,^^it\ (23J
Now let an instantaneous current, whose total quantity ia — Q,
be transmitted through the coil at the instant when the magnet is
at the zero point. It will change the velocity v^ into r,— c, wbeie
MO
V =
Q
m<
I
ir Q is gwtteT than Qo^~^^> ^^^ '■^w velocity will be negative and '
CHual to iffi
The motion of the magnet will thus be reversed, and the next
elongation will bo negative,
<?,=ff(Q(2„r») = e, = JS:c+^r^\ (25;
The magnet is tlien allowed to come to ita poaitive elongatioa
e^ = «,** = <*, = ** {A'Q«,«**). (26)
and when it again reaches the point of eiinilibnum a poativ«S
current whose qoantity is Q is transmitted. Tbia throws ihg'
magnet back in the positive direction to the positive dongation
$,=KQe^e»'; (27)
or, calling this ihe first elongation of a second scries of four,
n,= A'efle") + fl,('\ (28)'
Proceedbg in this way, by obecrving two elongations + and
then sending a pontive current and ohscrting two alonga
i
SERIES OP OBSERVATIONS.
8C9
— and +,tben Bcuding » positive current, and bo on, we obtain
K aeries consisting of sets of four elongations, in ea«h of wliieli
a—c
= «*.
and
(30)
If n series of clongntione have been observed, then w« find tbe
logarithmic diorement from tlie equation
and Q (Vom tbe equation
= S.(ai_c + d)(l +B**)_(^.a,)('/,«,)*«\ (32)
The motion of the magnet in tbe metbod of recoil is graphicaily
represented in Fig. 50, where tbe aliscissa rcpreacnU tbe time, and
the ordinate the deflexion of the magnet at that time. See Art. 760.
Jlfg/ioil of Miilliplication.
751.] If wo make the transient current pass ercry time that the
magnet pacvcw through the zero point, and always so bjs to increase
the velocity of the unngnct, then, if tf,, tf^, &c, ore the successive
elongations, e^=—KQe%, (33)
0s=—KQe'$..
(34)
The ultimate value to wiiicb tbe elongntion lends aft^r a great
vibrations is found by putting $^ = ~^iiii whence we find
If A is small, Uic value of the ultimate elongation may be large,
iMit «iue* Uiis involves a long continuwl cxin>riinent, and a careful
determination of A, and since a small error in A introduces a large
enxpr in tlie determioatioD of Q, this method is rarely useful for
I enxpi
tJbs tiiae of th« p«ss^ of the g nrrat vick sBttv.
W
CHAPTER XVn.
COVPASISOK OF COILS.
Erperimmfal Dttfrmina/ion of the EUdrkal ComtanU
^a Onl.
752.] Wb hnve eeen in Art. 71 7 that in * sensitiTe walvanomeUr
tlie coils ebould \k of small radtus, and should contain many
windings of the wire. It would b« extremely diflScult to determine
the electrical constants of such a coil by direct measurement of iti*
form and dimensions, even if we could obtain access to every
winding of the wire in order to measure it. But >n fiict th«
grcntvr number of the windinf^ arc not only completely hidden
by the outer windings, but we are uncertain whether the prvscnre
of the outvr windings may not have altered the form of the inner
I ones after the coilinpr of the wire.
It 18 bettor therefore to determine the electrical constants of the
^Coil by direct dectrieal comparison with a standard coil whose con
, stunts are known.
Since the dimensions of the standard coil must be determined by
I actual mea«ureinent, it must be made of t'oosiilerable size, so that
Uie unavoidable error of measurement of its diameter or circum
ference may be as small as jioBsiblc compared with the quantity
measured. 'Hie channel in which the coil is wound should be of
rectangular section, and the dimensions of the section sboald be
•mall compared with the radius of the coil. This is necessary, not
I M mnch in order to diminish the correction for the size of the
I ae«tion, as to prevent any uncertainty about the position of those
[windings of the coil which are hidden by the external windings*.
Large Uupat nhMKcnoUn w* tniiutlniM made with * aUigl* oirculkr Mo.
dnctliif ring of ecmuhnble IhicknEw. whioh i« fufnciutUr Uitt t« luiiiUuD itii fonn
sithnit KliT *ii>pi)rl, Thii ii not > good plsn for * ttiuiiiiuil tontnlDettt. The cU»>
tribulMti u( tlia eiURSt niUiiii Um conductor ilqiwiib du lb* r « h i ti ?o coiidaotMtj
^
COKPAfilSOK OP COILS.
Tlic principal poDKtante which we trah to det<>nnin« u«—
(1) Tho magnetic force at the cintrc of the coil due to ■ imtt
current. This is the quantily douotod hj G, in Art. 700.
(2) The magnetic moment of the coil due to a anitrormit.
Tliis is the quantity (f^. ^
753.3 ^** dfiermitu O,. Since the cofla of tlw worlciog' gn]«'
nometer are much smaller than the ittandard coil, we place Um
galvanometer within the standard coil, so that their centres coincide,
the planes of both coils being vertical and parallel to the earlb's
magnetic force. We have thus obtained a diflerential galvanometer
one of whose coils is the standard ooil, for which the value of 6',
is known, while that of the other coil is C/, the value of which w* h
have to determine. H
The magnet suspended in Uie centre of the galvanometer coil
is acted o» hy the eurrcntri in both coils. If the strength of thr
current in tlie titandard coil is y, and that in the galvknomctcr coil
y, then, if these currents flowing in opposite directions prodooe a
defleiion t of the magnet,
J/tani = G,V<?iy, (I
where J7 is the horizontal magnetic force of the euih.
If the currents are so arranged as to produce no deflexion, w«
may find G^' by the equation
(
<?.'=^C,.
a
I
I
We may determine the ratio of y to / in several ways. Since the
value of f?! is in general greater for the galvanometer than for the
standard coil, we may arrange the circait so that the whole current
y Bows through the standard coil, and is tJieit divided so that }^
flows throug)) the galvanometer and resistance coils, the combined
resistance of which is It,, while the remainder y— y' flows through^
uiotlier set of resistance coils whoee combined re«istanc« is S,.
of ba vsrknM'pvlj. UniM »aj wniMabd flkv fai iIm CODtioatty at tb« MStol
a«M th* nuia itnun o( el«<rtnrit7 to Sow nititvr tlem» lo lb* «M4Ua «* doM to
ImUt of thn circDljLr ring. Tbu tho tno paik of tbe oumM Wmm* ttnoarula.
BmUn thK wbea tbe curmM flow* oolj taiM rooBd the tinU, <ii«oiKl taiv h
■CMBHij (« ftraU MDj AcUan oc the nutfuAei nrngnct daa ia Oi» cmtmit o« Im
way lo or tKtu Uw tinit. If ■iiim th* cuR«)tl In U« elecUndM U oqiud t« tLat to
tba dido. In Iji* otautiwiUaii of maay hntnuntBU Uio anloB ol tlui put of Ua
comnt •••nw (n ki>r« Bm •ItogKlHr tort a/fla i>t.
Tba iDo«t ptifpct molliod U to nwik« mw of tbo elwtnxl** in t^ (ami <f • nwi
tnba, and lb* odm » win oortrod whli bniklinc nuttrial, urf UMtd InM*
(ab« ^id eooMMfie frith II, TIiaoxl«nuJ kM)an«f theelBeUoilaiwhimtliua
b UN^ bjr An M3.
COEFFICIENTS OF ISDCCTIOS.
S68
We hare then, by Ait 276,
or
and
y
y
(8)
is Mny unceHainly nboiit the nctiial resistance of the
iet«r coil (on account, Kny, nf itn uncertainty as to its tem
peratore) we may add resistance coils to it, so tliat the resistance of
e galvanoinet«r itself forms but a small part of 7?i, and tUos
introduces but little uncertainty into the final result.
754.] To determine ^, , the magnetic moment of a small coil due
a unit •current flowing through it, the magnet is still suspended
il tliv centre of the etnndard eoil, but the small coil is movod
lid to itself along the common axis of both euils, till the same
current, floning in o)iiosit« directions round the coils, no longer
deflects the magnet. If the distance between tbe centres of the
coils is r, we hiivc now
By repeating the experiment with the small coil on the opposite
of tbe standard coil, and measuring the distance between tbe
ptions of the small coil, we eliminate the uncertain error in tbe
lination of the position of the centres of the magnet and
' the email coil, and we get rid of the terms in g^, ;«, &c.
If the standard coil is so arranged that we can send tbe current
[.tbrough half the number of windin<^, so ae to give a dilTerent rnltic
&,, we may determine a ncvr vnlue of r, and thus, as in Art. 431,
we may eliminate the term involving y,.
Il \» often poEKible, however, to determine j^ '^y direct meanirc
it of the small coil witli sufltcient accuracy to make it available
caloalating tbe value of the correction to be applied to ;, in
lie equation
9^=\0^r^^
(?)
vhcre
y,= — ira»(6a' + 3f'27'), by Art. 700.
ComparitOH i^Coej^eitnU oflndnetion,
75S.] It is only in a small number of eases that the direct
GalculatioD of tbe coefBcieDts of iDdnction from tbe fonn and
poKitton of the circuito can bo easily porfoniUMl. In order to mttaia
a sufficient decree of aoRuracy, it is neoesMry that tbe dtstaimJ
botwwn the oircuita should be capable of exact measurement.'
But whiii the distance between the circuits is eufficient to present
errors of measurement from introdacing Ibi^ errors into tho result,
the coefficient of induction itsdf i» uecowarily very much reduced
in rouf^nitudc. Now for many eipcrimeots it in ncocsHiry to make
the ciwfficient of induction larg«, and we can only do bo by bringing
the eirouitit close togitther, lo that tlie method of dir»ct mcAsufv
mcnt becomeo impossiblo, and, in order to dctftmune the cocfScient
of induction, we must compare it with that of a pair of ooiU ar*
rang«d so that their coefficient may be obtained by direct meuure
ment and calculation.
This may be done as followt:
Let .( and a be the standard l
pair of coils, B and b the ooila to 1
be compared with them. Con
nect .f and B in one circuit, and {
place the clcetrodc« of tlic gal 1
vanomcter, G, tA P and Q, eo I
that the resistance of PJQ is
B, and that of QBP is S, S
}imag the renKtanoc of the ga^
Tanometer. Connect a and i in
one circuit with the battery.
Let the current in ^ be ir, 1
that in B, /, and that in the gdnaemlbn, nfr— jr, that in the battery
circuit being y.
Then, if J/, is the coefficient of induction between 4 and a, and
iff that between B and 6, the integiitl induction current tfarongb
the galTan<nnet«r at breaking tlie lottery circuit is
By adjusting the resistances S and S till there is no cnimi'
through tho gnlvanometi.^r at making or breaking tJie batte:
oircnit, tho ratJo of i/^ to if, may be determined by measuring ihsA
otStoS.
Fig. do.
SEUIITODCTIOJI.
860
H * [Tho expTMeion (8) may be proved aa follows : Let i^,j^, A'&nd
BT be th« coeffiviente of selfinduction of the ooils A, S, ai and the
Kgalvanomctcr respeotivelf. The kinetic energy T of th« Hyst«m i«
HthCQ up proximately,
H The dissipation function F, i.e. half the r]tt« at which the energy
Hof the currents is wasted in beating the coils, ie (see Lord Rayleigh't
BlW^ o/S<fUnd, i. p. 7S)
Vwbere Q is the resistance of the battery and battery coilf
H The equation of currents corresponding to &iiy varisble * ifl then
B~of tlie form j 4T ^T dP
wkece f w the oorreiiponding electromotive force.
IHenoa we bavA
i,* + r{«jr)+ jr,y + .ff* + A(i» = 0,
iJIr(isf) + af,y+5^A'(*^) = 0.
These equations can be at oncti integ:rat«d in regard to t.
Obfltmng: that T., x,y,y, y are zero initially, if we write x—y = z
we find, OD eliminating y, an equation of the form
■ AS+Si+Ci=Dy + EY. (8')
H A short time after battery contact the current v ^1 have become
^Ll^^y and the current i will have died away. Hence
H This gives the expression (8) above, and it shews that when the
^ total quantity of electricity passing through the galvanometer is
zero we must have jE = 0, or i/,ff— JJ/,5 = 0. The equation (8')
further shews that if there is no current whatever in the galvano
meter we must also have i> = 0, or if^.^)— JfiX^ = 0.]
Com/>an40* of a CofJ^eni <jf SrffiiuluiTlioK teith a Co^jKurBtaf
Mutnal jHilnction.
756.] In the branch AF of 'Wbcaitstone'a Bridge let a coil he
inserted, the coefficient of selfinduction of which we wish to find.
Let OS call it L.
• [Tlie iarMttekUan b KjnMebrMkets, IjiIwh ttoia Mr. Fltaihio'iiiotwoTrD/iwat
Cteffc M&ivcU'i XMtuN^ pOMiMM M Diotuiii.'hi'ly iiiiumt m Muz pMt nf tb« Ual
iMturs dclir<T«l bv thn Profcaaca. Id Mr. FlMnlag'* notw tha (Jiu «f tiw coEpari
BMot dlffcn ftoat thai ^voa In lh« Uit in hiring th« batuny and gslvanorndter
IaUnihM>c«d.]
366
OOMPABISOH' OP COTLS.
I
Id the oonnectinff wiru betwceo A and tlic bstii^ry another Mil
is iDfiert«d. The covllicicnLt of nmtuiit iiMluotioo between Uiix coil
and the coil in AF in M. It may be meaaared hj tlie metliod ^
described in Art. 753. fl
If the cuiTCnt from J.io Fiax, and that from J U> JliBjff th«t ~
from 2 to A, through S, will he t+j.
The cxtcrnul clcctromutivo force fron
^itoi'w
\ AF=P.^lpM(^.±).p)
The external eleetromotive force
alonff All is
JJr= Qy. (10)
If the galvanometer placed betw«w
F and U indicates no currvot, either
truH«ii.Dt or iwrmanent, then by (9)
and (10), siBCc U—F= 0,
Pr=.Q.y: (11)
M
Fis 61.
and
whence
(U)
(13)
I
«
Since/' i» always posilive, jV muet be negatircnnd tbereforo the
current must flow in ojipoKito directions through the coils plnved
in P and in B. In making the experiment we mfty either begin
by adjusting the resistances so that
PS=qR. (H)
which is the condition that there may bo no permoiuint curreat,
and then adjuxt tliii di»t«iice between tlio coil* till the galvanometer,
ceetca to indiciite a. transient current on making snd breaking the
batteiy connexion ; or, if thin distance is not capable of adjuitmeot,
we may get rid of tlie transient current by altering the restslAooes
Q and S in such a ^ay that the ratio of Q to 5 remaina constant.
If ihis double adjuntment is found too troublesome, we may adopt
a third method. Beginning with an arrangement in which thel
tnuuieiit current duo to sclfinductton is slightly in exoees of that
due to mutual induction, we may get rid of the inequality by in
Betting a conductor whose resistanoe is T between A and Z. The
condition of no pinnanent current through lhi> tn>lv.tnomebor is not
aficct«d by the introductioa of ff' Wo may therefore get rid c^
^i^^
a ci 1
757] SBLFIKDrCTION. 367
the traniiiciit current by ndjaetiDg the roUl^tce of ft' ftlooe. Whoa
liii! w douc the vnluo of L is
(15)
CoMpariaon of the CofJJieiaili of Seffmlnclion of Two Coil*.
757.] Insert the coils in two adjacent branches of WheatBtone'fl
Bridge. Let L and N be the coefflcieots of selfinduction of the
o(»1b inserted in P and in R respectirely, then the condition of no
g»!T8nomet«r current is
and
PS = QS, for no permanent current,
^ = =■, for DO transient current.
r it
(16)
(17)
(18)
I Hencf^j by a proper adjustment of the resistances, both the per
nuuient and the transient current can be got rid of, and then
the ratio of £ to iV can be deti^rmined by a oomparisou of the
kiesistuioc*,
366
OOMTABISOK OP C0IL9.
In the connectiDg wire bctwoen A nnd tlu> Hstteiy
a inserted. The ooefficiont of miitmil induction br
and tho coil in AF is JA. It ma; be mtiiutuTei*
described in Art. 756.
If the current from A\o FUx, and tUat fr
from SSUt At thr
The external c'
AU>F\»
M
V
AF=Px
The .
t« along /
STASCX
w .
Fi7.«l.
and
wheDoe
Since L \» aI\Tay
current must t!i ^
in P and in B.
by adjosting tf
which IK tl
and then n'
ceases to i
batterr c
we nius
tC antl
Ift^
a thit
tra"
*^ «rtor is defined as tlie ratio of thr
, live force to that of the cuitenl
,.jotor. The determination of tit
tf .ruiiffociie nieaHure can be made by
vMr, when we kooiv the value of the
.. determination of the vnltic of tht
□lEciilt, as the only cniw in wliich ire
. J,' is when it arise* from tlic relative
..•■^■ict to a knoTvn magnetic Hjvtem,
^^Btion of the ri}«i»1an«c of a wire in
«ai made by Kir«hhoir*. He employed
•^ J.and Af, and calculate thoir coefTiciest
of mutual induGlion from the geo
^ metric*] datA of their form and
apoation. "Hieee coils were jiUced
a in circuit with a (^vanonietor, G,
= and a battery, B, and two potntc
\ oftho circuit, /*, between the coils,
.^■' and Q, between tlie battery and
[galvanometer, were joined by the
^ C^ was to be measured.
^''wly it i§ divided between the wire and
. imd produces a certain permanent de
r. If the coil ji, is now removed quitikty
iinbbM •1i« iDlnulUit Mocirtor tlrlcniailiw
CHAPTER XVin.
ELECTROMAGNETIC CKIT OP RESISTANCE.
(h lie J)«iefmiiuition of fie HetUUtnee of a CoU in Sleetn
magneik iteature.
768.] TiiK reRiHtance of a conductor in deGned ae the ratio of I
numerical value of the clcetromotivi' force to that of the
which it produces in the conductor. Tlie dGtenninatJOD of
value of the current in elcdromagnetio measure can be matle
means of a standard galvanometer, when we know the ralue of
eartli'e magnetic foroe. The determination of the vslne of
electromotive force is more difficult, as the only case in whidi wf^
can directly calculate its value is when it arises from the relative
motion of the circuit with respect to a known magnetic tystcin. ^
759.] The first determination of the reastanoo of a wire ifl^
electro maniple tic measure was made by Kirchhoff*. He employed
two coils of known Ibnn, ^j and A^, and calculated their coefficient
of mntual induction from the geo
metrical data of their fonn and
position. These coils were placed
in circuit with a galraDompter, G,
and a b«ttery, S, and two point«
of the ciicuitj P, between tho coils,
and Q, between the butU^ry and
galvanomDlcr, were joined by the
wire whose resistance, R, mu to be measured.
When the current is rtoady it is divided between the wire
the galvanometer circuit, and prt^uces » certain iwrmanent
lloxion of the galvanometer. If the coil Ai \a now remorcd quickl^
F1k.CS.
[ckl^
■ • BaitimnaiB dtr OctutentM tod wslohar din IbMuUH bdadtttf •UtirliB
SbNlineaUiiMfl.' i'oyjp. daik. )»ri (April 1M»>.
KIBCiniOPF*S METHOD.
Ota A^ Aiul placed in a position in which tlic coefilciont of mutuftl
[induction between A^ and A^ is zero (Art. 53S}, a current of induc
bioa i* produced in both oircuita, and the galvanometer needU
ivw an impulse which produce* a certain tmneiient deflexion.
The rcsiiitnDce of the wire, H, ia deduced from a comparison
etween the permanent deitexlon, dae to the steady current, and th^
Dsti^nt deflexion, due to the current of induction.
Let the resistance of QGi,/" be A", of PA^BQ, B, and of PQ, R.
Let X, M and A' be the coefGoients of induction of A^ and Aj .
Let i be the current in G, and y that in £, then the current
from P to Q is i—^.
^Lei E be the electromotive force of the batter/, tlien
(^K+B)xBy+~(U+My) = 0, (I)
Wl
~Bi+iB+Ii)y + ^(Mx+Ny)=E.
dt
(2)
When the currents are constant, and everything at rest,
(ir+i?)Afly = 0. (S)
If M now snddenty 1>ei?oraes z^ro on account of th« separation of
Ai from Ja« t''*'*') intt'gratiug with respect to t,
(K+Il)x~RyMy = G. (4)
Rx+(B+S)y^3fi =JEdl = 0; (5)
Substituting Uie valoe of^ in terms of « Irom (S), we find
x_M (B^Il)(K+Ii) + B'> ,.
i~ B {B+B}{K+B)B' "'
h
'When, ns in KirchhofT's experiment, both B and K are large
punxl will) B, this (quation it reduced to
i R ^*'
thiM quantiticd, x is found from the throw of tlic gahaoomctcr
) to the induction current. Svc Art. 718. The permanent cur
rent, A, i* found from the ])vrmAncnt dcHcxion due to the eteiuljr
cumeDl; mw Art. 746. M w found cither by direct caliiilntion
from tiro geometrical data, or by a comparifion with n pair of coiU,
for which this cadculation bae boon made; see Art. 75S. From
■VOU 11. B b
wa
mflT OF BEeiSTANCI
[760.
tbwe three quantities R can b« dctorminod in eloctronuigo«tic
mcaeare.
TI10BC mcthoda involve the determmatioD of the period of ribra
tion of the gahanometei ma^aet, and of the logaritlimic dvcranunt
vritM oscilUtions.
freier's Method iy Traiu!fHt Currentt*.
760.] A coil of considerable etze is mounted on an axle, so as to
be capable of revolving about a vortical diameter. The wire of tbu
coil 18 connected with that of a tan^nt ^Iv&nometer so as to fonn
■ nng'le circuit. Let the reeistance of this circuit be It. Let the
large coil be placed nith ita positive face perpendicular (o the
mogDctic meridian, and let it be quickly turned round half a
tulton. There will bo an induced current due to the earth's
Di^tic force, and the total iiuimtily of ck'ctrieity in this curre nt il
electromagnetic miusure will be
where ,7, is the magnetic moment of tiie coil for unit current, whi*
in the catso of A large ooil may be dctcnnined dirMtly, by nten^
euring the dimeaaioas of the coil, and calculating the sum of the
areaa of it« winding*. // is tlie horixoutal oomponoDt of borrcstria)
ma^ctiem. nnd R is (he rpHistance of the vireuit formed by the
ooil and giilviinomcter together. This current sets the magnet of
the galvanometer in motion. jH
If the magnet ia originally at rest, and if the motion of the coil^
occupies but a Kmall fraction of the time of u vibration of the
magnet, then, if we neglect the resistaaoc to the motion of the
magnet, wo have, by Art. 740,
// T
Q = ^2mn\0, (25
where is the constant of the galroDometer, T » the time of
vibration of tlie magnet, artd $ is tbe obMTved eloogaljon. Prom
these eqaationH we obtain
The value of U docs not appear in tlus remit, provided it ia
same at the poeitien of the coil and at that of tJie galTanotnetcf
This ehould not be asauuied to be the case, but abonld be tested
comp&ring the time of vibration of the asame magnet, flr«t at one of
tfacco places and then at tbe other.
* £Ml. lUaa.: at Fogs., •^" ^'^='^ >>' (1851).
f
(3
N
I
I
>
871
701.] To mnlw n »crie« of observation)) Wcbcr began with the
vriil jKintllel to the magmtic miTiiHan. Hu tinn turned it with its
Kj(iitive face nortli, Hiul observed llic (iret eloni^tion due to the
negative current. He ttien observed the second olongution of (Iio
fn>eiy g^^ini^ing magiict, and on tLe return of the magnet through
the point of equilibrium he turned the coil with its positive fnou
Kouth. This eauned the magnet to recoil to the positive side. Tho
eeries was contiuiicd as iu Art. 750, and the result corrcuted for
feAistauce. In this way the value of the resi^tunce of the conibinccl
circuit of the coil and galvanometer was ascertained.
[n all Biich exporiments it is neceaaary, in order to obtain «n0i
tiently large deflexions, to make the wire of copper, a metal which,
though it is the best conductor, has the disadvantage of altering
conaiderahly in wsistaui'e with alterations of temperature. It is
also very difficult to aeccrlain the temperature of every part of the
apparatus. Hence, in ortlcr to obtain a result of permanent value
from such iin experiment, the rcKistuncc of the experimental circuit
ebould he compared witii that of n cttrefully constructed reeistance
ooil, both before and aftc^r eJich vxperimi.'nt.
fFeber^t Method b^ ohervhiif the Decrement of the Otcillations
of a MaifHtt.
762.] A magnet of considerable magnetic moment ie suspended
at the centre of a galvuuomuttr eoil. The period of vibration and
the logarithmic decrement of the oscillations is ebserveil, firnt with
the circuit of the giilvanonmUT open, and then with the circuit
cloaed, and the conduLtivity of the galvanometer coil is deduced
ftom the effect which the currents induced iu it by the motion of
tiie nttgnet have in resisting that motion.
If 7' is the obiterved time of a single vibration, and A the Na
pierian logarithmic decrement for each single vibration, then, If we
write ff ,,.
and a = y (8)
I the equation of motion of the magnet is of the form
^ = Ce*' eos {i»l+ 0). (3),
[Thi* expresses the nature of the motion as determined by obser\';i
ftioa. We mu»t compare this with Uic dynamical equation of
kmottOD.
D b 3
872
[76J.
Let M be tbe coiifficit.'iit of induction between the galvanometer
coil and tlic Ku${>codud mngnct. It is of the form
where (?,, &,, &c. are coefficients belonging to the coil, ^,,^, ft«.
to the magnet, and P\{6), 1*,{fi), &e. uv zonal harmonics of the
angle betweea the ax<>s of the coil and the magnet. Si>e Art. 700,
By a proper arrangement of the coiU of the gulvanometcr, and by
building ap the suspended magnet of several magnet* placed side by
side at proper distances, wc may canso all the ternu of M »n«r t'
lirst t(j become insensible comxHirod with the first. If we uW i
^ =: 9~^> ^^ ^^3 writ«
11= Gm»m^, (5)
whore is the principal coeffiirient of the galvanometer, m \t
taagnctic moment of the magnet, and lit the angle between
axia of the mugnet and tlic plane of the coil, whicli, tu this
periment, ik always a small angle.
If J, in ilie coetflcieiit of seirinductioii of tli« coil, and R
resistance, and y the current in tlie coil,
^(£7 + ilf) + fi). = 0, (I
or 2^+J?y+(?jfflC08<6:^ = 0. t
at at *
The moment nf the force with which the currrent y acts on 11
mag^Dct is y ^ , or Gm y cos ^. The angle ^ ia in this experime
Ki small, that we may Mippose cos ^ = I.
Let us BuppoM titat tJio equation of motion of th« mag^nefc '
the vireuit is broken is
I
^^^'^?^^* = 0'
(8)
where A is tlte moment of inertia of the suspended apparatus, B
expresses the resistance arising from the viscosity of the air an
of the suspension fibre, &c., and C^ oxprwnMs the moment of the
fone arising from the nuth's magnetism, the torsion of the sua
pcnsion nppnraius, Stc. tcniling to bring the magnet to its poaiti<
of equilibrium.
The equation or motion, u affected by the tnirrcut, will be
infl
Ihe
1
ial
WRDER 8 IfKTHOn.
873
I
To d«tennuie the motion of the magiiei, we have to combine ttiU
oijiuition with (7) and eliminate y. The resnit is
linear difiercotial equation of the third order.
Vie hnve no occnsion, however, to solve this equation, liecauM
the (Iiita of the problem are the observed elements of the motion
of the magnet, and from these we have to determine tlie valae
ofS.
Lot 00 and wg he the values of a and w in equation (3) when the
ircuit i» broken. In this cnso S is intinito, and the eqnation is
doccd to the form (8). Wo Ihtu find
Solving^ equation (10) for ^, and writing
j7=— (a+iw), where i = •/—!,
at
(12)
find
aI«
Since tbc value of w is in general much greater than that of a,
the beet value of R in found by equating the terms in f m,
& =
2'<(«''o)
■Oo
W« maj also obtain a value of £ by equating tbfi termH not
Iving I, but as theee terms are small, the equation is useful
Hjr M a means of testing the accuracy of the observations. From
tfaeae equations we find the following testing equation.
Since LAn^ is very small compared with G^m*, this equation
id equation (t4) may be written
In this expression G may be determined either from the linear
messnremcnt of the BJilvnnometcr coil, or better, by comparison
with a standard coil, according to the method of Art^ "53. /( i«
the moment of inertia of tbc magnet and its suspended apjuratu^
which is to be found by the proper dynamical method. <u, o,,, a
ad Og, are given by obocrvntion.
TTKIT OP RESISTAKCB.
The detenninntion of the valne of m, the magnetic momcat ef
tlip euspcnded magnet, is the moit difficult part of the invcstigatioii,
becAUfre it ib iiffect«d by temperature, by the dtrtLV ma^tivtJc force,
and by mLchantcal violence, so that ^n?at cnro must be taken br
measure this quantity when the mugnet u in the vcty same ctreuiQ
stances as when it ig vibratiug. .
The second term of H, that which inToIvw L, in of teat import I
ance, as it te generally xmall compared with the first term. Tbt
va\u6 of L may be determined either by ejilculation from the koown
form of the ooil, or by an esperiaient oq the extracurrent of to
ductioD. See Art. 756.
Tiomion'a Metkod f^ a Sevolviiig CoiL
768.'] This method was suggetited by Thomson to the Committee
of the British Aiisociation on Electrical Standards, and the ex
periment was made by MM. Balfour St«wart, Fleeming' JeDldo,
and the author in 1863*
A circular coil is made to revolve with uniform velocity about a ,
vertical axis. A Bmall magnet is eosponded by a silk fibre at the
centre of the coil. An electric current is induced in tiic coQ by
tho earth's magnetiEm, and also by the suspended magnet. This
current is periodic, flonnng in opposite dtn.ictioR8 through the wir«
of the coil during dif^rent parts of each revolution, but the effect of I
the current on the Kuwjiendcd magnet is to produce a deflexion from]
the magnetic meridian in the direction of the rotation of the ooil.
764.3 Let Jl be the horizontal component of the earth's nu^J
netiem.
Let y be the strength of the current in the coil.
ff the total area inclosed by all the modiDgs of the wire.
G the magnetic fotce at the centre of the ooil due to uoitr^
current.
Z the coefficient of selfinduction of the coil,
il/thc magnetic moment of the suspended magnet.
the angle between the plane of the coil and tho magnetio
meridian .
^ the angle between the axis of the «a«peadcd magnet andJ
the nu^nctio meridian.
A the moment of inertia of the suspended magnet.
ifffr the coefficient of torsion of the suspension fibre.
a the azimuth of the magnet when there is no torsion.
S tho resi«t«nc« of the coiL
• 8m Jbpori (/ At nMA AmtUOon/wr IMS.
765]
THOUSONS UETHOD.
376
I The kinetic f acr<;^ of the system la
The firet term, \Ly^, expresses the enei^y of the current lui
dependiD^ on the coil itself. The second term depends on the
mutual nction of the current itnd terrestrial tDagnetism, the third
on thftt of the current und the magnetism of the saepended magnet,
tJie fourth oil that of the magnetism of the suspended magnet and
ifrrestrial mngrictijim, and the Inst expresses the kinetic energy of
tbc matter eomposing the magnet and the suspended apparatus
which moves with it.
The }»otential energy of the suspended ftpporatus arising from tlie
torsion of the fibre is
Mil
r=
»2*o).
(a)
The electromagnetic momentum of the current is
jj = .p = LyIIg sin 6MG sin («*), («)
Pud if S is the resistance of the coil, the equation of the current i«
(5)
(8)
dtdy
, Mnce $ = iot,
(B+JL^) y~ nso)C09$ + 3fG{u—i))cos{6—it>)
765.] It is the result alike of theory and obeerratton that 4> ^hc
azimuth of the magnet, is subject to two kindu of periodic variations.
One of these is a free oscillation, whose periodic time depends on
e intensity of terrestrial magnetism, and is, in the experiment,
seconds. The other is a forced vibration whose period is
that of the revolving coil, and whose amplitude is, as wu shall
insensible. Henoe, in determioiDg y, vtv may treat ^ tut
eennbly constant.
We Uius 6nd
Hff*
S^ + L'^*
{Scos0 + LoiBin0)
MG<.
WTL*^
{RcQa{6—^)^lua.ix.{$—^)),
+ C/
{9)
The last term of thix exprcsuoa soon dies away when the rot«
ioQ is continued uniibrm.
376
UHIT OF BESiSTAVCK.
[766.
The equation of motion of the Huapended magaet is
d^ T dT
dld^
dV _
(10)
wbeooe A^—MQyQOi{0~^)+MII{faa4t^r(^—a)) = 0. (11)
Substitutiog: the value of y, and arranging the tcrmti ncconling
to the functions of multiples of 0, then we know from ol>*erTati'
tbat
^ = ^„ Jrbe" cos 11/+C cos 2 (tf— (3), {\i
where ^^ ifi the mean value of ^, and the second term expnwi
the free vibrations gradnally decaying, and the third the forced
vthrationa arising irom the variation of the deflecting current.
Soginning with the terms in (ll) whiob do not involve 0,
which must collectively vanish, we find approximately
g^^ J77^ (^ cos *o + /,» sin *,) + 0J«
= 2J/Zr(8in4o + r{^o)). (1$)
Since L tan </><, is generally small compared to 6g, th« »olati(m rf
tlic quadratic (13) gives approximately
3
ced
1
A =
.^^.ec,,^(l^,)Un.*.
9U
Og^Gi
I
(^Vdf')'*.} (M)
If we now employ the leading term in this expreesion in cqua^
tionR(7), (8), and(l]), weshall find t at the valitoof « in cciuatioD
(12) is A/ ~7~ scc^o That of c, the amplitade of the force^^
vibrations, is i^sin^g. Hence, when the coil makes m«Qr
revolutions during one'frcc vibration of the magnet, Ihc amplitnde
of the forced vibrations of the magnet ie very email, and we ma;
neglect tJie tenns in (1 1) which involve c.
766.] The roBistanco is thuK determined in electromagnetic m<
sure in terms of the velocity w and tb« deviation ^. It is n<
neceeeary to determine H, the horizontal tcrroitrial magnetic foreej
provided it remains constant during tbc viperiment. J
noi^
M
To determine js we must make use of the suspended magnet to
deflect the magnet of the magnetotneter, as descrilxd in Art, *54.
lu Ibis experiment M should be aaaall, 00 Uut thia correcUou,
comes of Bcoondaiy importance.
JOI'LB's METUOfi.
S7T
For the other corrections required in this experimeDt see th«
1 ^the BnVuh AteoeiaHon/or 1863, p. 1B8.
■Joale'a Calor'meiric Method,
767.] The beat geDeratod by a current y in possinj; through ft
leoDductor whose resistance is £ is, by Joule's law, Art. 242.
i = jfRfdt. (I)
where J is the equivalent in dynamical measure of the unit of heat
employed.
Hence, if It is constant during tlie experiment, its valae ia
ff.lf
This method of determining R involves the determination of i,
[the heat generated by the current in a given time, and of /, the
square of the stren^rth of the current.
In Joule's experiments *, A was determined by (he riae of tem
eniture of the water in a vessel in which the conducting wire was
It WHS corrected for the effects of radiation, &c, by
ltd experiments in which no current was passed through the
wire.
The etrength of the current was measured by means of a tangent
IvBOomotcr. This method Involves the determinatioa of the
intensity of terrestrial magnetism, which was done by the method
in Art. 457. These meiisurt^ments were also tested by the
. weigher, described in Art. 720, which measures y* directly.
be most direct method of measuring i y^ dty however, is to pass
the current through a selfacting eleotrodynamometer (Art. 725)
vitJi s scale which gives readings proportional to y, and to make
Itiie observations at equal intervals of time, which may be done
'approximately by tatinjj the reading at tlie extremities of every
vibration of the instrument during the whole course of the esperi
aent.
• Report of ih* JlritUb JMtortatien far IS67
CHAPTER XTX.
ooiErXKBON of the electrostatic with tub electro*
magnetic ukit8.
Determination of the Nnmher of EUdroitaiic Units of Etectrieitg
in one Eleelromagnetic Uitit.
768.] Tub absolute magnitudes of the electrical aaits id Wtb
^steins depend on th« units of length, time, nod maM whicit we
adoptj and the mode in wliich thoy tkpcnd oh these units i«
different in the two systems, so that the ratio uf the (tiectrical nnitt
will be expressed by a diSeront number, ncoording to the different
units of length and time.
It appears from the table of (limenKions, Art. 628, that the
mimlK'r nf electros t^itic unite of electricity in one electromaignetic
unit variett inversely as the magnitude of the unit of length, and
directly as the ma^itiide of the unit of time which we adopt.
If, therefore, we determine a velocity which is ropreeentcd nn*
merically by this number, then, even if wo adopt new unit* of
length and of time, the nomher representing thiw velocity will still
bo the number of electrostatic units of electricity in one <;Iectro>
magnetic unit, iMcording to the new system of measurement.
This velocity, therefore, which indicate* the relation betw<
electrostatic and electromagnetic phenomena, is a nataral quaoti
of definite magnitude, and the measurement of this quantity is one
of the most important resoarcbes in electricity.
To shew that the quantity we are in search of is really a velocity,
we may observe that in the case of two parallel currents the attzac
tion experienced by a length a of one of them is, by Art. G86,
F=2CC'%,
' where C,C *n tine numerioal values of the eorrenta tu
tro I
?69.]
RATIO EXP8E88BD BY A VBMICITT.
379
I magnetic mensiire, itnd b the distance between them. If we make
is=2a, then jp_ CC'_
Now the quantity of electricity transmitted by (hi' current C 'm
the time / is Ct in olectroma^Detio measure, or nCl in electrostatic
meiisure, if n is the number of electrostatic units in one clectro
uagiietic unit.
Let two smail conductors be charged with the ^uantitaec of
electricity tmngniittcd by the two currents in the time t, KoA
placed at n distance r from each other. The rejiolsion between
thetn will be CC'k*t'
F' =
Bib.
H Let the distance r be so chosen that this repulsion is e^ual to the
H&ttractioD of the currents, then
■Hence r=«(',
Bor the distance r must increase witli the time i at the rate n.
Hence ft !b a velocitj*, the absolute magnitude of which is the
—^siunc, what«vcr units we asaiime
H 769.] To obtain a phytrical conception of this velocity, lot uh inuf
■^ne a piano surface charged with electricity to the clcctrosUitic snr*
■ faocdcnsity (t, and moving in its own plane with a velocity r. Thia
moving elwitrificd surface will be equivalent to an electric current
Kvhcct, the Rtrcngth of the current flowing through unit of breadth
ll
■tro
lof the eurJace being av in electrostatic measure, or  trv in elec
magnelic measure, if n ia the number of electrostatic units in
one eIet:troma^netie unit. If another plane surface parallel to the
first is eleotrilied to the surfacedensity a, and moves in the same
direction with the velocity r', it will be equivalent to a second
oarrentsheet.
He electrostatic repulsion between the two electrified surfaces is,
Hby Art. 12i, 2ir<ro' for every unit of area of the opposed surfaces.
The electromagnetic attraction between the trto curreutshceta
is, by Art. U53, 2irMK' for every unit of urea, w and a' being the
surfacedensities of the currents in eleelromagitctic meuKure.
But
1
and w'= irV, so that the attraction ia
The ratio of the attraction to the itrpuldion U ocioal to th»i of
vv' to n'. Il^noe. tiinoc tliu attraction nod tho ropulaiuii are qoan*
titles of the same kind, n muNt W a ()tinDtitj' of ttie same Icitit] mi r,
tliat is, a velocity. If vif. Dow HupjiuKe tlic velocity of esch of Ibe
niDvlQ^ planes to be e^uai to », tlic attraotion nill l>e erjtml to the
repulsion, and there will bo no RMtclianical action between tltem.
Hence we may define the ratio of the electric unita to be a velocity,
such that two electrified surfaoes, moving in the samo direction
with this velocity, have no mutual action. Since this velocity ii
about 288000 kilometrcB per second, it is impossible to nuike ihit
experimeut above described,
770.] If the electric surfacedensity and the \Trlocity cao bo made
so great that the magnetic force is a measnrablo iinantity, we may
at least verify our supposition that a moving I'lcGtrifivd body it
equivalent to an ciccti'ic current.
We may assume * that an electrified sarihec in air n'ottld
begin to discbarge itself by sparlcs when tho electric fonoc S««
reaches the value 1 30. The magnetic force due to the currenLaheet
is 2jr(r ■ The horizontal magnetic force in Britain is about 0.1 T5.
Hence a surface tloctrified to the highest decree, and moving with
a velocity of 1 00 metres per second, would act on » magnet with a
force equal to about onefourthousandth part of the enrth'e hori
zontal force, a quantity which can be measured. Tlic eloetrified
surface may be that of a non>conductiDg disk revolving iu the plam
of the magnetic meridian, and the magnet may be placed «Iom to
the ascending or descending portion of the disk, and protectwl frnm
its eloctroatatic action by a screen of metal. I am not aware Uiat
^^ tlii:< experiment has been hitherto attempted.
^^P I. Comparison <^ Uniii of EfedrieUjf.
W 771.] Since the ratio of the electromagnetic to the electrostatic
I unit of electricity is represented by a velocity, we sliall in future
I denote it by the symbol o. Tlie first numerical determination of
I this velocity was made by Weber and Kohlraosch t.
■ Their method was founded on the measurement of the same
■ quantity of electricity, first in electrostatic and then in electro
I magnetic measure.
I The quantity of electricity measured was the charge of a Lcydei
^^^ jar. It WM nieuured in electrostatic measure as the product of
i
I
i
* Sir W. TlwmBca, & 8. Pme. ovB«pcfat, Atf. alx.
t f MArMfyMMMe MaaA^ntmmtm i and Pogg. Aim. Mix (Avf . 10, H
77I]
METHOD or TTEBES AKD EOHLBACSCn.
381
»
capacity of tho jar into ttic difforenoc of potential of its coatiogs.
The. cjijuwjity of thejiir was ditermiiicii by compiirisDi) with that of
a sphtire BUBjwiidrtl in an oicu s]iace lit n ilistuncc from other
Iwdios. The capacity of hiioIi a sphere is cxjinased in electrostatic
m>>aEiirc hy it« radius. Thus the capacity of tlic jar lany be found
Knd cxpn^^ed a^ a certain leu^.h. See Art. 2S7.
TIic dilTerenci' of the potj^ntials of tJte coatings of the jar was mca
cured by connecting the coatings with the plectrodcs of un electro
meter, the constants of which were carefully determined, so that the
differenee of tJie potentials, E, became known iu electrostatic measure.
By multiplying this by c, the capacity of the jar, the chu^ of
tlie jar was expressed in electrostatic meosarc.
To determine the value of the charge in electromagnetic measure,
the jar was discharged through the ooil of a galvanometer. The
; of the tranfiient current on the magnet of the galvanometer
unicated to the magnet a certain angular velocity. The
maginet then swung round to a certain deviation, at which its
Telocity was entirely destroyed by the opposing action of the
earth's magnetism.
By observing the extr«me deviation of the magnet the quantity
lof electricity in the current may be determined in electromagnetio
easare, aa in Art. 748, by the formula
rhere Q is tJie quantity of electricity in electromagnetic mMson.
^e have therefore to determine the following quantities: —
//, the intensity of the horizonljil oomponeul of terrestrial mag
aetism; see Art. 4 S6.
0, th« principal constant of the galvanometer; see Art. 700.
T, tbc time of a tingle vibration of the magnet ; and
$t the deviation due to the transient carrent.
The value of v obtiuned hy MM. Weljcr and Kohlrausch was
T = 310740000 metres per second.
The property of solid dieleetrios, to which the name of Electric
Absorption has been given, renders it difficult to estimate correctly
'the c«pacity of a Leyden jar. The apparent capacity varies ao
ooidtng to the time which clap^^cs between the charging or dis
charging of the jar and the miAsiircmont of the potential, and tJte
longer the time the greater is the valno obtained for the capacity of
lie jar.
COMPARISON OT irarira.
I
HoDCO, eiuce the tim« occnpiod in o1>Uiniag a reAdiog of tin
elcctroiueter is large in comparison witti l.lie time during which the
diidiargv tUroup^h the galvanometer takes ptace, it is probable that
the estimut« of the dicchargo in electrostatio measure is too h^,l
and the vuluc of v, derived rrom it, ih probably alao too high.
IL » txpretsed as a SeaU(anM.
772.] Two other methods for the detenaination of v load to an
cxprcKsion of its value in terms of the reaistance of a g:iven con
ductor, whieb, in the electromagnetic sjrstem, is also expressed ae a
velocity.
In Sir Willium Tliomson's form of the experimeoit, a ooDstaot
current is made to flow through a wire of great reeistonoe. VIm
electromotive force which urgeit the current through the wire is mea
sured electrostativally by connecting the extremities of the wire with
the eleetrodcH of an absolute electrometer, iVrts. 217, 218. Ute ,
strength of the current in the wire is measured in eloctromagnctic M
measure by the deflexion of the suspended coil of an elcctrodyna "
ntomotcr through which it passes, Art. 72S, The resistance of tJi*
circuit is known in electromagnetic measure by comparison with a
standard coil or Ohm. By multiplying the strength of the current
by thin resistance we obtain the electromotive force in clcctro
niftgnetic measure, and from a comparison of this with the elcctio
stutic mcasun^ the vuluc of v ts obtained.
ThU method rdiuircs the eimoltaneous determination of two
forces, by means of the electrometer and clcctrodyoamometer re
spectively, and it is only the nitio of these forces which appears in
^ the ivsult,
^^P 77S.] Another method, in which these forces, instead of being
^^^ separately mciistirt^'d, are directly opposed to each other, was em
■ ployed by the present writer. The ends of the great rcsistjince coil
I are connected witli two i>arallel disks, one of which is moveable.
B The same difference of potentials which sends the current through
H the groat resistance, also causes an attraction between these diaka.!
■ At the nme time, an electric current which, in the actual experi
H ment, was distinct from the primary currents <>■ tent through two
B coils, fastened, one to the Ijnck of the fixed disk, and Uio other to
B the l>sck of the moveable didc. Tlie current Bows in oppgsit^
B directions through these coils, to that they repel one another. By
B adjusting the distauoe of the two disks the attraction Is exactly
^^^ balanced by the rcpolsion, while at the same time another ohBenrer,
^^^^^^■B^HiM^^M^^i^rt^^H
»
SyiT METHODS OP THOMSOK ASD MAXWELL. 883
ipy BMftDS of a difTtTODtial f^Ivanomef«r with shunts, tlet^rmines
the Tttlio of the primary to tho secondary cumnt.
In this experiment tlio onlymcasiiromeut which most ho roferrod
to a material irtandurcl in that of the g^reat resistance, which must
be detcTmiDed in ahsoluto meiisnre hy comiiarison with tlic Olim.
The other meaaurGmcnts are required only for the deti;rmiiiution of
ratios, and may therefore be determined in tcmis of any arLiitrary
unit.
Thus tile ratio of the two forces is s ratio of eqnnlity.
The ratio of the two currents is found by a comparison of r«i(iHt>
ancea when there is no deflexion of the dtiferential galvanometer.
The attractive foroe depends on tlie stguare of the ratio of the
diameter of the disks to their distanee.
The repulsive force depends on the ratio of the diameter of the
coiU to their distance.
The value of v is therefore expressed directly in terms of the
rttsistance of the great coil, which is itself compai'ed with the Ohm.
The value of c, as found by Tliomson's method, was 28.2 Ohms * ;
by MaxwtU'g, 28.8 Ohmsf.
III. Elecfrotlalic CujiacUy in EUetromagnelio Measure.
774.] The capacity of a condenser may he nsctrtaiued in electro*
aetic meaoure hy a compariiiOD of the electromotive force which
liKes the charge, and the quantity of electricity in the current
' discharge. By means of a voltaic battery a current is maintained
'through a circuit containing a coil of gnuit resistance. The con
denser is oharged by putting its electrodes in contact with those of
the resistance coil. The current through the coil is measured by
tlie deflexion which it produces in a galvanometer. Let ^ bo this
deflexioB, then the current is, by Art. 742,
ys=gtiui0,
'here If is the horizontal eompooont of terrestrial magnetism, and
I'C ia the principal constant of the gnlvanometer.
If A is the resistimoe of the coil through which this current is
to flow, the ditTereace of the potentials at the ends of the
■ Btjiori t4 Hn'tUh Anwialimi, 1 fiflO, p. 4S1.
i PhU. TtaM., ISeS, p. 61S ; Mxl £qwn of BriMi Aitoeialion. 18S9, p. tS«.
coupARisoH or oyiTS.
and tlie charge of electricity produced in the condeaser, wluw
cApacitj in electromagnetic m«iieiirD is C, will be
Now let the electrodes of the condenser, und then those of Hw
^Iranometcr, be disLooDocU'd from the circutU, and let the magnet
of the galvanomotcr he hroiight to rest at its position of equili
brium. Then let the electrodes of the condenser be connected witb
those of the gnlvanomeb.>r. A triinsicnt current will flow tbron^fa
the galvanometer, and will oause the ma*nct to swinff to an ex ,
trcmc deflexion 0. Tlien, by Art. 743, if the dtMhargv \a equal to I
the chmjje, }£ f
We thus obtain as the value of the capacity of the coodi
electromagnetic measure
_ T 1 2m\\&
^ ^S ^~ ^^ ^ ■ •
T It tan <t>
The capacity of the condenser is thus dctennined in terms of th«
following quantities '—
T, the time of vibration of the magnet of the fralvanometer &om
rest to rest.
J?, the resistance of tho coil.
6, the (Txlreme limit of the ifwing produced by the discharge.
0, the conKtant deflexion due to the current through the coil R.
This methoil was employed by Profeinor Flecmiufir Jcnhin in dflt«r
minin^ the ca[)acily of condenaent in electroma^ctic meaaorc*. I
If c be the capacity of the same condenser in electroetatic meo ^
sure, as determined by comparison with a condeu^er whose capaci^
can be calculated from \U geometrical data,
T ism\0
T!ie quantity v may therefore be foond io this way. It depends I
on the determination of R in electromagnetic measure, but as it "
involves only the square root of i?, an error in this determination
will not alTect the value of f so mnch as in the method of Arts.
772, 773.
Inlrmiilcnt Current.
776.] If the wire of a batterycircait be broken at any point, and
• fi9M ofStkiA Jbtotiatiim. ie«7.
I
WtPPH.
S85
tli« broken eii^a connected with tti« oU'c^trodot of it coiiKlpn«cr, tin
I current will flow into the condonwer with a «treng:th which dimin
isliee as the diSerenoe of the pot«ntiuU of tbv vondcneer incrciucs,
so t]iat when the condvnser luw roceivcd the fiiU chikrj>o com
spondin^ to the electromolivv force acting on the wire the current
I oeBMfl ontirelr.
If the electrodea of the condeiuvr btc now disconnected from the
ends of the wire, and then again connc^et^Hl with them in the
■i reverse order, the condenifer will <Iiii«tiar^ itself thiwigh the wire,
and will then become roe)iurg«d in tho opposite wnj, eo that a
traosietit current will flow through the wire, tlie total quantity of
* which is equal to two chitrgCH of the condenser.
By menns of a piece of inechnnijim (commonly called a Commu
tator, or KJppe) the operation of revcning the connexione of tb«
condenser can he repeated at regular int«rvaU of time, each interval
being equal to T. If thia interval is sufTiciently long to allow of
the complete diw^hargo of the conden^r, the quantity of electricity
tnin«nitt«d by the wire in each interval will Ik* 2 KC, where E k
iho electromotive force, and C is the ciijnwity of the condenser.
■ If the magnet of a gj1vunometerinclud«<l in tlic circuit is loaded.
BO as to swing eo slowly that a great many discharges of the eon
I denser occur in the time of one free vihrntion of the magnet, the
Bnccession of discharges will act on the magnet like a steady current
whoee strength is 2/.'C
If the condenser is now removed, and a resistance coil substituted
for it, and adjusted till the §teady current through the galvano
meter produces the same deflexion as the succession of discharges,
and if i? is the resistance of the whole circuit when this is the case,
m. ^^1^. Ml
■ yUe mt
B motion to
R
R =
T
T
id P")
We may thne compare the oondenser with its commutator in
* motion to a wire of a certain electrical rcBistanec, and we may make
use of the different methods of measuring rcBistance described iu
Arte. 315 to 357 in order to determine this rcsisfanoo.
776.] For this purpose we may suhstitutv for any one of the
wires in the method of the Differential Galvanometer, Art, 340, or
in that of Wlieatstone's Bridge, Art. 347, a condenser with it* com
mutator. Let na suppose that in either case a sero deflexion of the
VOL tt. C C
COWPABISOS OP TTNnS.
[777
'a
galvanomoter lias bcvn ol>t«tn(i], firsl with the cotvleoaer and com
tDutfttor, and then wiUi a ooil of rvtirtancc R^ io its place, tl
tli« qtiautity —^ will bo measured by the renstaooe of tlio circui
which thf coil /?, forma part, aiid which in oom))l(rt«d by the n
tnaiodcr nf thfi oonducting system including tho battery. Uma
tli<! rcBistanw, If, which we have to ealcuinte, i* wjuaJ to 7?,, tint
of the nKiittAnce coil, together with H^, the re^isltuice of th«
mainder of the system (including the battery), the extroinitjef
tliv ri'»istiiDCe coil being taken as the electrode* of the oyKtvtn.
In the cases of the diflercntinl gahanumet«r ii»d Wbentcl
Bridge it is not neoeaaary to make a second experiment by su
tuting a resislanoe coil for the coudcnscr. The value of tlic Tv»i
auoe required for this purpose may be found by calculation (nta
the other known resistances in the system.
Using the notation of Art. 347, and supposing the condenBTf
Aod oommutntor substituted for the coaductor JC in WheatMt<me'ii
Bridge, and the g»1 vniKunctir inserted in 0.^, and tltal the dtiflt'xion
of the giitvuuomctcr is xcro, (lien we know that the resistance of
coil, which placed in AC would give s zero deflexion, is
The other part of the resistance, R^, is tliat of the sj'^tem of co'
ductors AO, OCf A£, BC and OB. the points A and C being con
sidered as the electrodes. Hence
g __ ;i(o + a)(y4a) + w(y + a) + ya(g+a)
* (c + a)(y+o)+j9(fl + a + y+a)
$
In this exprettsiion a denot«s the internal rvviKtitnee of the bftttery
and its connexions, the valne of whieh cannot be detormined with
crrtuinly; but by m;)king it small comiwrcd with the other re«i»t>
ances, tliis uncertainty will only uligbtly afTiwl the value of &^,
The value of the capacity of the condeniier in eluclromagn
measure is "p
777.] If the condenser Itas a lai^ capacity, nod tlie commutator
is veiy rapid in its aotioo, tlie conden«er may not bo fully discbai;g^U
The equation of the electric current dorii^ t^lfll
4
at each reversal,
discharge is
« + ^C^?+.jSC«0,
(6i
where Q b the charge, V the capacity of the cotulenBcr, It. tl
WITH COIL.
387
reoistaDoe of the rest uf the system betne«n the vlt^ctrodos of the
ooDdoiiser, and E the eleetrouaotivo force due to the cuniivxiouH
' with the battery.
t
Hence q = {Q„i BC)e'^EC, (7)
whore <?„ is the initial value of Q.
If T i& the time during which eontaet is maintained during each
dtBchorge, the quantity in each discharge is
q = 2Ec
it 1^
(«)
1 + ^ «.c
By makings e and y in equation (4) large compared with ^, a, or
tt, the time represented by Rfi may be made so email compared
with r, that in calculating the value of the exponential expreseion
we may use the volne of C in equation (5). We thus find
where B, is the reeistance which must be substituted for the con
deoficr to produce an equivalent effect. R^ ia the resistsnce of the
reat of the system. T is the interval between the beginning of n
dischargu and the beginning of the next discharge, and r is the
duration of contact for each discharge. We thus oUtaiu for the
] corrected \aIue of (7 in electromagnetic measaro
1— fl *» '^
IV. Gmparitm of t/ie Efeetroatalie Capaeiiy <f a Cond<m<r with
lie EUetromagnftk Capacity of Sflfinductitm of a Coil.
_ .J76.] If two points of a conducting
circuit, between which the rc«iHtnace is
S, are connected with the electrodw of
a condenser whose capacity is C, then,
when an electromotive force act« on the
cironit, part of the current, instead of
passing through the resisluncc R, will
be employed in charging the condenwr.
Tie carrent through R will tlierefore
rise to its final value from iMtro in a
I gradual manner. It appears from the
athematical theory that the manner in which the current tJirough
cea
u
Tig.t9.
3S8
cosTPABisos OP fjwra.
It rises from ziyro to its fina] value is czprcMcd by a fononla U
exactly tbe sunv IcinO u« tbat which vxpri^se* tbo value of a cur*
rvnt urfreU by a uoQKtunt electrroniutive (otou tbrougli the doU of aa
el«ctromii{^nct. Henoe vie may plncc a condenser uad nn electn>
tnftgnetr oti two opposite mtMnbers of Wheatslone'ii Hrid^ in exh
a wuy tUal. the current thraogh the galvanometer in alMniys xen,
even lit Uu iuptmit of making or Itrealcing tlic buttery circuit.
Id the 6giire, let P, Q, R, $ be the resistaDces of the four mcin
bere of Wbeatetone'e Bridge respectively. Let a coil, wIiom &» Hi
cieut of 8clfinduetion is L, be made part of the member AI/, wImv
resistance is Q, and let the electrodes of a condeneer, whose capacity
is C, be eonuected by pieces of small resistaoce witb tbe iMiinU F
and Z. For tlio sake of simplicity, we shall assume that there in on
current in the ^galvanometer G, the electrodee of which are con
nected to F and //, We have therefore to determine tbe condilioo
that the potential at /' may be equal to tbat at II. It is only when
we wish to estimate the degree of accuracy of the method that we
require to calculate the current through tbe galvanometer wh^a
this condition is not fulfilled.
Let « bo the total quantity of electricity which haa pancd
through the mcmlicr AF, and e that which has passed through FX
at the time /, then x—: will be the charge of the condenser. Tlie
electromotive force acting between the electrodes of tbe eonden«ci
is, by Ohm's taw, R jr , so that if the capacity of the oondenscr
at
0)
Let y be the total quantity of electricity which has passed throng
the member A/l, tbe eicctfomotive force from A toll must be wjtial
to that from A to /', or
(2)
^tie^^dt' ^ dt
Since there is no current through the galvanometer, tbe quantify
vhioh has patu.ed through ///must be abo^, and we find
(3)
Substituting in (2) the value of r, derived from (t), and com
paring with (3), we find as the condition of no current through Ibu
galvanometer
m
CONDESSER COMBINED WITH COIL.
889
r
The conilition of no 6nni cumeiit is, as in tlie ordinary form of
I Vh^aUtone's Bridge, Qg _ $p_ /^\
The condition of no current at making and breaking the battory
oimexioD ia /:
•^=XC. (6)
Here V. and AC arc th« timccoiisUnts of the merabem Q and R
respectively, and if, by varying Q or H, we can adjust the members
of Wheatstone's Bridge till the galvanometer indiciit«M no iiirrent,
cither at making and breaking the circuit, or when tliB current ia
steady, then no know tliat the timeconstant of the coil is equal to
that of the condenser.
The coefficient of wif induction, /, ojin be determined in electro
magnetic mensiire front a compiirison with the coefficient of mutual
induction of tiro drcait«, whose geometrical data are knowD
(Art. 7oG). Tt is » (juantity of the dimensions of n line.
The eajmcity of Ihtj eoi)den:«er can be determined in ilcctro^latic
mcftKUrc by comiiarison with a condenser whose geometrical duia
are known (Art. 229), This quantity is also n length, e. The vioc
troinagnctic mcaoure of tiie capacity is
c=4.
(')
I
■ Subatitnting this value in equation (C), we obtain for the value
where e is the capacity of the condenser in electrostatic measure,
»/. the coefficient of selfinduction of the coil in electromagnetic
meaiiure, and Q and II the resistances in electromagnetic mca«uru.
ITie value of r, as determined by this method, depends on tho
determination of the unit of resistance, as in the second method.
Arts. 772, 773.
K
V. QmliinttfJoH of tie BUcfro^attc Capacity (f a C<md<nur teitk
ike EUeffomagK^ie GipaciSy of Srlfindtietitni of a CoU,
779.] Let C he the capacity of the condenser, the *urfaces of
which are connected by a wire of resistance R, In thii: wire let the
coila L and L' be inserted, and let L denote the xum of their ca
pacities of selfinduction. The coil L' is bung by a bitiUr snspen
sioa, and consiste of two coils in vertical planes, between which
390
OOWPARlSOlf Of UKITS,
[77^
pn^cfi a virticnl axis which carries the tnagnet 3f, the axis of wbiek
roTolvog in a horizootal [ilaae between the coils L'//. The coil I
hoe a large coefficient of yclfinductian, and is fisetl. The eik
pended coil L' is protected from th*
currents of air caused by fbe rota
tion of thv nuLgitct by encloeing tb
rotating yaii» in a hollow esse.
Till,' motion of the moguet
ctirrent« of induction in the ^oil, and
tiltHe are acted on by the
so that the plane of ike si
coil is deflected in the direction of
the rotulion of the magnet, lirt
us determine the Klrcngtli of tic
induced currents, and the mitgnit
of the detk'xioD of the 8wpei>«]<
coil.
Let X he the charge of electricity
on the npper Btirface of the condenser C, then, if £ is the electro
motive force which produces this charge, we have, by the theory of
the condenser, g — crjf. (j
Hg. •}*.
tic
51
We have also, by the theoiy of electric currentR,
1
(«)
where 1/ is the electromagoetic momttntum of the circnit I/, when
the axis of tlie magnet is normal lo the plane of the coil, and in
the angle between the axiei of the magnet and this normal.
The equation to determine x is tlierefore
aP at at
If the coil is in a position of cquilihrinm, and if tlic rotation
the magnet is uniform, the angular velocity being a,
e = n(. (4)
The exprowion for the current consists of two parte, one of wbidi
is in<te>cndei)t of the term on the righthand of the equation,
iiod diminislieM acoording to an exponcutial function of the time.
The other, which may be called Hiv forced curroot, depends entirely
on the term in 0, uud may he written
COSDESSEE COMBINED WITH COIL.
Sdi
Finding llie values of A ami £ by »uWtilutiou in the wiUJitton (3),
pe obtain JtCnco9d(lCf.n^«a0
Tb« momtrnt of thv force with which the magnet acts on (he coil
V, in wliicli the currvnt m it flowing, heinf* the reverse "f that
KDg on the mu^oet Ibe coil bcini* by supposition fixtd, i«
1nahy
Intpgrating this expression witli respect to ( for one revolution,
kod dividing by the time, we find, for the mean value of 0,
= l
(«)
If the coil has n considerable moment of inertia, its forofd vibra
tioaa will be viiyemiiU, and its mean deflexion will be proportionui
0.
Lut i). i>j, D^ be the observitl deflexions corrcfrponding to nn
iikr velooitius n, , n^, Wj, of the mngTiet, then in gcnenil
P~ = (^C£ny + R'C\
(9)
[where Pisa constant.
Eliminating P and R from three equations of this form, wo find
IC'L*
^K'V)+J^(V«.')+J^{VVJ
«»' ">*«.* »i
(10)
If «ig is «ueh thnt CLti^' = 1, the valne of ^ will be a minimum
Ifor this vnliic of ii. The other vnlucs of a should be tnken, one
Igrvater. und the other Ivss, than i«^.
value «f Cf/, determined from this c^tmtion. is of the dimca
I of the Kqitare of a time. Let u« call it t',
IF C, he tlie elect rusLitic measure of tho capaeity of the oon
r, and L„ the electromajrnetic measure of the selfinduction of
» coil, both C, and X„ are lines, and the product
C,£. = ^C.L. = v'C^L„ = v't'; (H)
tr= —s—i
UHl
(12)
whew r* ifl the value of C*£*, determined by this experiment. Th«
k
392
[7^
experiinetit here so^i^fibtd as n method of drlermmin^ r is of tlw
aatntt luitur^ as odo dcecribed by Sir W. R. Grovp, Piil. H/^^
March 181)8, p. 184. S«e also rcmurks on tliul *.xi>unmvQt, by
ynteat writer, in the ntimbcr for M»j I8G8.
«
'J
VI. Elffctro$ia(ic MeaMrettttttl of ItetiifttHCf. (See Art. 385.)
780.] Lt't u condenser of capactly C be (lisctiarged thron^b a
conductor of remittance R, then, if « is tlie diarge at aujr iostant,
Hence » = Xf,e *'.
If, hy any method, we can make contact for a tihort time, whJvli
i» accarat«ly known, so as to allow the current to flow throug'h
conductor for the time I, then, if Eg and £, are tlie readings of .
electrometer put in connexion with the condenaer before aod
the operation, IiC(iog. S^ log. A',) = t. (3)
If C is known in electrostatic meamirt! as a linear <)uantity, X
may be fonnd from this equation in electrostatic meawirv an the
reoiprooal of a velocity.
If jK, i» the numerical value of the resiiiUnco as tlius detorniinod,
aocl Ifm ^he numerical value of the resistance ia elect roma^etie
measure, ff ^^
Since it is noceesary for this exporimcnt that S shoold be very
great, and since H most be small in tho electroma^etic experi
ments of Arts. 7li3, Sic., the es)criinei)t« must be made on e«i*Tuti)
conductors, and the rceistaneo of these uoodugtors compared by tho
ordinary methods.
CHAPTER XX.
BLBCTBOMAONETIC TBEOBT 0? UOHT.
■ be at
^ftttrib
78).] Ik WvithI parts of this trentie? aa attempt lias beffn rand«
explain «U>ctroniii^i'tic ph«nomeiia by means of mfichanicRl
Ktion trntiMiiittAiI fr«m one tMnJy to another by meimii of n mLtUiim
oceupjing Uik ajiaw between tiwm. The UDdiilutory theory of liglit
alHO aiisuincs the existence of a medium. Wo have now to Hhvw
tliat the pro[ierttes of iJic clectrom noetic medium are identical
with thoBe of the luminifiTOti* mt^diiim.
I To fill all space with a new medium whenever any new f\te
nomenon is (o be eitplniiiiil ih by no mcitOK philosophical, but if
Uie study of two diQeriMit branches cif ecienoo bax inde{>endently
Buggeated the idea of a medium, and if tlie propcrtitw which nauat
be attributed to the minliiim in order to iKVount for eleclro
etie phenomena are of the Hnmc kind ns tlioae which we
ttribute to the luminiferous medium in order to account for the
phenomena of liffht, the evidence for the physical existence of the
medium will he considerobly stren^hened.
But the properties of bodies are capable of quantitatirc meaflnro
meoti We tJmefora obtain the numerical value of oome property of
the medium, sueh ae the velocity with which a disturbance is pro
pagated thron^h it, which can be catculnted from electroma^iictio
expcrimentfi, and aleo observed directly in the ease of light. If it
should be found that the velocity of propagation of electromagnetic
dietorbancee is the same as the velocity of light, and this not only
fn air, but in other transparent media, we shall have strong reasons
for believing thiit light is an electromagnetic phenomeuou, and the
uombinatioD of the optical \^ ith the electrical evidence will prodiic*
a conviction of the reality of tJte medium similar to that which we
obtain, in the case of other kinda of matter, &om the combined
evidcBCtt of the Kcmn's,
894
EUCTBOUAGHETIC TUEOBY OF LIOIIT.
[7
782.] When light is omiUed, a oerUin amouui of coerg;
exitendod by the luminous body, and if lh« liglit im aUxorbcd
another body, tliiit body heoom^a heated, ahewiD^ tiuit it hu
ceived energy from without. Durio}; the interval of time afi«r
light left the first body and before it reaebed the second, it
have cxist«d as oner^* in tho intervening space.
According to the theory of emiFsion, the tranMmieHion of on
U effected by the actual tranHfercnec of ltghtcor}>nKCu1ca from
luminoue to the illuminiited body, currying with tlivm tlieir ki:
encrjjy, tfsctber with any other kind of energy of which they
be the rt'cepfaclcs.
Accoi'<lin<7 to the theory of undulation, there is a material medim
which fills the space between the two bodies, and it is by the actiw
of contif^oiis parts of this medium that the energy i« paaeed on.
from one portion to the next, till it renches the illuminat«d body.
The lumiuiferous medium is therefore, during the poMRgc of ligbt
through it, a rcceiitacle of energy. In the uudulatory thtwr}', »
dvveloi>ciI by Huygciis, Frcsncl, Yonng, Green, Sec., this euerg;'
is suj)])OKcd to be partly potential and purtly kinetic. Tlie potential
energy iK Hup[)u«ed to bo due to the distortion of the eltmcnlMj*
portions of the medium. We must therefore Tiegaid the mediutn h
ebisliv. The kinetic energy i$ jitippoMHl to be due to the vibratory
motion of tlio nicilium. We mu«t therefore regard the medium a *
having a 6nite density, ^M
Id the theory of electricity and mtgnettmn adopt4>d in llii^^
treatise, two forms of energy are recognised, the e)eetruiitatie antl
the electi'okinvtic (see ArU*. 630 and 636), and tlieae are suppotHul
to bare their mat. not merely in the electrified or magnetized
bodies, but in every part of the surrounding space, where electric
or magnetic force is observed to act. llenoe oar theory agrees
with the unilnlatory theory in assuming the existence of a medjiun
which is capable of becoming a receptacle of two forms of energy*.
783] Let ns next determine the conditions of the >ro]ngMtion
of ail electromagnetic disturbance through a uniform medium, which
we shall suppose to be at re«t, thut is, to have no motion except th:
which may bo involved in cloctromagiKtic distnrbftnce*.
J
* ' Far my own )>*rt. coiialdnriiii; lliK rHtilioli •>t k vicunin (o Om majpieUc fi
sad lliQ KuUHral cltkmrtiir i>f iiiunistio phuianuak •xt(<rti>il to Ihv inaMI, I un
inclinad to tliv antiiiu tliitt ia tha truumiBiiMi of tho forca Ihera u asdi wi tetiaa,
•xMfittt la the ma^rt. thui that cbn •ffocto a(« iuanl;r aHtxi i na sad itfmUaa ■> >
dUtWNo. Stub sn action nmy bu a (nuutluD ^ Uie nlbcr ; W U U a»t M ■!! nnlikei*
Ihul, IT iian ba ka Mthrr, It ilioiilil lixvn oUiw met tbu Ktiofilf ik* cuiiBjiataoe td.
ndiMSuiu.'— PmmI>^'b I'^f^malal /ionircAct, 9»i6.
fa
jS^.] PBOPAOATION OF RLKCTROMAOSETIC DIPTl'RBANCES. 395
Lot C be the specific conductivity of the medium, K its speciGe
[capRcily for cloctrostjitic induction, and n its mai^netic 'pcnne^
ibilily.'
To obtain the general <>>{iiut Ioiik of cIcotromafTDctic disturbance,
Ifte shall exprcM the true current G in terms of the rector potential
fil and the electric potential 4*.
The true current 6 is made up of the coniltiction current j^ and
the variation of the electric displacement Ti, and since both of these
I depend on the electromotive force @, we find, as in Art. 611,
But since there is no motion of the medium, we may express the
:electromotive force, as in Art. 599,
I g=_aU1'. (2)
H«,o. 6 = (c + ±;.)(f+v*). ■ (3)
I But we may determine a rehition between S and SI in a diOVrent
way, as is eIicwd in Art. 616, the equations (4) of which may W
twritten 4ir*iS = 7»?I+ W, (4)
. . iiF dG dii
Combining equations (3) and (4), we obtain
^(4«C+A'^)(2^ + V*) + V*a + V/=0, (6)
rhioh we may express in the form of three equations as follows—
,.(4
(0
dt^^dl ^ ds
Ihwp arc the g<'ncral M{n»tions of elwtromagnetic di:sturbaiieoa.
If we difTerentiate these equations with reiipect to jt, y, and t
recpeetivcly, and aM, wn ohlain
H^ If tltc medium ia a nonconduct<Mr, C = 0, and 7'^, whieh is
^'proportional to the volnmodenjtity of free electricity, b indejiendent
gf t. Itence J must be a linear function of /, or a constant, or zero,
and we may therefore leave 4 and '\ out of account in concddering
eiiodic disturbances.
396
BLKCTROMAGSETIO THEORT Of LIGHT.
{.7h\
(»)'
Propanatioit ^ Unduiatioiu in a Non'<:imdtielittg Medium.
7&4.] Id this case, (? =: 0, And the cxiujitions become
The etiuatioDs in thia form ar« similar (o those of th« motiou of
an elastic solid, and when the initial conditions an* {^iven, tbp,
solution cao be exprtwed in a form gircn by Potsson *, uud appliedl
by Stokes to the Tlieory of Diffraction t
' (10)
Let us write
r=
^rz
dF
If tb« vnluea of f, O, ff, and of j .
., t T7 are giT«a at erVjl
point of space at the epoch (/ = 0), then wo can determine theii]
TaluM nt nny eitV>sc[ucnt time, /, as folloWB.
Lot be the point for which wo wish to detrnninc the valae
of /"lit the time /. With as centre, and with radius ft, dtwrilie
a sphere. Find the initial value of /'at every point of tlie opluriral
surfaa.', and take the Huan, F, of ail thofo values. Find aivo tbo^j
initial value:* »f
rff
the mean of these values be
at cvcrj' point of the spherical surfiico, and lebl
r«
Then the vnlue of /'at the point 0, at (he timo ', ix
'=i(K)^'f
4i
dt
Similarly
=
s(^')'f
^=^(^0^4'
785,] It appears, t]>eTief<>re, that the condition of thing* at tb€
point at any instant de{iends on the c<»idition of thing* at
distance Vt and at an interval of time t ptevioosly, so that anj
disturhance is propagated through the medium with the velotity
Let us suppose that when t is zero the quantities ?I and 31
■ MtM. dr TArtot.. Una. Ul, p. IM.
■t UamhrUtgt rnmtattiotu, vol. ix, {>. 10 il830>.
nmk
rs?]
■nttocnr of iTGrrT,
escevt tritbin a cerUin Rpace S. Then tKeir vu1uk« at at
tlie time ' will be usero, unless the spherical surfiK'e duHi'tilied aliout
lO as centre with Tsdius Fl lies in whole or tn part withia the
I ^. If is out«ide the space S there will be no disturbance
■t until n becomes equal to the shortest distanee from O to the
B)a«e S. The disturbance at will then beg'in, and will go on till
\Fl is equal to the {rreatest distance from to any part of S. The
Idisttirbancc at will then coase for ever.
786,] The qtiantitr F, in Art. 784, which expresses the velocity
of propa^tion of electromagnetic disturbances in a nonconducting
Kxnediuui is, by equation (9), e<iual to
■/Kl
If the medium b air, and if we adopt the electrostatic system
it meaanrcmeat, K = I and fi = — , so that T = », or the velocity
of propagation is numerically equal to tlic nninber of electrostatic
■unite of electricity in one electromagnetic unit. If we adopt the
eUctromagnetic system, A'= y and (4= I, so that tlie equation
r= II ie still true.
On the theory that light is an electromagnetic disturbance, pro
pagated in the same medium through which other electromaffnetio
action* lire transmitted, F must be the velocity of light, a quantity
Illio vuluv of which haa been estimated by several methods. On the
other haud, v is the number uf electrostatic unita of electricity in 0D«
electionuif^ctic unit, and the metliiidG of determining this quantity
bave been described in the lust cha[>ter. They are quite inde
liendcnt of tJie methods ol' iindiug the velocity of light. Hence
the agrocmont or disagreement of the values of Tand of v furnishes
a te«t of the electromagnetic theory of light.
787.] In the following table, the principal results of direct
observation of the velocity of light, either through the air or
through the planetary spaces, are compared with the principal
results of the comparison of the electric units :—
Vslvdtj oT Li[)it (iiilitrw par Hoomt],
Kzeau 314000000
Aberration, &c., and )
Sun's Parallax J'"
^'oDcault 298S6O00O
. 308000000
FUUo at Eloctric Vnitn.
Weber 310740000
•Maxwell... 288000000
•Thomson... 282000000
[Tic nperimenU of (b« ComiDiUv* Of Um Dril^h AvMUtion fur thq Jcttr
(ID of tlw uiut of raiMtanM ia abtolut* UMuun ban) natndj bMO irpeatoJ
89e
KIECTROMAOITETTC THEORT 07 LtOHT.
It is maniPcgt tlmt the velocity of light and the ratio of (Jw*
nri> <uim(itii« of tho samo order of magnitude. Neither of i
can 1)0 »iul ti be detcrtniacd us yet with such a de^ee of ac
M to CDRhle tis to assert that the one is greater or Ices than
other. It is to Iw hoped that, by further ex{)criinents the relttk
between the mitgDiiudcs of the two quaiititici) niuy bo more
CQnt«ly deLermininl.
In th<> meantime our theory, which assertit that thcso two qoaa*
titles are equal, and ossig^DH a physical reason for thin cjiinh'ty, ii
certainly not coutnulicled by the corapurison of these results rodi
aa they are.
788.] In other media than air, the velocity f i» inversely pro
portional to the (iquarc root of the product of the diflcetric and the
ma^etic tnduotive eapaciliex. According to tlie undulutor^' theoij*,
the velocity of light in difTcrent media is inversely proportiooa] to
their indioee of refraction.
There are no transparent media for which the magnetic cnMicilj'
diSers fi'om that of air more than by a very amall fraction. Uence
the principal part of the dilferenee between these media muat depend
on their dielectric capacity. According to our theory, therefore,
the dielectric capacity of a transparent medium should be equal to
the square of its index of refraction.
But the value of the index of refraction is different for light of
different kinds, being greater for light of more rapid Tibi»tioi)&
We must therefore select the index of refraotion which correeponda
to waves of the longest periods, because theae are the only wavea
whose motion can be compared with the slow processes by which
we determine the capacity of the dielectric.
789.] Tlie ooly dielectric of which the capacity has been hitherto
determined with sufficient accuracy is paraffin, for which in thu solid
form MM. Gibson and Barclay found *
K= 1.975. (12)
Dr. Gladstone has found the following values of the index of
refraction of melted paraffin, sp. g. 0779, for the lines A, D and i?: —
T«inpwMuM
54 '€
A
1.4306
D
1.4357
B
1. 4499
a7'C
1.4294
1.4343
1.4493
ti Dr. KchiuM' Kt til" CkVMdliA l^ihunl/nj. wMk tb*
, . , , , ^t" eont, (mailer thui it *<■ iutuiiifed Uihr. Ttw afbet
IBpOk tb iattt of Uie t^eeuiu nnlu ■■ fi*ca bj M«i*rdU uul Tbanuna woalil
mini's tibta lij t»t pe> cnnt.]
• mi. Tnuu..lS7l, p. SJi.
1
?90.]
300
b^
I
which I fiii^ ttiat iim index of ttfnwtioii for wave* of iiirmiU
ig^h w<iul<l bo ubotit t.432.
e square root of K is l.*05.
c (linVrenot bct.wittin thoMO nunibeni in greutiiM thttn can be ae
>uated for by errors of observation, and shew^ Hint our thoorit* of
e structure of bodies must be much improved before we .can
deduce their optical from their electrical properties. At the same
tim«, I think that the agreemeDt of the numbers is such that if no
greater discrepancy were found between the numbers derivetl from
ihe optical and the electrical properties of a considerable nimiiier of
lubstauces, we should be warranted in coocludinjf that the square
root of K, though it may not \te the complete eipression for th«
iodex of refraction, is at least the most important terra in it *.
I
Pittne ffaoet.
790.] Let us DOW confine our attention to plane waves, the front
of which we ehatl suppose normal to the axis of s. All the quan
ties, the variation of which constitute)) such waves, arc functions
of s and i only, atid aro independent of x aud y. Hence the equa
tions of ma^Ktic induction, (A), Art. 391, are reduced to
dO
'=di'
dF
J. 6=jr' '^ = 0.
da
(13)
br the mat^nctic dirturbanco is in the plane of the wave. This
agrees with what wc know of that difiturbanoe which constitutes
Putting' po, fi,S and ny for a, (i and e respectively, the eqnattona
;l«Gtnc currents. Art. 607, become
rfS
d*F ^
iv^^ = ^= —
dz
da
i^^p= _=__,
(»)
Hence the electric disturbance is also in Ihe plane of the wave, and
if the magnetic disturbance is confined to one direction, ray that of
* pn ■ iMHr md to the Rojal Booitty on Jane li, 18TT, Dr. J. UepUnam givM
i« ronlti or expcrimonla dihIo for tha purpoM of detPnniDtng tile ipoelfic ItiiiiK^tiro
itir* otruioot kind* of sfi^. TticM renilM do not vurlfy tli« ibmndlial ouii
oliMMu urliti) hi III Uio text, liiii valii* of K Mtix in •oir'h ram in "lora at Uial of
Uio aqiiani of tlm r»rnul!v<> !i»li». In * mlmwiurat jnyn to thu KnykI i^ialy, Ra4
on Jkn €. tfisl, Dr. Hopkimixi find* tiuit. if tim dmolc the Jnjci of Tefr.iction bv
«a«H of iiifinhv Unc^, [lien K • fi'w fbr hydnoarLoiiL but for uJuul and <cDvUl>la
dil<K>,.<«i.)
400
tiTOHT.
M
r.tiie electric tlisturbanoc la confiaed to the perpcmliculnr dii
or that of jr.
Dut vre may citlculitte tho electric disturbaooe in auotber
for ify, j/, i UTC the coiiipoDCnt« of electric displacement in a
couilactin^ mt^liiun,
d/ df M
1£ P,Q, R an) tliv compooentit of tli« electromotive force,
A' K ^ . K
•^=n^' '=4^«
^=o^=
(M
and Kince there is no moUoD of the medium, equations (B), Art. 59^1
become
?=
dr
O ^^
R=
du
Uenoe
K d*G
CMuparin^ these r»)ii«e with those ^ven in eqnatioD (11), wv Gwl
d^F .. d'r
dz^^^" de '
^G .. d»G
Q = K,i
dfi"
The Grst and fiec^ind of these equations are the equations of
pagatioQ of a plane nave, aod their wlc
tiou is of the welUkuown form
^
^
(20)
The solution of tlic third equation is ■
Jl^A + Bt, (21)
where A aud Ji are functions of ;, // to
therefore either couatant or varies directly
with the lime. In neither case ean ij
(die part in the proportion of waree.
791.] It appears frotn this that the
directions, both of the ma^ettc und tlie
cloi'tric ditAurlMWOH, lin in the plane u:
lhi> wave. The malhvmaticnl fonn of
diKtiirhattcc therefore, Of^rrnF wi(li thai
the disturbance which concliluleii light,
being transverse to the direction of propagation*
Fig. GS.
BNEBOT kVD STBE89 OF BADIATION.
If we suppose G = 0, the diaturbauoe will correspond to a plniic
slarizi'd ray of light.
The magnetic force is in thia case par&Ilel to the axia of y aiid
1 IP
vaa\ to ~ ^, aiid the «lcclromotivc forco in parulU'I to the axis of
^aod equal to — ^. The magnetic force is therefore in a plane
Ipcrpvitdiciilur to Lhitt which contains the electric force.
The value:* of the ina^Detic force and of the electromotive force
1 itt a ^iveti inhalant at different points of the ray are represented in
rig. 65, for the case of a simple harmonic disturbance in one plane.
This corresponds to a ray of plane polarized light, but whether the
plane of polarization corresponds to tho plane of the magnetic
Lirbance, or to the plane of the ckctric dieturbaoce, remaing to
kteen. See Art. 797.
Energy and Slreta of PaJialion.
79~2.] The electrostatic energy per unit of volume at any point of
ie nave in a nouoonducting mediuu] is
a
(22)
lie cloctrokinclic energy ut the same point is
(23)
8;r inp, ivfi dx
virtue of equation (20) these two expressions are eqaal for a
single wave, so that at every point of the wave the intrinsic energy
,of the medium is half electrostatic and half electrokinetic.
Lct/> be the value of either of these quantities, that i«, cither the
Blectrostatic or the electrokinetic energy per nnit of volunie, then,
^in virtne of the electro^tatic etat« of the medium, thert \<s a Un«ioD
whoMi magnitude is p, in a direction parallel to x, combined with a
priwsnre, also equal to/), pamllcl to j/ and :. See Art. lOr.
iln virtue of the electrokinetic state of the medium there is a
cnKJoD equal to /i in a direction parallel to j', combined with a
treffiure e(]uiil to p iu directions parallel to x and .*, See Art. 643.
Henoc the combined effect of tlie electroKtalio and the clcctro
[inolic rtlrecws is a pramre equal to 2p in the diixction of tho
iropAgation of the wave. Now 1p also expresses tho whole enCi^
R unit of volume.
HencL in a medium iu which waves are propugaletl tlieie ia a
VOL. 11. D d
lOirenC THEORT OF uonT.
pressure in Uie direction nonnal to thv warcKj adiI untni
equal to the enerj:) in nnit of volume.
793.] Thas, if in strong eunligbt the energy of the light «hi«4
fftlle on one eqaare foot ie 83.4 foot pouiidit per second, the mtn
ener^' in one cubic foot of sunliglit is nhoot 0.00000008^2 of a fix*
pound, and the mean prcsauro on a Kiuare foot is 0.0fl0oo00d83af >
ixinnd weight. A flat bodjr exposed to sunlight would cxtenMtt
this pressure on its illnminalwi side only, and would th<'refore tt
lepcUcd from the side on wliith the light falls. It is prol>ablc thil
a mnch gnnt^r energy of ntdiiition might W obtained by mems at
the oonctatrat^d rays of tlw vlectric lamp. Such rays falling on a
thiu oiflaliicdisk, delicately suspended in a Tacuum, might >erhai>t
produne an obstTvabU; mechanical eOect. When a diatarbane* of
any kind eonniits <^ terms involving aiiies or cMioes of angtn
which vary with the time, the maximum energy is double of Um!
mnui energj'. Hence, if /* is the mazimnm electromotive force
and ii the uaximnnt in^>n«tic force which are calkKl into pby
during the propagation of light.
— P^ = — 3* = moan energj" in unit of voluniei,
8 ff Hit
(RP
With Pouillft's data for the energy of siiulight, as quoted by
TliomsoQ, Tmat, H.S.E,, 1854, this givej in electromagnetic m'
ture
P = 60000000, or aboat COO Daniell's cells per rodtre ;
j3 = 0.193, or rather more than a tenth of the hotizoutal
netio force in Britain.
ru 111
PropaffaiioH vf a PlatK Ware in a CiytfalliuJ ifediam.
7M.] To calculating, from data ruriiish<;d by ordinary electr
magnetic esperimonts, the electrical phenomena nhieh would result
from periodic distDrlumoes, millionii of millions of which occur io
second, we have already put our theory to a vert' severe test, eve
when the medium is supposed to be air or vacuum. Itut if
attempt to extend our theory to the case of dense media, we U
involved n<tt only in oil thi' onlinsTT difficulties of molecular tbcorii
but in the deeper mystery of the relation of the molecuUs to
elcctromagnctio medium.
To cviidr tlii'si* diflicidlics, wc fhall assume that in cerl^iin mtilij
the specific cnpucity for elvclTostntic induclion is ditrenml in dif
ferent dlrecttouB, or in otlier words, the ekotric diephiccnieat, it
BOCBLE REFHACTION.
■d of bein^ in the same direction as the; electromotive force, un<l
proportional to it, is related to it hy a system of linear c(UiitioDS
limilar to those given in Art. 297. Il may be shewn, tw in
Lrt. 436, that the systi'm of «oefncients must he symmetrical, so
Itnt, by » projier choioe of axes, the eijuatioiis become
/=T7^^
5 = Aa;«2, i = ~K,s, (1)
pwbere ^, K^, and Kj »re the principal inductive capacities of the
medium. The equations of propagation of disturbances are therefore
d^J^ ^ d^G
tfiG d*G
dx' dy*
dxdjf
dUI _ ,d^F d^*.
= ^■f' ( rf^  d^J
d.'djs
d'F
= ^^^Kdfi~A.dV'
dydx djrdy ■' ^dfi Jydi
dzdx dylf: ~ "*" W/* d£dt>
i
(2)
795.] If /, m, n are the direction cosines of the normal to the
I wuvefront, and V the velocity of the wave, and if
h^my^nz— Vl = w, (3)
knd if w« write f", 0", //", +" lor th« second different ial ooeffi.
[ei«nt« of f\ Q, II, ♦ nspcetively with resjiect to v>, and put
^iM = ::5' ^j'* = ^. A*)fi = 3'
I wh«rc a, h, e are the three principal velocities of propagation, the
' equalionc become
(«a+„E_ ^) r'i»icrnm"~ r*"^ = o,
n
ImF" i (»» + ;» ~) ff'mnir f*^  = 0,
«//"'«nG"+(/»+M»^)fl''r*"^ = 0.
796.] If wc write
1^ «* •«
(«)
F"*^
==£'.
(»)
(7)
' wc obtain from these ecjuatiouH
rV{VF''i^') = 0,
rv(rji"n¥')
Uenee, eitb«r Fss 0, in wbicb caee Uiu w«ve is not propagated ut
Dda
ii
404
ELBCTBOSIAONBTIC TRRORY OF LrOHT.
[79J
all ; or, V= 0, which leads to ILc cciuntion for f givon hy FresofJ ;
or the quantiUn within briirkcU vaninh, in whioh caiw th« vector
whose componcntv arc t", O", H" U nnrnial to the wavofronl and
proportioniil (o th« electric volumedensity. Since the medium is
a non conductor, the electric density at any given point ia con^toiit,
and tlicn^forc the disturlmnee indicated by these equations i» not
periodic, and cannot constitute a ware. We may therefore con«d«r
•V'= in the investigation of the wave.
797.] The velocity of the propa^tion of th« wave is therefi
complett'Iy determined from the equation P = 0, op
I*
M»
I
= 0.
(8J
There arc therefore two, and only two, values of f^ correspoodio,
to n {fivcn direption of wavefront.
If A, fi, r are the direction^oosines of the electric corrent wl
component* are «, v, «p,
then ;A + m;t + iif = O; (1
or the carrcnt is in the plane of the wavefront, and its dinctioi
in the WKvcfront is dfrtcrmined by tlie equation
These equations are identical with those given hy FresncI if we
dftltne the plane of polarization as a plane tlirough the ray per
pUHJicular to the plane of the electric disturhaitce.
According^ to this electromagnetic theory of double refraction the
wove of normal diKturf>anoe, nhielt conctitutex one of the chief
(lifficnlties of the ordinary theory, do<M not exist, and no new
aaaiimption is required in order to account for tite fact that a ray
polarised in a principal plane of the crystal is refracted in the
ordinary manner*.
Refatitm ielwcm S?«irie ConducticUf aitd Opaeif)/.
7^.] If the medium, irutead of being a perfect insulator, ia
conductor whoso conductivity per unit of volame is C. the di
turbonce will consist not only of electric displocemcnte but of
cnrreat* orcondnclton, in which electric energy is transfoTmod into
heat, to that the unduliition is abfiorb^d by the niedinm.
* S«t SlolMa' ' Itcpoet on Double RrfbaUen.' Brii. J^te. Btrot*. ISOi, p. US.
J
■th
SOO.] COKDUCTIVITr ASD OPACITY. 405
ir the distarbaBc« is expressed bj it circular function, we nuy
write /?= «"«.«(«( J.), (1)
for this nilt Katisfy ttie eiiuit)OD
d^f „d^F , „dF
Bridcd y"/)* = p.Kn\ (3j
[ud 2jt7 = 4irjxC7«. (4)
The Telocity of propagation is
and the coefficient of abaorptiOD is
p=2ii^Cy. (G)
Let R be the resistiince, in elcctromagnolic measure, of a plate
whose length \b t, bresidth 6, and thickauss ;,
The proportion of the incident light wliieh vrill be tniusniitted by
tlii« i)Ule will be
799.] Mo»t transparent solid bodies are good insulators, and all
^ood comluc'lors are rery opaque. There are, however, many ex
cq>t!ofU( to the Uw that the o]>acity of a body is the greater, the
grcat^tr ilti conductivity.
Kleotrolytes allow an electric current to pass, aud yet many of
them are transparent^ We may supjmse, however, that in the oase
of the rapidly alternating forces which come into play during the
propagation of light, the electromotive force acts for so short a
time in one direction that it is unable to eOect a complete sejiaration
between the combined molecules. When, during the other half of
the vibration, the electromotive force acts in the oppoiite direction
it simply reverHca what it did during the first half. There is thus
no trne conduction through the electroljte, no loss of electric
energy, and consequent!}' no absorption of Hgbt.
SOO.] Gold, silver, and plitttinmi are good conductors, and yety
hen formed into very thin ptuten, they attow light to pass through
th«ni. From experiments which I have made on a piice of gold
leaf, the icsixtanee of which was deU^rmiiuHl by Mr. Hockin, it
appeuv that its tranitpareQcy is very much greater than is cou
nrtcDt with our theory, unless we suppose that there is less loss
406
ELECTROMAGNETIC THEORY OF UaRTT.
of Anergy when the dectron)otiv« forces are reversed for every seoi
vibretioQ of lig'ht than when they aot for sensible timeis, u bi osr
ordinary experiments.
801.] Let ua nest consider the aise of a nie<Iium in which lh<
conductivity is Inr^ in proporfion to the iniluctive capacity.
In this case we may Iottv< out tht; term involvinjj K in the eqii»
tions of Art. "83, nrnl they tlion hecoine
<fF
Each of these equntions is of the seme form M the equation of
diffusion of heat frivcn in Fourier's Traifede Oaietir.
803.] Takini; the firet as an example, the eomponent /* of
vector potential nill vary according to time and podlion in the skta
way na the tnnpcrahire of u homoireneoiis solid varies according
to time nnd position, the initial nod tho sarface conditions beinfj^
made to corresptrnd in the two csokx, and the quantity 4xiiC bein:
numerieully equul to the reciprocal of the thermomctric corulitctivit
of the inihftliincc, that is to say, the numitr of unit* of rofnmt
tht tuhitance teiUi would (tf^ ieaUd one degree iy tie heal tchuh /Mittt
through a ittiit eiihe of ike tuLifanee, Iteo ofipotitefaeet o/" which differ
bjf one degree of temperature, wiUg Ihs other facet are imfiemeable to
heal*.
The diiferent problems in thermal conduction, of which Fou
hnx given the solution, miiy be tnuififormed into problem* in
diifuiiion of electromngnet ic quantities, remembering that /', O, ff
are the componeiila of a vector, whcreu« the tcmpemture, in Fouricr'i
problem, is a wahir (juantity.
Let UK t«ke one of the case« of which Fourier has given a com
plete volution t, that of an infinito mMlium, the initial state of
which is given.
I
It
fc/o
iricifl
1
[ nf Btat, p. ISS Snrt adkloD, p. 2ii tmrlli »JUkn. I
An. asi. Tlin •<UMi«) wbiMdvUnBiMa 111* (oNpontanh '
r a tima ( In t«niui Dr/(a, 0. y>, tik* isitial tsmpcmtaMafeJ
• 8«a UmwiOI'* Theory «/ Btat, p. ISS Snit adkloD, p. aOA tmrlli »aUkn.
1 TroiU it. la CAdl'ur. An. 881. ~ . 
*, M ■ ">fut <j, y, i) alWr i
tbc pgitti (a. «, 7), !■
wbtn i !• Uie iharawm*tria MnlocUTity.
'/i'.B.yX
!04.] ^ESTABLISH UKKT OF THE BrSTlSCTIOS OF rORCR. 407
The stslo of any point of the meilium at th« timo i it fonod
by takiD" the average of the state of every part of tlii> mediant,
ifche wci^ht assigned to each part in taking the average being
I
where r is the distance of that part front thepcHnteonsidered. Tbia
average, in the case of vectorquantities, ia niost ooiivenicaUy taken
iby considering each component of the vector separatt^ly.
803.] We have to remark in the first plaoe. tliat in thi* problem
the thermal conductivity of Fourier's medium is to be taken in
Tetsely proportional to the electric conductivity of our n»ediuitt,
00 that the time rL^plirvd in order to reach an a.iiii^'iteil vta^ in
the procees of diffusion is greater the higher the electric conduct
ivi^. This statement will not appear paradoxical if we remi'mhcr
the result of Art 655, that a medium of infinite condiietivity fi)rmn
^a complete barrier to the process of diffusion of ma^etic force.
H In the next place, the time re<iuisit'> for the production of an
Vsasigned stage in the process of diffusion is proportional to the sijuara
Bof the linear dimensions of llio system.
B There is no determinate velocity which can he delincd as the
velocity of diffusion. If wo attempt to meusurc this velocity by
ascertaining the time requisite fur the production of a given amount
of disturbance at a given distance from the origin of disturbance,
wc find that the Miiiillcr llie Kelcetcd value of the disturlwince the
greater the velocity will apiKar to Ihs for however great the distance,
and however small bhu limi;, the value of the diaturbance will differ
I mathematically from xoro.
This peenliarily of diffusion disfinpruishes it from wavepropaga
tion, which tukcx ptace with a definite velocity. No disturbance
take* place at a given point till the wave reaches that point, aud
wlien the wave Iiasi passed, the disturbance oeasca for over.
801.1 Let tis now investigate the process which takes place when
tan electric current begins and continues to flow through a linear
circuit, the medium surrounding the circuit Wiug of finite electric
DOlulootix'ity. (Com>nro with Art. OGO.)
When the current begins, its timt effect i» to produce n current
of induction in the parts of the mtdiuni close to the wire. The
direction of this current in opposite to that of tJie original current,
and in the first instant iitt total quantity is equal to that of the
original current, so that the electromagnetic effect on more distant
,rU of the medium is initially aero, and only rises to its final
L
ELECTROIIAOSETIC THSORY OF UOHT.
vilite sm the inductioncurrent die^ away on accoant of tbe el«etra
reHiiitance of the muJinin.
But aa th« ii)i]»c1 loncnrrent close to the win dies away, a tie*
inductioncitrrent in ^nerated in the medium heyond, so that the
•pace oocupiod by the induotioncurrent is continually becoming
wider, while iU intensity is continually diminUhiR]*.
T\m diiliuion and decay of the inductioncnrrcot is a plirao
menon precisely anatcg^ous to the difTusion of heat from a part of
the medium initially hotter or colder than the n^t. We most
remember, however, that since the current is a viTtor qoaulity,
and since in a circuit the current is in opjMUto directions at op
posite points of the circuit, we must, in <4tlcuUling any ^iren com*
poDcnt of {he iiiductioncurrent, compare the problem nitb oae
in which equal quantities of Heut and of cold are diffusod bam
neighbourin'^ places, in which rase the effect on diiitant points viJI
be of a smaller order of ma^iitude.
805.] If the current in the linear eircnit is maintained oonstaat,
the induction current*, whieli depend on the initial chanf^ of state,
will g:radua]ly be diffused and die away, leaving tbe medium in its
perninnent state, which ist auato^us to the permanent state of tbe
flow of heat. In thiti statv we have
throughout the medium, except at the part occupied by the cireoit,
in which vt/^4,^
rtO=<„r. I (3)
Tlicsi! cquntions are snffieient to delOTmine the vatnes t>{ P.G.M
throughout the medium. They indicatr that Ihirc arc no cunvots
exoej)t in tJio circuit, and that the mngnctic force* nre simply those
due to the current in the eircnit aecor<lin^ l« the onliniiry theory.
TTic rapidity with which this permanent nUiU i» csdihlishcd is go
ffrcat tliat it oould not be measured by our ex«erimenl«l methods,
exwpt perhapii in the case of a very brge mas* of a higlily con
diicling medium nuoh as copper,
NoTK — In a paper puldishcd in PogrsendorlTs Annals, June 1867.
M. Lorenr. luw dedooed from Kirebhoff's equation of clietrir cur
TvnU (I'offg. ./«»,cii. 185G), by the addition of certain tcnu* which
do not nflWt any experimental rcstilt, a new set of e(]uati<ins, tiMli
cating that the distribution of force in the electromagnetic 6cld
may be Coticeirml as ari»ing from tbe mutual action of contignoua
<
>5]
L0BKSZ8 THEORY.
409
jplements, and that waves, consietiDj* of transverse electric carnmt*.
ay be propagated, mth a velocity eomparaWe to that of light, in
^noneondtictini* mcdiii. He tbcrcforo rc^rds the digtiirbanoe wlitcb
^ COD eti tilt es It^ht ns identical with these electric oiirrents. and he
[ shews that condiioting media must bo opaque to such radiations.
ThcHi' conehisions are cimilar ffl those of this ohnpter. thoiif^h
Fobtaincd hy an entirely diflVTeiit tncthfxi. The theory given in
this etiapter wns fir«t published in tbc PMt. Trant, for 1803.
CHAPTER XXI.
MAOKETIC ACTIOIf OV LIOHT.
806.] The most important step in establishing a rclatioii between
electric snd magnetic pb«nDmet)a unci thoBO of liffht must be th*
discovery of some instance in which the one Get of phcnomeoft »
afTvctfid by the other. In the search for such phi>nomenn wo must
be guiJctl by any knowledge wc may bav« uliyjuly obtainetl with
r«N[ioct to the mathciDaticiLl or geomitricul form of th« qoantitie*
which wp wish to compare. Tliu», if we ftndciiTour, as Mr*. Sonutr
ville did, to magnctixe « needle by nKonit of light, wc must re
member tliuL tlie didinction 1>etween magnetic north and itoutJi it
ft mere matter of direction, and would be at onee rereraed if
reverse certain eonventiona about the use of matlieiuatical eigaa.
There is Dothiug in magnetism annlof^us to those phenomena
chK^rulysis which enable us to diatiiiguieh positive Irom negative
rloctricily, by obscrvins that oxygfin appears at one pole of a cell
and liydnigin ut the other.
Hence nc must itot cxpoct that if wo make light fall on on« end
of a needle, that end will become a pole of a ixTtain naniv, for thu
two poles do not differ as lit;ht docs from dnrknecs.
"Wo might expect a bctt«r result if we cauMil circubrly polariz<MLi
light to fall on the needle, rightbandit light falling on om^ end
and Ieft<lianded on the other, for in some reepecte these kinds of
light may be said to be related to each other in the same way as
the poles of a magnet. The analogy, however, is faulty even here,
for the two rays when combined do not neutralize each other, but
produce a plane polarized ray.
Famday, who was ac^iuainted with the method of studying thit
strains produced in transparent solids by means of polarized light,
made many cxporimenta in hopes of detecting some action on polar
i£<d light while passing through a medium in which electrolytic
conduction or dielectric induction exists *. He was not, iiowever,
■ StftHtrntaf Stmrdim^kdlH smI SSI62nO.
I
1
FARADAY 3 DISCOVERY.
411
(o d«tect an} adtoD of this kind, thoug^h the experimenU were
nged ia the way b«st adapted to di§oorer effects of tension,
!« eleitric force orciirrent being at right anglofl to the direction
the ray, and at an angle of fortyfive degrees to the plane of
polarization Farsdar varied these experiment* in many waya with
out discovering any action on light due to eWtrolytic ourrenta OP
to static (.'leclrie induction.
I! lie succeeded, however, in eetabliahin;* a relation between light
^od mignetisra. and the experiments by which he did so »re <1e*
■scribed in tlie nineteenth eeriee of hifi Experimentai Bc*Mr<^Jir». Wo
•hall tak« I^raday'a dieooTery as onr etartingpoint for further
investi •Ration into the nature bf ma^etiem, and we ehatl tborcfon)
I describe the phenomenon which he obeervcd.
807.] A niy of planeimlarizod tight ie transrDitt4.'tI through a
traiivjinrcnt diumiignctic medium, and thv plane of its polarizjition,
wIh'H it emerge* from the medium, is ageertain^'d hy obftcrving the
position of an unuljRcr when it cuts off the my. A magnetic force
is thin made to att to that the direction of the force within the
tranHjiar^'iit medinm coin<id«! with the direction of the ray. The
liglit at once rcapt>o(int, but if the auiilysiT i» turned round through
■ certain angle, the h'ght is ngnin cut off. This iihen^ that the
eflVet of the magnetic force is to turn the plane of jwlarizal ion,
round the direction of the ray as an axin, through a ei'rtain angle,
im>a«ur('d hy tlte angle through which the analyser must be turned
in order to cut off the light.
80H.1 The angle through which the plane of polarization is
turned i« projiortional—
(1) To the diftance which the ray travels within the medium.
Hence the plane of polarization change* continuously from ita posi
tion at incidenne to ita poaitJon at emergence.
(2) To the intensity of tbe resolved i<art of the mngnetio force in
tlie direetion of the ray.
(3) The amount of the rotation dejwnda on the onturo of the
medium. No rotaljoa Itaa yet been observed when the medium is
air or auy other gaa.
These tiiree statements are included in the more general onv,
that the angular rotation is nnmerically equal to the amount by
which the magnetic potential increases, from the point at which
the ray enters the medium to that at which it leaves it, multiplied
by o eoeflieient, which, for diamagnetio media, ifl generally positive.
SOd.J In diamojfnetic aubetanoea, the direction in which the plane
412
MAQKETTC ACTIOS ON UGITF.
of polariution h miwle to rotate ts the same as the direction in viaA
a pocitive current must circulate round the ay in order to pi
a magnetic force in the same direction aa that which actually rniU
in tlie medium.
Verdet, however, disooTered that in oertnin fcrroma^trtio medi^
as, for instance, a strong solution of percliloridc of iroo in wm^
spirit or ether, the rotation is in the opposite direction to the eujntf
wfaieb would prodaoe the ma^Dctic forc«.
This shews that the diSerence between rerromn^ctic and dia
magnetic iiibstanecs doi!« not arise merdy from the ' nm^ctic per
mcnbility ' Wioff in the first case greater, and in the twoond l«as,
than that of air, hut that the pioperttn of the two claaws of bodies
arc really opposite.
TIr powtrr acijuirwl hy n Kubstanco under the actioa of magnetic
foroc of Totatinff the plane of ]>olariz»tion of light i^ not enutlyfl
proportional to its dianiagnetic or ferromagnetic magiiiitimilnlity. ^
Indeed there are cxeet)tion!i to the rule that the rotation in ixwilive for
diamagnetic and negative for ferroiaagitutie aubstanoca, for neutial
chromate of potash is diamagnetic, but produces a n^ative rotatloB.
810.] There are other substances, which, independently of tbt .
application of magnetic force, cause the plane of polariEation to'fl
turn to the right or to the left, aa the ray travels throu;^ the sub
vtance. In some of these the property is related to an axis, as in
the case of qaartz. In others, the property is independent of
direction of the ray within the medium, as in tnrpentinc, soluti'
of sugar, Jtc. In all these substantcs, however, if the plane
polarization of any ray is twisted within the medinm like u nt;ht>
handed screw, it will still be twistwl like a rig'lithunded si^nw if
the ray is transmitted through the medium in the oppositft direction.
The direction in which the observer has to turn his analyser in order
to oxtingiiisb the ray aAer introducin<^ the medium into ita path,
is the same with reference to the observer whether the ray coroe^^
to him from the north or front the south. Tlie direction of thefl
rotation in space in of courw reversed when the direction of tltc ray is ''
reversed. Bnt when the rotation is produced by magnetic action, its
direction in spooc is the same whether the ray l>e tmveUiug north
or sonth. The rotation is always in the same direction as that of
the electric current which produces, or would produce, the acinal
magnetic state of the field, if the medium belongs to the (lOsitJTO
elaas, or in the oppoaitc direction if the medinm bvloug» to the
negative cbus.
IS m
tio^
3 olH
I
;i2.]
STATEMEST OV THE PACTS.
^B It follows from this. Unit if tlic nty of lij^ht, alter passing throug^h
^BJi« m«diiim from norlli to Month, is ri'llocti'd hy a mirror, eo ns to
^Bvtiini through the mpiliuin from eoulh io north, tho rotatiou will
^H^ doubled wh«n it re^ilbt from magtutic iiclion. When thi; rota
Htion dqipo'l* on tho nntiiix nf t hu ineilium aloiu*, as in tiirpoi»tiii«, &v„
Hthc ray, whtii rcflcctitl bai.k throiif^h the mwiium, emerges in Hut
^HuM plnD« la it entered, the rotatinn during thu lintt poMOgft.
^Hhroiigl) the medium having been exaetlj' reversixl (lurio^ the
^B 811.] The phy«ioal esplanntion of the phenomenon present* con
siderable diflicultics, which can hardly be said to have been hithi;rto
overcome, either for the magnetic rotation, or i'ur that which
certain media exhibit of themselvefl. We may, however, prejxirt!
the way for such an explanation by an analyeia of the observed
lots.
It is a wellknown theorem in kinematics that two uniform cir
ular vihrutions, of the same amplitude, having the same periodic
lime, and in the same plane, but revohing in opposite directions,
are equivalent, when compounded together, to a rectilinear vibra
The periodic time of this vibration is equal to that of the
Pcircular vibrations, its amplitude is double, nud it« direction is in
ithe line joining the pointii at which two. particles, dt^'Scribing the
Bironlar vibrations in opposite directions round the same circle,
rould meet. Hence if one of the circular vibrations has it« phase
lecclerated, the direction of tlte rectilinear vibration will be turned,
the same direction as that of the circular vibration, through an
angle equal to half the acctIomtion of phase.
It can also bo proved by direct optical csperintcnt that two ray«
of light, cimilarlypolNT7zed in opjwsite directions, and of the same
intensity, become, when united, a plane polarized ruy, and that if
by any means the phase of one of the cireHlarlypolarized rays is
accelerated, the plane of polarization of the remiUaut ruy in turned
mind half the angle of acceleration of the ph«»c.
812.] We may therefore express the phenomenon of the rotation
of the pinue of polarization in the following manner: — A plano
>Inri7.cd my fallo on the medium. This is equivalent to ttvo cir
bnlarlypolarixcd ravK, one righihanded, the other teflhanded (as
(ganin the olmerver). After juiasing through the niediuni the ray
I still planepolarized, hut the plane of polarization is turocd, eay,
I the right (a.* reg;ird=i the observer). Hence, of the two circularly
rized rayo, that which is righthanded muat have had its phase
Kisnnc AC
[8r
witL
to the
obDok IIS
tbroi^lki
mediatn.
In iOhKT wsiia, th* ri gh llMMJrf i^ bas perfbroMd a giwbf
BHobn tf libntiMH.
hw
laglk.
wit^B tbc Brdioat, tkaa tfce hft l—iM nr which baa Um
Tlua BHid* of (Utia^ what takes plaee im qait« iailffpeikdeat tf
aay tivMrf of 1%^ for tkoagk w* am toA Ions as wsnlea^
cnrnlar^polafiiatiaB, ke., wbidi onj bi aapoented in oar ntadi
with a i«rti«dir Smm of tk« nada l atoty Xheorj, the rgaaoaiiif »
iaiepadMA of thu Mwciition. aod depends unljr on fiwrts pnnd
by cxpRUBvnt.
SIS.] lit i1 iiiaiiidnr tiir irraligTmliT nf rrnr iiniiiiiii iiji
st a gimi iartant. Anv Bntlolatian, the tnotiaD of which at mth
point is areolar, mxj W rapnaeBtcd by a beGx or aemr. If Uw
■ovw is made to nrolTC about its axis witbovt anr longitnltiul
aoUoii,caelt particle will deanibe a ciide; and at the mmt tunstlM
ptopagatsoa of tfas aod d at io a will be fcpw aeBte J by tb« appannt
lawitudiasl notion of tbe nmilariy HtaBted pBCts of tbe tbraad of
the screw. It t* easy to see that if tb« screw is rigfatiiandnl, aad
th« obsenr«r is placnl at that end towards which the undoUtioB
tiarel^ the motaoo of the senw will appear to him leftI
that is to say, in the opposite i
rectioD to that of the baods ■
wsfe^ Hsaee such a ray bas
beeo called, onginally by Fmuih
writei*, bat non lij the whole
eeieotifie worid, a lefiba
colarlypobrized rsy.
An^bthiuided nrcoUrlyi
ized ray is rrpreseoted in
natiDrr by a lefthaodifd
In t^. 66 the right luln■l*^i K
J, oa the rigbthiinil or Uie H^fi':
represents a leftluuided ny, luul
tbt^ lefthanded hrlix /Aon thr li(l'
band, repraeats a rightbawled
ray.
814.] Let us nun rfiaaider two
KOth Tvyt which bar* the same
Tbey wv i^eometaually aUka in.
h— de^_l
•oaitedfl
ods4>f ^^
wareleagth within the nedinm.
m
I
fl&l
415
fall rc¥p<.ct«. oxcc'i)! that one is t1i« pervernon of the other, like it«
uma^o in x luokiiiggIii»s. Ouu of thom, houev«r, siiy A, baa a
uliortcr period of rutntion thnii the other. If the motion is entirely
rau0 to the forces culled iato piny hy the displacement, thJa ebetvs
kbkt greater forces are cnllcd into play hy the same ilitplaeement
hrfa«n the configiimlioit is liki^ A than when it is like B. Ilenco in
Blii* ouc the le(Vhuiided ruy will be aroclerated with rE«>ect to the
pight.'hitnded ray. and thi* will be tlio case whether the rays are
hraveUiiig from S to S or from S to A'.
1 This therefore is the explanation of the phenomenon av it is pro
Muccd by turpentine, &e. In these media the displacement caused
my n circularlypolanzed ruy calls tnl^ plav grcjiti^r forces of resti
Itution when the onli^uration is like A than when it Js like B,
The forces thus defiend on the eoiifiguration alone, not on the dir«c*
tion of the motion.
But in a diamagnetie medium acted on hy magnetism in the
direction S/V, of the two screws A ami B, that one always rolales
with the greatest reloeity whose motion, as seen by an ey« looking
from S to y, appears like that of a watch. Hence for rays from S
no X the righthanded ray B will travel quickest, but for rays
from N to S the lefthanded ray A will travel quickest.
816.] Confining our attention to one ray only, the helix B has
[exactly the eame configuration, whether it reprcHints a ray from S
isK one from N to S. But in the first instsncu tho ray travels
^ftod thervfonc the helix rotates more rapidlj. Ilcnee greater
forces are called into play when the helix is goin^ round one way
tluiD when it is going round the other way. The fbrct*, tlierefore,
I do not depend solely on the configuration of the ray, but al«o on
the direction of the motion of its individual parta.
816.J Tlie diatiirbance which constitutes light, wliatever it»
physiml nature may be, is of the nature of a ve<;tor, perpendicular
lo the din«tioii of the ray. This is proved from the fact of the
interference of two rays of light, which under certain conditions
producea darkmiiu, combined with the fact of the ood interference
of two rays polarized in pinnee perpendicular to each other. For
since the interference depends on the angular position of the planes
of polarization, the disturbance mnst be a directed quantity or
Te<)tor, and since the interference ceases whi'n the planes of polar
ization are at right angles, the vector representing the disturlmuce
must be ])erpendieular to the tine of ioWnoction of these pbnes,
m tliat is, to the direction of the ny.
416
MAGNETIC ACTIOS OS LIGHT.
[8.]
817.] The disturbance, hemg & rector, can bo resotved into cna
pon«ats pamllel to x and jf, ihn txis of 2 bein^ >ara)I«I to
diivcttoD of tbe my. Ix.t ^ and 7 b« tlwse componiuU, then, lo^
case of a my of homo^neous circuUrlypoUrized ligrht,
f = r cos tf. If = r Ein 0,
wborc = Ht—q:ia. (ij
III tkft^e espreasions, r denotes the loa^Uudc of tfa« vector, wij
6 the angle which it Duike« with the direction of Uie axis of*.
The ]H;rii>dic time, r, of tbo disturbance k snch that
iar = 2ff. (X
Tho wavcleo^tb, k, of the disturbance is such that
;A = 2ir. [4
The velocity of projMigation is  •
The phiute of the didrtiirbancc nheo t and r are both zero is a.
The circularlypolariziKl light i» rigrhthandcd or lvlYhandi:J
uec'oiding aa q \» wgalivc or jioKitivc.
Its vibrations are in tb« positive or the negative direction of_
rotation in the plane of [x, y), according as a is positive or n^aliv
Tlie light \» propag.ited in the prisitire or tho negative direction
of the axiH of f, uccording as n and j are of the Game or of opposite
signs.
Id all media n varies when q varies, and y it ain'ays of tlie same
Ei^ with ■ H
Henee, if for a given Dtimtrical value of a the valu« of  is
greater when it is positive than when * is motive, it follows that
for a vahiv of q. f>tven both in inx^iitiidc and sig^, the posittva
value of a will be gr(«(«r than the ntgative value.
Now this is what is observed in a diamaguetic medium, aeted on
by a magnetic forcff, y, in the ilircction of ^. Of the two circularly.
p»lariz«d mys of a given periud, that is avcelfmti>d of which tli^j
direction of rotation in the plane of (•b, ^) is positive. Hence^ o(
two circiilurlyiiolartitid rays, both lefthanded, whose wave lengtl
within the medium iK the mme, that has the ahortsst period wl
direction of rotation in the phtne of ty is positive, that is, the raji
which is pro)Mguted in the pokitive direction of e from south till
north. Wc have therefore to account for the fact, ttiat when in thaj
equations of the syntcm j and r are given, two values of m
f8i9.]
ENEROT OF THE MKDKTU.
417
^^w
SBtisff the equatioDs, one poiitire ami the other oe^tive, the
positive raluc beinjS' oumerically g^reater than tho nej^ative.
818.J We may obtain the et^iialioRH of motinn from a conaidera
ioD of tbo potentifti and kinetio ener^es of the medium. The
]>ot«ntiul energy, r. of the system depends on iU confii»uration,
tlint is, on thv relative poeition of its parts. In so far as it d^peods
on the dii^litrhanoe due to circularly polarized lig:ht, it must be a
fuDctioii of r, the amplitude, and ^, tbo coclficii'Dt of tonion, only.
It muy h( diirtTont for positive and ne^^tivo values of f of oqnal
Dumerieal value, and it probiihly is so in the catw of mudi« which
of themselve* rotate the plant of polarization.
"Hie kinclie energy, T, of the system \» a homogeneous function
of the second degree of the velocities of the systflm, the eoelficienta
of the different terms heing fiinctioiis of the coordinates.
8IO.3 Let us voiLsider the dynamiuil condition that the ray majr
Ite of constant intensity, tJiat if, that r may b^ constant.
Itfgrange's et^uatJon for the fort^e in r become*
41 dr df^4lr~^'
ISioee r is conrtant, the 6rst term vanishes. We have thrrt^fore the
'in which q is supposed to be j^iven, and we are to determine the
value of the an^Ur velocity 6, which we may denot* by its actual
value, n.
The kinetic enert;y, T, contains one term involving n' ; other
terms may contain prodacts of n with other velocities, and tho
re«t of the terms are independent of n. The potentiid energy, /'. is
entirely iDdcpcndcnt of «, The equation is therefore of the form
AKfl + B»^C=(i. (7)
This Wing a quadratic equation, gives two valnos of n. It 8p{>Enr(t
from exjieriment that both values are i^al. that .ono is positive mid
I the other negative, and that the positive value is aumericidly the
greater. Hence, \^ A w positive, botli B and C are aegativc, for,
if K, and «, are the roots of the equation,
J(a, + >4)+/t = 0. (8)
The coeflicient, ?, tJiorHorc, is not zero, at least when magnetic
force a«tK on the medium, We have therefore to consider the ex
[presaion Bn, which is the part of the kinetic energy involving the
[finA power of n, the angular velocity of the disturhanoo.
VOL. It. £ e
meiTETic Acnov or uobt.
[83C
820.] Every term of 7 ia of two dineiuiioQe u rOf^rds Teloctl]
ileacv tbe t«nn> iDvolrmg^ n must iaTolve Mino other vdt
TTiis Telocity cuutot be »■ or j, becauie, io the cmtK we consider, '
r and ; are coiutaDt. Uenoe it u m velocity wbivh exists in tbt
medium independentlT of (hat motion which cotutitattv light. It
roust also be a velocity related to x in such a way tluit when it ii
multiplied by • tbe resnlt i» a scalar (aanlity, for only mxlar ^van
titics can occur as temu in tbe valoe of T, which i« itMlf ecsltr.
Hence this veloci^ must be in the same direction as «, or in tb»
f^poeite direction, that is, it mt»t be an anyutar veiocity nboot tlit
axis of t. I
Again, thif velocity cannot be independent of tbe ma^otic force,
for ir it were related to a direction fixed in the medium, the phe
nomenon would be different if we turned tbe medium end for end.
which is not the caae.
We are therefore led to the eonclusioa that thin velocity is an
invariable accompaniment of tbe magnetic force in those media
which exhibit the ma^^netic rotation of tbe plane of polarization.
831.] We have Ik«q hitherto obliged to ose language which \t
perhape too euggeetive of the ordinary hypotheeis of motion in the
nndnlatorii theory. It is easy, however, to state our reenit in a
form free from this hypothesis.
Whatever tight is, at each point of space there ir something
going on, whether displacement, or rotation, or sometbin^^ not yet
imagined, but which io certainly of the nature of a vector or di^
reoted quanti^, the direction of which is normal to the dirccliooH
nf the ray. This is completely proved by the phenomena of inter
fercDOe.
In the cnM of drenlarlypolanzed light, the magnitude of this
Tictor remains always the same, but its direction rotates round tbe
direction of tbe ray so as to complete a revolution in the periodic
time of the wave. Tbe uncertainty which exists as to wbethf r this
vector is in tlie plane of polarisation or perpendicular to it, docs not
extend lo our knon'Iedge oflhe direction in wbiob it rotates in rif>ht
faandcd nnd in Icfthnmled circularlypobrixed light respectively.
The direction and tbe angtilar velocity of this vector are perfectly
known, though llie physical nnluro of the vector and its absolute
direction at a given iniilant are uncertain.
When a my of oircularU'poiarized light falls on a medium under
the action of magnetic force, its propagation within the medium
is affected by the ralation of t> '' *>od of rotation of the light to
MOLECirLAR V0KTICE3.
419
lie direction of Lbe mag^tiv^tic force. From thi> we conclude, by tlic
>ning of Art. 817, Uiat in the medium, when under th« action
'^of magnetic force, some roUtory motion ia going i>n, tlie axis of ro
tation heing in the direction of the magnetic foroea ; and tliat the
rate of propagation of circnlarlypolarized tight, when the direction
of its vibratory rotation and the direction of the magnetic rotation
of the medium are {he same, is dificrent from the rate of jtropaga
Uon when these directions are opposite.
The only reseiublance which we can trace between a medium
iroogb which circularlypolttnzcd light is propagated, and a me
linm through which lines of magnetic force pass, is that in both
Ifre is a motion of rotution about an axis. But here the rct>cm
cc HtojiH, for the rolut.ioQ in the opticjil phenomenon is that of
be vector which represents the disturbance. This vector is »1wny«
erpcndicular in the direction of the ray, and Totate^ abotit it a
'known number of times in a sicoud. lu the magnetic phenomenon,
that which r(>t»tes has no properties by which its eides can he iia
tingut«he(i, (to that we euiinot determine how many times it rotates
^_ in a iteeond.
^K Thera is nothing, therefore, in the magnetic phenomenon which
^Hcorresponda to the wavelength and the wavepropagation in the
^■optical phenomenon. A medium in which a constant magnetic foroe
^iis acting is not^ in consequence of tliat force, filled with wnvc«
travelling in ojie direction, 8s wlien light is propagated through it.
The only resemblance between the optical and the magnetic pheno
menon is, that at each point of the medium something exists of
I the nature of an angular velocity about an axis in the direction of
itbe magnetic force.
On lie Ifypo/Jifti* <jf Moleeuiar Fortiea.
822.] The contti deration of the action of magnetism on polarized
light hsds, iu> we have setn, to the conclusion that in a medium
iiuder the action of mngnctic force Kometbing Iwlonging to the
nme mathemntical clans as an angular velocity, whose axis is in the
■direction of the magnetic force, forma a part of the phenomenon.
Thia ang:ular velocity cannot he that of any portion of the me
dium of RCHHible dimeuMona rotating us a whole. We must there
fore conceive the rotation to 1>e that of very small portions of the
Itncdinm, each rotating on ita own axis. This ie the hypothesis of
molecular vortices.
The motion of these vortices, though, as we have shewn (Art. 576),
Be z
4S0
KlQKfinC ACTI05 OUT LTQBT.
[M
it doBB not aseiblT kSect tbn vinbli motinna of large bodies, bs;
W swh as to afcot that vibntory motion on wbJdi tbe propagatmi
of fiflrtt a eetr Sia g to the nsdalatory tbeoiy, dopends. The Aa
flMCBteato of the nxdiam, diinng tbe propoi^tioti of light, trill
prafaw a £Btoriaii« of the Torti«c% and the vorlioes when m
distanlicd wmj raaei on the medium m u to affect the mode o(
fnfu e a tix ta of tlie ny.
8StS.] It is inpoMable, in oor pnwDt ttato of t^oranoe aa to the
natitre of the Tortices, to awign the form of the law which oonnectt
tbe dufilaoeakent of tli« mediom with Uic viuiatioii of the vortices
We shall tbeieTore assome that tbe Tariution of the rorticea caoud
bv tbe disf U<«nKat of the medium U subject to the same conditioiui
whMh Ilelmbottz, ia his great memoir on Voitexmotion *, hu
efaewQ to regulate the Tariation of the vartioca of a M>Tfcct liqoid.
Ilelmholtz's law may W stated as foUows : — Let P and Q be t«o
Deighboaring particles in the axis of a vortex, tlicu, if in oona^
^Beaee of tbe motion of the fluid tlwae parttcles arrive at the
poiat* ^^, the line P^Q" will represent the oew direction of the
az» of the Toitex, aad its strength will be altered ia th« ratio of
/^^toPQ.
Ilence if a, p, y denote the components of the strength of a Tor
lex, and if (, ih C denote the displacements of the medium, tlic valiu
of a will become
1
o'=«+BTi+fl^+y3i,
We now aasome that the same condition ia satisfied during thoj
amall displacements of a medium in which a, ^, y lepreaeut, not
the component* of the strength of au ordinary vortex, but tb«
components of magnetic force.
824.] The comionen(« of the angular velocity of an element o(
t}iQ medium are , il /4C dn>.
(%y
• CnlU't Jantml, ToL 1*. (IS^S). Timiuktwl bj 1UI. PIUL Mag . Jul; , 11167.
WOLFCUUR VOBTICES.
421
The next ntcp in our Iiypollicsis is tbe assumption that tliv
kinetic enei^ of tbe medium contuins n term of Ok' form
l!nitt is eqniTalciit to supposing that the anf*u1ar velocity acquired
by the element of Ihc medium during tho proi>a<fation of li^hl ia a
[qoMttity whieh mAv enter into combination with that motion by
which nugnetic pbcnomona are rxplninod.
Id order to I'nrm thu eiiiat.ions of motion of the medium, we must
[expreBs its kinetio energy in tLTm» of the velocity of its parts,
the components of wliich are f, i], C We therefore integral* by
parts, and find
[2C J Ij {a»j + fin^+yoif) dx dy d»
= cff{r,H)di,dz^cff{aCyt}dzdz^cff{pi<in)ds<}y
The double intt^rals refer to the bounding surface, which may
tie mippOHMl at an infinite distance. We may therefore, while in
vextigatin^ what takes place in the interior of the medium, confine
our attention to the triple integral.
825.] The part of the kinetic energy in unit of volume, expressed
by tliis triple iatej^nil, may bo written
I iilC{imirtv + («y, (5)
where «, r, w are the components of the electric current as given in
equations (E), Art. G07.
It appears from this that our bypotlieais is e([uiTalent (o the
imption that the velocity of a particle of the medium whose
poncots arc ^, ^, f, is a quantity which may enter into com
bination with the elcctrie current whose components are », w, w.
826.] Returning to the expression uuder the sign of triple inte
gration in (4), substituting for tbe values of a, )3, y, those of
w'ljS', /, as given by equations (l), and writing
If
Tk ^" ":&+'
dy^^di'
^■tbe expression under the sign of iutegratiou become*
(6)
fn
the cau of waves in planes normal to the axis of < the displace
422
MAOSETIC ACTIOK OS U6I1T.
[Sa;
tatfata sre fiuictions of * aod t otiljr, so that ~ = y — ^ ami Uw
expieesioB is reduced to
Cy
(«J
The kinetic energjr per unit of volome, so &r as it d«p«nda «
the vrtocitiei of dtsplaoement, may now be written
wlwre f> is tb« density of the mediom.
827.] The <y)mpoii4'ntd, X and }', of the inprfntcd force, refemd
to unit of Toliune, may be deduced from this by LdgniQ^'s ecjos
tioDa, Art. 561. We observe tbat by two suecewiTe int^catiaw
by partd tti regard to ;, and the amission of the double int«j^k at
tli« boundiog surface, it may be shewn that
H«ice
Tlie cxprcwioD for Ute forces are therefore given by
^icse forces arise from the artion of the remainder of (lie mcdiam
on the element nnitcr consideration, bik] muat in the ease of
isotropic medium be of the form indicated by Oauchy,
J = J.^+^,J+*C
,^+.*.^ + &C (13)
828.] If we now take the owe of » circularlypolarized ray for
which f = r oo« (»/— yr), 7 = rsin(i»(yr), ()^H
we Bod for the kinetic enei^y in noit of Tolume ^^
r= ipr'H'—CYf'fM;
and for tlie potential energy Jn unit of volume
vrbeie Q ia a funotioo of ^ '.
829.]
RXPBESSIOK POE THE ROTATION.
423
The oondilion of frw j»roj)iigalion of the my givuti in Art. 820,
equation (6), in df _dT . .
which gives p»*2Cyy''» = Q, , (XS)
whrace tlie value ofn may be found lu terms of j.
But in the cnse of a ray of ^iven wave[ieriod, «ct«d on by
I maeiiclic forci^ what we want to determine ia the value of ^ , when n
da "*
■e constant, in t«rms of j^> when y ie conatant. DiiTerontiatinff (18)
{%^*—2Cyq^)dn—{J^^'iCyqH)dq2Cq^Hdy = ^. (13)
We thus find J =  ^'i'; , % . (20)
829.] If A is (he wavelength in air, and t the corresponding
index of rcfmclion in the medium,
q\ = 2!ti. H\ = 2ffF. (21)
The ohaijge in the vahie of q, due to magnetie action, is in every
I an exceedingly email fraction of its own value, so that we may
it*! dq
? = ?o + /j,y. (22)
where fo U the value of q when the magnetic force is sero. The
LIMgI<', 0, through which the jihme of jiolarizntion jg turned in
sing through a thiikness c of the medium, is half the sum of
the positive and negative values of jc, the nign of the retmlt heing
changed, becauiie the sign of q ia negative in equations (14). We
thus obtain
' dy
{lX^\
1
' FpA
The second term of the denominator of thia fraction is approx
imately equal to the angle of rotation of the plane of polarizntion
during its passngu through a thickoesv of the medium equal to half
a wuvolen^h. It is therefore in all actual ciuen a quantity which
wo may nirglcvt Jii com]>unson with unity.
Writing 1^=01, (25)
vp
we may call m the coefliuiunt of roftgnctie roUtion for the medium,
a quantity who»ie value muxt he detenu iuMl by observation. It is
found to be positive for most <riuDiugnetie, and nt^tire for some
424
UAGNETIC ACTIOS OS LIOBT.
[Sja
panunagnetic luedia.
th«orv
Viv hare thervforc u tlie Goal
result of xm
9 = MCy'l{iX^), (26)
where is Uie aiif[tilAr rotation of the plane of polarization, » ■
oonsUnt determioed by obaerration of the metiinm, y the intensi
of the iBag:nebc foree resolved in the direction nf the ray, c
length of the ray within the raediam, A th» wavclcngUi of
light in air, and > ite index of refraction in the medium.
830.1 The only test to which this theory has hitherto heen ffl
jectcd, i« that of comparing the valnes of for difTvnnt kind«
light passing through the same medinm and acted on by the eame
magnetic force.
This has been done for a oonnderable nnmher of media by
Verdet*, who has arrivc^l at the following results : —
(1) TIk' magnetic rotations of the planes of polarization of
TKyt of diflVtrent cuK>un> follow approximately the law of the invi
squan' of thv wnre^lejigth.
(2) The exact taw of the phenomena \» alwa}e such that ihti pro
duct of the TotatJon by the leqnare of the wuvcloiigth increaacs from
the le«»t refrangible to the most refrangible end of the spectrum.
(3) The aubstances for which thia increase is most seoaiUe are
also thuRc which have tlie greatest dispervive power.
He also found that in the eolation of tartaric ncid, which of itself
produces a rotation of the plane of polarization, the mi^netic rotation
is by no mtans proportional to the natural rotatiou.
In an addition to the same memoirf Verdet has given the results
of very careful experiments <m bisnlphide of carbon and on cn»fiote,
two snbstaoces in whioh the departure from the law of the inrervG
square of the wavelength was very apparent. H« has also cmn
p«rc<i tJiese results with the numbers given by Ihne difiereut foi^
(I) ««.y^(,i^);
(II) '' = yjJr('^^)i
{III) $ = mcy
(.<4y
The first of these formula^ (I), is that which we have already ob
tained in Art. 829, e<iuation (26). The seooad, (II), is that which
• B*cti«rdMt lur Im pNtiritftA optlqun il><Telopf^(B 6aiu Iw ooorp* tn4uiM«Ui
par l^uUoii <1u nuffndiJim^ «** putJaL Cvnptn lUndui. X. t>L p. «M (4 Apifi, IBM).
t Cempla OhmNw, IrU. p. «» (19 Oc*.. 18«>.
830.]
PORMULA FOB TH8 ROTATION.
425
' Tcralte from substituting in tho cquuUoiui of motion. Art. 826, cqua
4»(
fPn
tions (10), (U), tenns of the form ~^and — '^~, mst«ad of ^rj.
I am not aware tlint this form of th« equalioti bus
suggested by any pbysical theory. The third fortnuta, (III),
reeulU from the pbysicul thi'ory of M. C. Noumaon'*, in which the
equatioDs of motion contain terms of the form p aud — jrt.
' lii III
U IB evident that the values of given by the forniuU (III) nrp
'not even nppr^ixiniatily proportional to tho invert gquure of the
wavflcnglh. Those ^iven hy the f'lnnulti (I) and {II) satisfy this
condition, mid give vahies of fl which agree toknibly well with the
< obccrrwl viduts for media of moderate disiiersive power. For hi«nil>
' >hide of OJirbon and creosote, however, the vnlues given by (II) differ
very much from those observed. Those given by (I) agree better
' viHh observation, bnt, though tho agreement ia somewhat close for
bisulphide of carbon, the numhera for creosote still differ by quan
tities much greater than con be accoauted for by any errors of
observation.
Mojfneiie liolafion "/ the Plane cf 'Polarization (from Frrdef^.
Buulphido or Cnrboa at Sl'.UC,
Llnca of tti* •{HKtmtn
C
D
B
F
Obaorvod rnlatiun
692
768
10(10
1234
1704
CUauUtod by 1.
S8B
780
1000
I'M*
17IS
11.
OUS
772
1000
I.'I8
1A40
III.
043
0(7
1000
1031
lOSl
1 Kotation of ths i*j .
K  as'.ss".
CnonMt at !t
".ac.
Linca of tho nptctrum
C
D
s
F
a
^DbaWTod raUtiao
573
758
1000
IMl
1721
CUcutalolby 1.
tir
7S0
1000
ISIO
1608
u.
6SS
7H9
1000
1.iOO
15U5
in.
»7«
993
lOOO
1017
1011
I
RotKtdon ol th* n>; E •• SIDES'.
* 'Enluwre tontiilur quomcKlo lUt at lucii pluium pol&rilatiotiii par vlret oloo
triow *cl DU^eticM diplinttur.' Ilali* Sajri'iiini. i^iS.
t Tb«« ihrot form* of tho uqi)»tiniiii iif million vtrn finl iniitB"ttfit liy Sir O. H.
Alij {Pim. Mail., June I<i40) u x uiiWiiH of muljiin); tlio iilioSdiiiuiion Ihiii r*nRtIy
diMDvamI by Fanulny. Hsu Cull*^ b.id pnrioiiiilj' tu^gwiwl vquMiona ooutaiuing
of tlw fbnn is ordsr M rtpraunl niBtlininftlicaUy the pheaomnis of nnutK.
ajr
Him* •quktiunii mm oiTarwl hy Mm Cullu^h utA Airj'. *iinl m ,'i<^iig s mecibMiiaal
nplMMlim of lh« pbvnoiiieru. but •■ ■h«<rui)' tbM tba phenamcnii mity b* njilftinvd
bj aqusliaiw. trhioa equuiotw Sppoar to be *uch m ralgbt poailily be deduiwil (ivm
t plsiiMbJo mtcluidGBl wBumplloiL ftltliauirlt no iiioh BMiuiipticui Iua i4t boon
426
MAOSETIC ACnOK OS UGWC.
[83'.
[II*
1
Wi) lire 80 little acquainted nith the details of the molecukr
coDutitutioD of bodies, that it is not probable that any satislaeton
theory can be formed relating to a particular phenomenon, wich u 
that or the magnetic action on lif^ht^ until, by an induction foumlftl
on a number of difTerent cases in which visible phenomena are fonod '
1o dcpcn<l upon actions in wbtch the molecules are concerned, wt
leant something more definite about the pro{)crties wbicli must bt
attributed to a molecule in order to satisf^' the conditions of eb
«rved fiwts,
Tiic tli<.>ory proposed in the procedinf; V^f!^ >" evidently of a i
proviKioniil kind, re^in;; as it does on unproved hypotheaee relatio^fl
to the nature of molecular vortices, and the mode in which they are
afFccU'd by the displace m<.>iit of the medium. We must therefure
regard any coincidence with observed facts as of much lees adeiitific
value in the theory of the magnetic rotation of the plane of polari
zation than in the electromagnetic theorj'of light, which, though it,
involves hjpotbeses about the electric properties of media, does do
tp<rulute ax to the constitution of their molecules.
831.] N'oTB.— The whole of this chapter may be rcigarded as an'
expimMou of the exceedingly important remark of Sir William
Thom«m in the ProeeediNSi o/the Reyal Socirty. June 1856 : — 'The
magnv'tic influence on light difcovered hy Faraday depends on the
direelion of motion of muviug particles. For instance, in n medium
possessing it, parf.iclfw in n utraight line {nrallcl lo the lines of
mn^uetic force, displaced to a helix rviund this line us axis, and then^
projected tangentially witli siicb velocities as to di^scribc eircles,fl
will have ditrtrent velocitien according an their motions are ronnd
in one direction (the aime u» the nominal direction of the gidvanic
current in thi magnetizing coil], or in the contrary direction. But
the elastic reaction of the medium muat be the Ksme for the same
displacements, wltatevor be the velocitieH and directions of the irttt
tides 1 that is to say. the forces which are balanced b; centrifugal
foroe of the circular motions are equal, while the luminiferouafl
motions are unequal. The ab§olute circular motiooB being there
fore either equal or such as to transmit eq<uil centTifugul forces to
the pnrtielee initially considered, it follows that the luniiniferous
motiims are only components of tJio whole motion ; and that a less
luminifirrouji component in one direction, Compounded with a mo
tion existing in the medium \vlw(i franitmitting no light, gii'cs aa
cqiuil rtKultunt to that of a greater himiniferoun motion in the oonJ
tnry dinxtiou compounded with the same nonluminr>u.> i>iiili.iti.f
AEGUMENT OF THOMSON.
427
I
I
I
I think it is not only impoHiibIc to coniseiro any other than this
; dyniitnicnl tixjiliinntiun of thv IwH: that circularlypolarizml Hg^hi
.^ransinittt'd Uinnigh mftgnctizeil glass pantile! to the linos of maff
netixiiig lori,'!', with the aame qnality. righthuiiOcd iilways, or hfk
handed alwnya, ie propiigate<) at dilferent mios ucconlinf* as its
coumc is in the direction or is contrnry to the direction in which a
north magnetic pole is drawn ; but I heli^ve it can be demon«l ruled
that no other explanation of that fact ie possible. Henee it appears
that P'araday's opUeoI discovery affords a demonstration of th« rfr
altly of Amp^r«>'8 explanation of the ultimate nature of magnetism ;
and gives a definition of msgnetization in the dynamical theory of
heat. The introduction of the principle of moments of momenta
("the conservation of areas") into the mechanical treatment of
Mt. Bankine's hypothesis of*' molecular vortioeti," appears to indi
cate a line perpendicular to the plane of reatiltant rotatory nio
mentam ("the invariable plane") of the thermal motions as the
magnetic axis of a magnetized body, and suggests the resultant
motneot of momenta of these motions as the definite measure of
the "magnetic moment." The exphination of all phenomena of
viectrotnagDetio attmction or repulsion, and of electromagnetic in
dnotion, is to be looked for simply in the inertia and pressure of
the matter of which the motions constitute heat. Whether this
matter is or is not electricity, whether it is a continuous fluid inter
prrmeating the spaces between molocnUr nuclei, or is itself mole
cularl}' grouped ; or whether all matter is continuous, and molecular
hcterogcucoutinesH consists in finite vortical or other relative mo
tions of contiguous parts of a liody ; it is impossible to decide, and
porfaapn in vain ti> spi'eulate, in the present state of (icicncc'
A theory of moiccuhir vorti(«8, which 1 worked out at consider
able length, was published in the PAH. Mag. for March, April, and
May, leci, Jan. and Feb. 1802.
I Utink we tiavc good evidence for the opinion that some pheno
menon of rotation is going on in the magnetic field, that this rota
tion is ]ierformed by a great number of very nniull portions of
maU«r, each rotating on its own axis, this axis being parallel to the
direction of the magnetic force, and that tlie rotations of these dif
ferent vortices are made to depend on one another by means of some
kind of mechanism connecting tliem.
The attempt which I then made to imagine a working mode] of
this mechanism mnet be t«kea for no more than it really is, a de
munetralion that mechanism may be imagined capable of producing
MAGNETIC ACTION ON LIGHT.
m^
B connt^sion mechanicallj eqcivalent to tlie actual conaexion of tbe
jiarte of the elcctroma^Gtic fiold. The problem of determining' the
mecbanism required to establish a given species of connexioo be'
tween the motions of the parts of a system always admits of an
inGnite namber of fioIatioQE. Of these, some may be more clniDiy
or more complex than others, but all most satisfy tbo eonditions of
mechanism in general.
The following results of the theory, however, are of higher
valne : —
(1) JIagnetic force is the effect of tbe centrifugal force of the
vortioce,
(2) Electromagnetic induction of currents is the effect of the
forces called into play when the velocity of the vortices ie changing.
(3) Electromotive force arises from the stress on the connecting
mechanism.
(4) Electric displacement arises from the elastic yielding of the
connecting mecbaniEm.
CHAPTER XXII.
FBRROUAQKETfSM AKD DUHADNBTIRM BXPLAIKED BY
MOLECULAR Cl'nRENTS.
0» EUelromagnetic Theoriet of MaffaetUm.
832.] Wb have seen (Art. 380) that the action of magnets on
one another can be accnrattIy reprcsfntt'il W the attractions and
repulsions of an imiiginary sitbstuiicu called ' mti^nctlc inatt«r,'
We have shewn the reasons wliy wo mii«t not suppose this ma^etio
matter to move from one part of a inng^iiet to another through a
wnsikk) dirtancc, as at fintt et^ht it a[)]«ars to do when wo msg
DVtue a bar, and we were led t" Poissou'a liypoUieHis that the
SiagDCtie matter is strictljr confined to single muleoulfts of the mag
imUo iubstAnci, so that a raa^uetized molecule is one iu wliich the
kinds of mugnvtie matter are more or less seipnratod to
wards opposite ))olcs of the molecule, but so that no part of cither
can ever bo actually separated from the moUcule (Art, 430).
These arguments complctt^ly establish the fact, that magnctiza
tton is H phenomenon, not of large masses of iron, but of molcculea,
that is to say, of portion* of the subsfancc so small that we cannot
by any mcebanical method cut one of them in two, so as to obtain a
north pole separate from a south pole. But the naturo of a mag
Detic molecule i» by no means determined without further invcsti
g*tioD We have seen (Art. 142) that there are strong reasons for
bdieving that the net of miignetixiug iron or sUx) does not consist
in imparting magnetization to the molecules of which it in eom
pose<l, but that these molteules are already magnetic, even in un
magnetised iron, but with their axcx placed indifferently in all
directions, and tliat the act of magiKtixation consi^a in turning
the moleeule« so that their axo* are cither rendered all parallel to
one direction, or at Uast are deflected towards that direction.
Bt.ECrmC THBOBY OF MAOSOTTSlt
8S3.J Still, bowwur, wo hkvo arrived mt do cxpUnatioQ nf U>v
nature of a magavtiit molociil*?, limt i«, wo have not rocognized tu
likeness to nny other Uiing of which we know more. Wo \tan
theiefore lo oonsider the li}'j>(>tlie«is of Aini)^rA, lliat the m»gnetim
of the molecule is due to an electric current constantly cinmlattng
in some closed path within it.
It is possible to produce an exact imitation of the action of nay
magnet on points external to it, by means of a aiiecX of clwtric
currents properly distributed on its outer eurfooe. But iJie action
of the magnet on points in the interior ia quite dilTercnt froin the
action of the eltctrio currents on corresponding points. Henco Am*
pdre conolnded that if magnetism is to be cxplaiiu'd by mvniM nf
electric currents, these currents must circuluttt within the molvculc»
of the ma^et, and must not flow from one molecule to anotJief.
Ah we cannot experimentally measure the magnetic action at a
point in the interior of a molecule, this hypothea> cannot be di»>
proved in the same way that we can disprove the hypothesis nf
currents of sensible extent within the ma^et.
Desidea this, we know that an electric current, in passing (Vom
one l>nrt of a conductor to another, meets with resistance and gene
nit<* bwit ; so that if there were currents of the ordinary kind round
pnrti'iiis <if the magnet of sensible size, there wouM be a constant
expenditure of energy required to maintain tliem, and a magnet
would be a perpetual source of heat. Dy confining the circnita to
gthe molecules, within which nothing is known about renstanoe, we
Day assert, without fear of contradiction, that the current, in cir
culating within the molecnie, meets with no resistance.
According to Ampere's tbeorj', therefore, all the phenomena of
magnetism are due to electric currents, and if we could make ol>
siTvaliona of the magnetic force in the interior of a magnetic mole
cule, we should 6nd that it obeyed exactly the same laws as the
force in a region surrounded by any other electric circuit
831.] In treating of the force ia the interior of magnets, we bavi
Bupposed the measurements to be made in a small crevasse hollowed
out of the substance of the magnet, Art. 395. We were thus led
to oOQsider two different quantities, ihe magnetic force and the.
magiuitic induction, both of which are snpposod to be obeerved in
ft space from which the magnetic matter it removed. We were'
not imppoaed to be able to penetrate into the interior of a
wHic molecule and to obeenre the force within it.
If we adopt Ampire^s theory, wc consider a magnet, not as
'36.1
HOtECtTLAS CBKRESTS.
I
ntinnoiw sabetonee, tlio mn^ctization of which varies rrom point
point acoonliii^ t<i in>an msily c(>ni.'cive<l law, but ns n tniiliittide
of inolcciilui, within ciich of which circulutcs ii Kjirtoni of electric
Current)!, tjivin^ rise to a dielributioH of mBgoetii force of CKtrcmo
complexity, the direetioTi of Iho forie in the interior of a inoleetilc
\mag generally' the rever»e of that of the average ioroe in ita neigh
bourliood, and the magnetic potential, where it ex'iaU at all, Imiiig
a function of as many degrees of multiplicity as there are molecules
in the miiKm't.
835.] But we shall find, that, in spite of this apparent complexity,
which, however, arises merely fiom the coexistence of a multitude
of simpler parts, the mathematical theory of ma^etism is greatly
Bimplilied by the adoption of Anip&re's theory, and by extending
our mathematical vision into the interior of the molecules.
In the firet place, the two deGnitions of magnetic force are re
duced to one, both hecomlDg the samo as that for the ^ncc outside
be magnet. In the next place, the components of the magnctio
everywhere satisfy the condition to which those of induction
luhject, namely, da d/i _ dy _ ^
dx
^y
'di = '
by
In other words, the distribution of magnctio force is of the
same nature as that of the velocity of an ineompre.i.tihle fluid,
or, as we have expressed it in Art. 23, the magnetic force has no
convergence.
I Finally, the three vector functions — the electromagnetic momen
torn, the magnetic force, and the electric current — become more
simply related to each other. They are all vector functions of no
nvergcnce, and they are derived one from the other in order, by
thi^ Niinic prociiits of taliin<f the space variation, which is denoted
by Hamilton by the symbol V.
886.] But we are now considering magnetism from a physical
lot of view, and we must enquire into the physicul proportic* of
tlte molecular currents. We assume that a currunfc is ciiculatiog
io a molecule, and that it meebt with no resistance. If £ is the
coefficient of aelfinduction of the muleeuhtr circuit, and ^the co
efficient of mutual induction between this circuit and some otlicr
cirvait, then if y is the current in the raolwul«, and y' that in the
other circuit, the e<aatiou of the current y i«
^iLy + 3ty') = ~Ry:
(2)
432
[8.^"
and sinoG by the hypotliesis there is do reaistAiicv, i? ^ 0. Bod ««
get hy int^ratioa
IiyTify'=: constant, = Ly^, say. (
l>ct us Giifipose tlint the area of the projeclioa of the tnolscnlu
circuit oil a place perpendicalar to the axis of th« molcculo is J,
this axis heingf defined as the normal to the plane on wbldi iht
projection ia greatest. If the action of other ciimnt« prodocec ■
magnetic force, X, in a direction nhose inclination to th« uxia of
the molecule i§ $, the qaantity My' becomes XA cos 8, and we haw*
as the equation of the current
IjyiXAe<M0 = Ly„, ('
whore ya is the value of y when J* = 0.
It appc^'nrs, therefore, Uiat the strvn^b of tbc molecalar cnrrent
depends entirely on its primitive value yg, and on the intensity of
the magnetic force due to other current*.
837.] If we supimse that there i* no primitive currunt, but tiiat
the current is entirely due to induction, then
y = — y cos tf . (5
Tha se^tivo sign shews that the direction of the induced cur
rent is oppoiiitu to that of thi indueicj; current, and its magnetic
action is such that in the interior of the circuit it acts in the up*
posite direction to the ma^rnctic force. In other words, the mole*
cular current nnU lilic a Kmall ma^et whose poles are turned
towards the poles of the iianie nume of the inducing magnet.
Now this is an action the reverse of that of the molecules of iron
under msj^uetic action. Tho molecular current* in iron, therefore,
arc not excited by indtietion. But in diamu^^etic suhstances an
action of this kind is observed, and in fact this is the explanation of
diamagnetic polarity which was lirst given by Weber.
I
4
Weiert Tieorjf of Liamagnetitm.
83B.] According to Weber's theory, there exist in the moleeules
of dianiaguctie substancea certain channels round which an elertri'
cumnt can circulate without reaistance. It is manifest that if
snpiKwc tlicse channels to traverse the molecule in every directi<
this amounts to making the molecule a perfect conductor,
Be{>inning with the assumption of a linear circuit within the m
Iccule, we have the strength of the eurruat given by c<[uutioD
PERPECTLT COSDUCTIXO MOLECrLES.
433
The mn^Detio momcut of tke current ie tJie product of iU strength
by the aroa of the circuit, or yA, aud tlie resolved part of this in the
direction of tlw magnetizing force is yd cos 0, or, by (5),
_^co8«tf. (e)
If there arc i such molecules in unit of volume, and if their axes are
distributed inditlercutly in all directions, then the averat^ vnlac of
txm'0 will be i,nDd the intensity of magnotiscation of the substance
will be ^ nXA' ..
Neumann's coefficient of magnetization is therefore
The magnetization of the substance is therefore in the opposite
directioQ to the magnetizing force, or, in other words, the substance
is diamafpietic. It is ftlso exactly proportional to the ma^etizmg
foroc, and does not tend to a finite limit, as in the case of ordinary
ma^etic induction. See Arts. 442, &c.
8S9.] If the directions of the axes of the molecular channels are
arr«nj^>d, not indifiertrntly in all directions, but with a preponder
ating uumber in certain directions, then the sum
oxtmded to all th» molecules will have different values according
to the direction of the line from which is measured, and the dis
tribution of these valueti in different directions will be similar to the
ditttributioQ of the values of momeuts of inertia about axes in dif
ferent directions through the same point.
Such a (tistributioa will explain the m.ignelic phenomena related
to axes in the body, described by Pliicker, which Faraday has called
Magnecry stall ic phenomena. See Art. 435.
840.] Let us now consider what would be the effect, if, in^trad
tile electric current beiiit; confined to a certain chanmd nithia
! molecule, the whole molecule wertt stippimod a poriVcl conductor.
Let us begin with the case of n b(tdy the form of which is acyclic,
that id to say, which is not in the fonn of a ring or ierfor.it«i
body, and let us suppoM that this Uxiy i* everywhere surrouuded
by II tliin ahcll of perfectly conducting matter.
We have pwvcd in .^rt,654, that a ilused sheet of perfectly coq
^—^ '"tter of any form, originally free ^m ounente, becomes.
434
BLEOTBIC TUKOBT OF UAONETISJL
[S4I.
when exiwsec] to external magnetic torve, a cnrrestsheet, the artion
of which on every point of the interior is Euch as to nuke thr
magnetic foroe Tero.
It may ai«ist as in understanding tJiis case if we obeerre that
the distribution of magaeiac force io the neighboorhood of aacfa ■
body 18 similar to the distribution of velocity in an iDcompresdUe
fluid in the neighbourhood of an impernuus body of tb« same farm.
It ia obvious that if other coiiductia}> shelU are placed within
the first, since they are not exposed to mu^^netic force, 00 cnrrentf
will be excited in thenL Hence, in a solid of purfoctly conducting
materia), the effect of magnetic force is to fp'ncratc a system of
currents which are entirely conBncd to the siirfncc of the body.
Sll.] If the conducting body i* in the forui of a inhere of nuiita
r, its magnetic momeot may he shewn to be
and if a number of such spherce are distributed in a mediam, «>
that in unit of volume the volume of the conducting matter is V,
then, by putting *, = I, i, = 0, and p = f m equation (l 7), Art
314, we find the coefficient of magnetic permeability, taking it as
the reciprocal of the rcsistunce in that article, viz.
22 i'
f =
2+i'
"M
(10)
<")
whence we obtain for Poisson'a magnetic coefficient
4 =♦*,
and for Ncnmann's coefficient of magnetizatioiHjy induetion
3 ^
Sinee the mathematical conoeptioD of perfectly condttcting bodies
leiuls to nwulta exe«cdingly different from any phenomeoa which
we ran observe in ordinary conductora, let ua pursue tlie subject ,
BOmowhat further. M
842.] Returning to the case of the conducting channel in th^^
form of a closed curve of area A, as in Art. 83f>, we have, for tha
moment of the electromagnetic force tending to increase the angle 9,
m ~~p siaOcoa0. /13)
Tliifl force is positive or negative according as ^ is lest or great
than a right angle. Hence the cffcet of magnetic force on a per
leotly Donducting chanr.Ll tri:!i tn lura it with its axis at ligbi
(12)
843] HODiPIEB THBOST OF TITOUCBR MAflJfBTISM. 435
I
aaglM to thv lin« of magrwtic forc«, that is, so tlmt tlie plane of the
ohannel bocomes punillcl to the lines of foKtf.
An efr«ct of n simikr kind may be ohserved hy pWiog a ptnny
or a copier ring bcttvoen the pnleti of an elect roinig^ot. At the
instant that Uie inignct in ezoil«d tJio ring turns its plane towards
ihe uxi.il diniction, \nil thiti tbrce vnniiithfg ax soon as tho ourrtrotK
are deadened by the resistance of the copper *.
843.] ^Ve liave hitherto conHidered only the owe in which the
molecular currents are entirely excited by the oxterna) magbetie
force. Let u» nt'Xt examine the bearing of Weber's theory of the
magnetoelectric induction of molecular currenta on Ampere's theory
of ordinary magnetism. According to Ampere and W^iber, the
moleuular currnnttt in magnetic substAnces arc not excitM by the
external magnetic force, but are already there, and the molecule
itself is acted on and deflected by the elect roma^ietio action of the
magoetie force on the conducting circuit in which the cmrent flows.
When Amjiirw devised this hypothesis, the induction of electric cur
nDtx was iiol known, and ho made no hypothceis to account for the
existence, or to determine the strength, of the molecular currents.
We are now, however, bound to apply to these currents the same
laws that Weber applied to big currents in diamagnetic molecules.
We have only to Biippose that the primitive value of tbe current y,
when no magnetic force acts, is not zero but Vq. The strength of
the current' when a magnetic force, .V, acta on a molecular current
of area A, whose axis is inclined to the line of magnetic forcei, is
XA „
(u)
and the moment of the couple tending to tnm the molecule so as
to iucreosc fl is „ X^A* . , .
~y^XA aintf + —^ sin 2 0. (16)
Hence, putting . A
i*y„= m.
'Yo
= S.
(16)
inibc investigation in Art. 443, the equation of equilibrium becomes
Xtiia0BX*mn0C960 = Ds[a{<t9). (17)
Tbe resolved part of the magnetic moment of thv current in the
direction of A' is
yAco»9 = YtfAcosff j~coif9, (18)
= tHCm&(l—SXoot$).
• 8oe Fkndjiy, Krp. Ha.. 2310, Jtc
(19)
438 StStnftlC TltKOBT OP 3UG!7EnS3r. [84^
844.] Th«e comlitionii difier from those in Weber's tiiooij of
nugnoUe indnctioD hj the terms involving the confficifnt S. IT
BX u nnall oumjiared with unitj*, the results will approxioulv U
1ho«c of Weber's theoiy of ma^etism. If £X is large oompORd
with unit}, the results will approxinwte to those of Weber's tbooiy
iff iiaBiigoistJxm.
Now the greater y^, the primitive vslae of the moIeeuUr current,
the HmiilliT will B become, ntxd if £ is also larffe, this will aUo
dimituMh B. Now if the cnrrent flows in a ring ohaanel, the vahie
of £ depends on log ^ , where S is the ndius of the mean line of
the channel, and r that of its section. The emalter therefore ihf
section of the channel compared with it« ana. the greater will be I,
the coefficient of selfinduction, ftnd the more nosrly will the plie
nomcnn agree with Weber's origiiiul <licr>rj'. There will be tbu
ditferencc, however, that us X, the magn*aixtiig force, increases, the
temporary magnetie moment will not only reach a maximnm, but
will afterwards diminish as X increase*.
If it should ever be experimentally proved that the t«tnporaiy
ma^etizatiun of any snWtanoe Btst inereaseii, and then diminii)h«a
as the ma^etizing force is continually increased, the evidence of
the existcncu of these molecular currents would, X think, be mised
almost to the rank of a demanstration.
845.] If the molecular currents in diama^etic mbetODon are
conGnod to definite channels, and if the molecules an capable of
being deflected like those of ma^etie subetanoes, Uiuii, as the mag
netising force increnscH, tlie diamagnetic polarity will alwavs increase,
bat, when the force ti« great, not quite so fast oa the ma^^etiziojf
force. The small abwolute value of the diamagneticcoeflidentBbewB,
however, that the di^Recling force on each molecule must be small
compared with tJiat exerted on a magnetic molecule, so that any
result duo to ihia deflexion is not likely to be perceptible.
If, on the other band, the moleoular currents in dinm^netic
bodies are fre« to flow through the whole sabstance of tlie molecules,
the diamagtivtic polarity will be strictly progwrtional to the mag^^
nctizing force, ant) its amount will lead to a determination of th^
whole space ocoipie*! by the perfectly conducting maates, and, if we
know th« number of the taolecnlai^ to the determination of ihi
of each.
CHAPTER XXIII.
THEORIES OF ACTION AT A DISTANCE.
On tie Explanation of Ampir^a Formu/a ykn Ij/ Oaiui and Wehtr.
846.] The attraction between the element* dt and d/ of two
circuite, carryiog electric curreut^ of intensity i and i', is, by
Ampdre's formula,
th.
itdsd^ . d'r
dr dr^
the curreatfl being estimated in electromag'netXc unit*. See Art. 526.
Hw quantittea, whose meaning as f hey appear in theee expreaeiona
we have now to interpret, ace
dr dr , d'r '
*~'' .UT/' ■""• d^'
lud the most obvious phenomenon in which to seek for an inter
pretation founded on a direct relation between the curronbs is the
relative velocity of tlic electricity in the two elements.
I 847,] Let 03 therefore consider the relative motion of two juir
ticlea, moving with constant vclociticE r and v' along the ekmenta
ds and d^ respectively. The square of the roktivo velocity of iheiie
particles IB tfi= i>*2vtfco9t+i^i
[ ftnd if we denote by r the distance between the particles,
<*f dr . dr
u'^'di^'^di"
^y^^2r^
d'r
dt'd^
^x^
d»r
di^'
(3)
438
ACTION AT A DinAKCE.
im
vrbere tlie symbol d indicates that, in the qtuatity OifTor^ntiatei
the cootdinates of the particles are to be expressed in t«nne of the
lime.
It appears, therefore, that the terms involvioi; tKe product rf' i^j
the eqoationa (3), (5). and (6) contain the (^nantities occurring i^H
Wu thcrcrore cnd^avoaT t«
But in order to
(l) and (3) which we have to intor]>ret.
exprow(1)nnd (2) in termaof a", jr\ , and ^.
do so we must gtrt rid of the first and third tcrm» of co^ili of thcM
cxpreaaioDf^ for Ihcy involve quiintitits which do not ap{>cAr m tli«
rormiifai of Aiin)Jre. Ht^nce we cannot esjilain the electric currenl
ns a tjnn»rer of electricity in one direction only, btit we must coi
hiae two opposit« streams in each current, so that the oombi
etTecl of the tcnns involving e* and r^ may be zero,
848.] Let us thcrof<»re rappo»c tltat in llic first clement, tit, we
have one electric purticle, f, moving with velocity r, and another, r, ,
moving with vctoeity t',, and in the same way two particW, ^ ani
f'l, in da', moving with velocities v' and u', respectively.
The tcnn involving r* for the combined actJon of tJiece partici
Similarly 2 (v'W) = {v'^/ , r'.V,) (* + e,) ;
■nd 2(cp'</) = (w + Pie,){»'«'+f',/,).
Id order that £(r'tt'] may be zero, we must have either
rvnt
1
I
(r
(9)
(9)
e'+/, = 0.
or
'< + r,
V, = 0.
According to Feehncr'a hypothesis, the electric current oonsis
of a current of positive electricity in the positive direction, com
bined with a current of ne^^tivo electricity in the nefritive direc
tion, the two current* being exactly equal in numerical magnitude,
both as respects the quantity of electricity in motion and the velo
city with which it i* moving. Henee both tlie oonditioos of (10)
arc sstiKfiecl by Kechner's hypothesis. ^^
But it is sufticicnt for our purpose to assume, either — ^
That tlie quantity of positive electricity in each element la on
merically equal to the quantity of negative clectricitj; or —
That the quantities) of tlie (wo kia<ls of elcctncity are invcrecl;^
a» the wjiiarce of their velocities.
Now we know that by charging tli« second conducting wire at
whole, wi can make tf'+«' either iKKtitivo or negalirc. Such a
cbari^ wire, even without a cun«at« according to this formal^
would act on the first wire carrying a carrcnt in whicfa ^tit^
FOHlIPtAE OF GAU83 AND WEDER.
43d
I
haB ft valiw difieriog from zero. Such an action has never been
observed.
Therefore, since the cjuantitye'ftf'i maybe shewn experimentally
not to be always zero, an<] since tb« quantity i^e + v'^e, is rwt
capable of betnf^ experimentally tcslwl, it is better for these specu
lations to aasume that it is the latter (juantity which invariahly
vanishes.
840.] \Vbat«ver bypothivis we adopt, ther« can be no doubt that
the total translVr of electricity, ruckooed algebraically, along the
first circuit, is rvprctii.'nt«d by
p<+r,e, = «'(/*;
where i; is the numlxr of units of elatical electricity which are
transroitted by the \init electric current in the unit of time, so that
we may write equation (9)
X(r^r^) = cHi'^ift»'. (11)
HcDfie the earns of the four valui.'g of (3), (5), and (6) become
2(^b2) =_2c*HV*.i«'co««; (12)
Jrdr
^'
(13)
Hi)
and we may write the two expressions (t) and (2) for the attraction
Wtween d^ and (//
■[?(:'i':«(l^)')] (■•)
650.] The ordinary* expression, ia t^e theory of statical elcctri
city, for the repul^ion of two electrical particles e and / is ^, and
(IT)
h gives the electrostatic repulsion between the two elements if
they are charged as wholes.
Hence, if we assume for the repulsion of the two particles either
of the modified expressions
we may dedocc from them both the ordinary cIrctro»lnttc force*, and
the forces acting between cnrrents as determined by Ampere.
440
ACTION AT A DBTANCB.
[851
851.] Th« first of these expremoita, (18), wiw dUoovered \ff'
Gauss* in Ja)y 1835, and iDt«Tpret«d by him as a runilameiital b*
of electrical action, that ' Two elementit of electricity in a state «f j
relative motion attract or repel one another, but not in the
way as if they are in a state of relative rest.' "niiit discorcry «» I
not, »o far as I know, puhliiibed in the lifetime of Uaaa», m that tlttj
aecond espreesion, which was discovered independently by W. Weber.!
and published in the first part of his celebrated E/eJtltvJfaamutie'
.ifa/islpfi/immiiu^eti t, was the first resolt of the kind made known j
to the scientific world.
S52.] Tlte two expressions lead to precisely the same rvsult whai <
they are applied to th« detemiination of the mecfaaoica) force be
tween two electric carrents, and this reeolt is identical with that ;
of ArapSre. But when they arc considered as exprewnonM of tlie ,
physical law of the action betw(«n two electrical particle*, we ate .
led to enquire whether they are consistent with other known bcU '
of nature.
Both of tbeee expressions involve the relative velocity of the!
particles. Now, in establishing by mathematical reasouing 1h«
wellknown principle of the conservation of energy, it is generally
asMnmed that the force acting between two particles is a function of I
tlie distance only, and it is commonly stated that if it is a rnnctiOB]
of anything else, such as the time, or the veloci^ of the particli
the proof would not hold.
Hence a law of elcrtricid action, involving the velocity of
particles, has sometimes been supposed to be incoosisteDt with,
principle of the conservation of onerpy.
(i53.] The fonnula of Gau«8 is inconsistent with this principle,
and muit therefore be abandoned, as it leads to the conclusion thai
energy might be indefinitely gmeratevl in a finite system by phystoal
means. This objection docs not apply to the formula of Weber, for
he has shewn ] that if we asHume as the potential energy of a system
coosistiDg of two electric jttrUeles,
(80)
tthe repulsion between them, wbioh is found by difiVrontiating this j
inantity with respect to r, and cfaanging the sign, is tltat given hyj
the formula (19).
*=^[^(^«)']'
• HM»(0llttlMM«dWin.inT),*nLr.B.«18.
+ JU. ZfOafM. Oa, Lriprif (1M«>.
: A«y. Amn^ luUL >. Sir(l«8).
854]
HBLH1IOLT2i) CRITICISM.
441
Ilence the worh done on a moving pitrticle hy tlie repnUioa of A
partide is ■f'o— Vi> vhcre ^„ imd ^, arc tiic vuIdck of ^ at the
Dning aad at the end of its ]jat,h. Now ^jl il([)cnd« ouly on th«
nee, r, and on the velocity re«rtlve(i in thu direction of r. If,
therefore, the particle desi'ribea any cloacd path, so that its position,
velocity, and direction of motion are the same at the end as at the
bffpDnin;;, ^, will be equal to i/f^, and no work will be done on tie
vvholc dnring the cycle of operations.
H HenO! an indcfiaite amount of work atnnot be generated by a
Bporti«le moving in a periodic manDer ondvr the sctioa of tbe force
Kmiumed by Weber.
854.] But HcImholtE, in his v«y powerful memoir on the ' Eqni^
tiona of Motion of EUetricity in Conductors at lUst*/ while htt
shews that Weber's formnla is not inconsiiitent with the principle
of the conservation of energy, as re^rds only the work done during
a complete cyclical otH^ration, points ont that it leads to the conclu
sion, tkftt two electrified particles, which moTC accordinj? to Wcher's
law, may have at first finite velocities, and yet, while still at a finite
di«lanc« from each other, they may acquire an inRnite kinetic entrgy,
and nay perform an infinite amount of work.
fc'To till* Weber t replies, that the initial relative velocity of the
rticlM in Hdmholtz's example, though finite, is greater than the
locity of light ; and that tlio distand.^ at whieh the kitietio energy
beoomcs infinite, though finite, is gmaller than any magnitiidt; which
weoan pcTX<eive, *o that it mny be physically im)o^ib1e to bring two
moleeulea so near together. The ej[ami>le, therefore, cannot be teated
by any experimental metliod.
Uelmboitz { has therefore stated a eaite in which the distanoea an
not too small, nor the velocities too great, lor experimental verifioa
ioil. A fixed nonconducting spherical surface, of radius a, it ani>
armly cliarged with electricity to the siirroccdensity <r. A particle,
Eof miuB M and carrying a charge r of ckvtricity, mores within tiw
sphere with velocity v. The cicctrudynamic potential calculated
trom tlie formula (20) is
4«r««(l^), (21)
and is independent of the position of the ]iartivle within the sphere.
Adding to this f, the remainder of the potential energy anting
I • CrtUrM Jouinal. 72 (1870).
I f BIrAlr. Maatk. inArwrnittr* Aler Jat Prinrip Jrr Brlnjlanff ihr JEiwrpfe.
I t UerhH JU(.aa(<brKcU. April 1872 i I'kU. Mag., D«a. 1672, J^mv.
442
ACTIOK AT A DISTANCE.
[85^
from tlic nction of otJier forces, and i imp', th« kioetio etier*7 o( ti«
portidr, wc find as the equation of ener^
1 («_ J '^) v*+ i wa<Te+ r= cotatL (S»)
Siac« ihc ecooix] tvnn of the coefficient of t>* taaj be tocreand id
dvfiDiUily \>y inffrin»in^ a. the radius of the sphcro, while the aarhcf
daoflty a remuins constant, the coefficient of i^ may be mnde nfigstiv*.
Accelurotion of the motion of the particle would then corrcspuoJ to
diminution of its m n'tvi, and « body moving in a closed path and
a£t4Kl on by a force like friction, always oppo«it« in direction to il*
motion, would continually incrouM in velocity, and that withoot
limit. This impossible reault iit n necefmry oontMiucnm oraaMiminii
uny formula for the potential which introduces negative terms iato
the coefficient of v^.
855.] But wo have now to conaider the application of WeWa
thcor)' to phenomena which can be realized. We have seen how it
gives Ampere's expression for the force of attractioD between twa
elements of electric currents. The potential of one of these ele
ments oQ the other is found by talcing the sum of the valoes of tlie
potential ^fr for the four combinations of the positive and oegativt
currents in the two elements. The result is, by equation (20). taki&K
the sum of the four vulue« of ^7
it
— » Mat  7 TT'
riti d*
where
and the potential of one closed current on another is
»'//;£^*'^'="'^ H
M=jf'^dtd^, ac in Art*. ^23, 524.
In the case of closed currents, this expru«non agrees with t
which we have already (Art. 524) obtained •. _
ffW/» The«ry oftAe Muetim 0/ Electric Currentt. "
85fi.] After deducing IVom Ampfire's formula for the KctJnn
lietwecn the elements of currents, his own fonnuU for the action
between moving electric particles, Wcher proceeded to apply hte
formula to the explanation of the production of electric corrents by
• In the Bbolii of Uib iDTMiiijpiiIoii Wotxr •dopt* tbe dtElndynamln ■jnuui of
unlu. In tU> tnuiae <*« dwajr* um (h* •IwitronMciiBtio tjttuta. Tha ^mtUo mH
lullc mih M cumoi b to tha deatrodjwmlr oali In lltr Tstla of VI to 1. Art. RSB
WEIlERfi THEORY OF INDrCRD C0BRENTS.
443
P>nfl^cto^f«tric induction. In this he wag cminontly ctcppssTiJ,
and we shsll ittdiratc tlie method l>y whinh th« lawi: of indiicccl
currcntec may lie deduced from Weber's formuU. But vie must
oUterro, tlint the oireumstance that a kw d«d»crd from the pheuo*
Mmcnii discovered by Ampere is abt« alito to uccautit for the ph«no<
mens afterwards discovered by Faraduy dois nnt ^v^ ko much
additions) weight to the evidence for the physicul truth of thv Uw
as we might at iiret sappose.
For it has beeu shewn by HelmholtE and Thomson (see Art. 543),
that if the phenomena of Ampere are true, aud if the principle of
th« coQBcrvatioo of energy is admitted, (hen the phenomena of in
duction discovered by Faraday follow of necessity. Now W'eber'a
^law, witJi the various assumjitioos about the nature of electric
^RUTTent« which it involvee, lends by mathematical traafiformatioos
to the formula of Ampere. Welder's law is also consistent with the
principle of the conservation of energy in so far that a potential
rxists, and this is all that is required for the application of the
principle by Ilelmholtz and Thomson. Uence we may assert, even
Ixiforo making any calculations on the subject, that Weber's law
will explain the induction of electric currents. The fact, therefore,
that it is found by calculntioo to explain the induction of currenta,
leaves the evidence for the physical truth of the Eaw exactly where
it was.
On the other hand, the formula of Gnuss, though it explains the
phenomena of the attraction of currents, is inconsistent with the
inciple of the conservation of energy, and therefore we cannot
< that it will explain all the phenomena of induction. Id factj
' it Gul« to do so, OS we shall see iti Art. 8S9.
857.] We must now consider the electromotive force tendinji* to
Bproduce u curTcnt in the element i/V, due to the current in ih, when
i/* is in motion, and when the current in it is variable.
According to Weber, Uw action on ^c material of the conductor
of which lit" is an element, is the sum of all the actions on the
electricity which it carries. The electromotive force, on the other
hand, on the electricity in ///, is the ilifferenne of the electric forces
aotiDg on the positive and the n<^tive electricity within it. Since
all these force* act in the line joining the elements, the electro
motive foi«e on ii»' is also in tliis line, and in order to obtain tho
electromotive force in the direction of lU we must iwsolvc the force
in that direction. To apply Weber's formula, wo must cu1cu!at«
L^e varioQB terms which occur in it, on the supposiliou that the
ACnOK AT
elemvnl. //« ia in motion relsttvely to i«', aad that the cottnU .
both rlcmcots vmiy with the time. The expicaawma tfaos fMiot
will contain tenns tnToliiDg r*, r/, ^, r, v', end temM not invn'
iD^ c or P*, bU of which are multiplied bv ar*. Bxaminin^, %& •^^
did before, the four tkIucs of tarh t'.Tm, ooii cottGideriagr firet '
mechanical forc« which artsr« fnin the mm of th« Foar Talnes, ''
find that the onlv t«nn which we murt take into account is iL>
iovolring the product rr'«'.
If wr then coaader tlie force tending' to prodooe a canvnt in Uu
epcond etcraent, artwog from tlie diflerenoe of the action oftlK Snt
element on the po«itive and the nt^tire electricity of the stcond
dnnent, vrc find that the only term which we have to examine ii
that which iavolvea vf^. We may write the four termt indnded a
S(r«/). Uitu
e'(«+P,e,) and «',(wfc^«i>
Since /+/ = 0, the mechanical force arising front tluwe temu it
z«ro, but the LlcctroRiotive force acting^ on tlie poaitire electricity i
is (rr + r, «,), and that acting on thfl negative electricity /, u eqiul
and opposite to this.
858.J Let UB DOW snppoM that the first element dt ia moving
relatively to </«' with velocity f in a certain direction, and let w
dwiote by f'd* and fdt', the angle* between the direction of Taad
tlioae of d* and of Ha reepectircly, then the tquare of the rdatiTe
velocity, a, of two electric particles is
a*Bt» + t>'' + n2pc'co»e+2ri;co«/^2r«i'coe;^'. (2SJ
The term in rr' ia the tame as in equatioa (3). That tn v, oo wbidi
the electromotive force depends, is
2rpcoe/^.
We have also for the valu4 of the time*Tariation of r in this case '
ir <fe .dr dr
(2Sl
vhere r^ refers to the motion of the electric particle*, and jr
that of the niatcrial conductor. If we form the sqoare of this qoan
litv, the term invwlring re', on wliicli the mechanical force dcpcndu,
ia the same as before, in e^joation (A), and that involving P, on which
the electnKBOtive force depends, is
drdr
i* dt
i
'.I
TBBEfiS THEORY OP ISDl'CED ClfRREKTS.
445
DifilTcntiatinfT (26) vrith respect to t, we 6ad
rfw'rfr
*'''';/^'*'^j^*dtd^
(37)
^o find that the t«rm involvings vi^ i« tho huiwi u before in (6).
Utrm whose sti'ii alters with that of v is 7 j •
at as
8B9.] If w^ now calculate by th« formula of Oiiuiiii{c(iuatian(18)),
Ite reetiltant electrical forco in the direction of the vecond dement
f, aaiiiag from the action of the Srat element <U, we oblAio
^tiid/ir(2cwrdt~3c<MrrcMrdt)mar^tl/, (38)
iAb in this expression thero is do t«nn involvini; tho mt« of ro
of the current 1, nnd sinoo wo know that the vnnation of
ihe printary nirrcnt prodocett an indactirc action on the Mcondaiy
circuit, we cannot accept the formula of Oaa« i» a true expreMioo
,of the action between electric partielea,
860.] If, however, we employ the formula of Weber, (19), we
or
drdr d
^fC±(l)d4d/
(80)
we iot^iBte tbia op rc M J on with raipect to t and /, we obtda
tbe eleetioinotive force 00 the weoad eircait
d .fCXdrdr . .^
Now, wbeo the Gut ctretiH ii dond,
riJrJr. CAdfir
^"^J'rdil/'^'i^iiZr''
JJ^d^d,
d*r
Bat
.)^=f^d..(Z2)
(")
dtd/
M, bf Art*. 429, 62i,
Baace wa oajr write tiw daetsDMotin lone «■ IW
which apcca with wkat we hare altaady flrtaMH^ad ty
»3Su
(W)
446
ACTIOH
A DIWANOBT
[861.
Om Weifr'i Formula, ex>MMidered at rfnUit^from an AeSitm trtm*mlit4
from ome Etectrie Particle to tAt otier «itA a Cotitlaitt FeUcttj,
861.] In a vory int«n>9iting leUcr of Gatips to W, Wel«er* l»
refers io the eleclrodytiainic specuIatioDs with wbich iu had been
occupied Ion; before, and which he would h^re puhlulu'd ifbecoaU
tlien have estahlitJied that nliich he considered the rc«l kefttow
of electrodynamics, namely, the deduotion of tb« force uctioff be
tween electric particles io motion from the considcntion of no actiiw
l)«twe«n thorn, not instaotaaeoiis, but propagated in limr, ra a
aimilar manner to that of li^ht. tie had not tiacoeedtd in makiit^
thi« deduction wh«n he gave up his electrndyoamic rcAf«rtihet^ tuA
he hud a subjective conviction that it would be necessary in the
firiit place to form a ooDnEtent representation of the manner in
which the propagation takes place.
Three emtnent malhematicimiB have endeavoured to s«pd; thij
ltcy>.loiie of elect rodyoamics.
B(I2.] tn a memoir present«d to tlie Royal Society of Gottingen
in 1»S3, but afterwards withdrawn, and only published in PoirgeB
durfl"<i Anna/en in 16t>7, after the death of the author, Bcrnhaid^
Biemnnn deduces the phenomena of the induction of electric
rent* from a modified form of Poiason's equation
d*F rf»r rf«r I jtF
s? + ;v '^^ +*"'' = ?5ii"'
where f is the electrostatic potential, and a a velocity.
This equation is of the same form as those which espresa tt
propn^tion of waves and other distnrbances in elastic media. Hie
author, however, seems to avoid making explicit mention of anj
medium throug^h which the propagation takes place.
The miitbentatical investiffation given by Riemann luia been ex*
amincd by Clausioat, who does not admit the soumlne*^ of the
mathematical processes, and eltews that the hypothesis th»t potential
is propagated like light does not lead cither to the formula of Weber,
or to the known lawn of electrodynamics.
863.J Ctausius ha» also examined a far more elaborate investifpi
tion by C. Neumann on the ' Principles of Electrodynanuoa {.' Neu<
mann, however, has pointed out ^ that his theory of the tranemiGsion fl
of potential from one electric [larticle to another is quite different V
fnim that proposed by Oaun, adopted by Riemann, and criticized
• M>rdi IP. IWS. WtHM, U. r. t».
KFlSTOSB OP EtECTBOBTSAMICS.
447
thy Clansius, in which th« propagatioa is like that of U^btL There
as, on the oontntry, thv grvaleBt possible difference between th«
nnsmissioQ of potential, accordiu;^ to Neumann, and the propaga
tioD of li^htk
A luminotis body tciwh forth hf>bt in all directions, the intensity
of ivhich dvpends on tho luminous body alone, and not on the
presence of tliv body which is cnlif^htencd by it.
^An ele<tric particle, on the other hand, sends forlh a polcntinl.
«/
I
of
h
■ at
be value of which, — , depends not only on «, the emitting particle,
hill on <^, th« r«cnving particle, and on the distance r between the
particle* at lAe irutant of^tMum.
In the case of light the intensity diminiahea as the li^t is pro*
''pogftted further from the luminoua body ; the emitted potential
flows to the body on which it acts without the slightest alteration
of its original value.
The lig;ht received by the illuminated iKdy is in ^neral only a
'raction of that which f;ill9 on it; the potential as received by the
attracted body is identical witli, or equal to, the potential which
arrives at it.
Bfsiiles this, the velocity of transmission of the potential is not,
like that of li^bt, constant relative to the aether or to space, but
nther like that of a projectile, constant relative to the velocity of
the emitting particle at the instant of emission.
*It appears, therefore, that in order to understand the theory of
Neumann, we must form a very different representation of the pro
cess of the transmission of potential from that to which we haro
been accustomed in oonsideiing tJie propa^fation of light. Whether
it can ever be accepted as the ' eonstruirbar Vorstellung ' of the
process of transmission, which appcjired necessary to Gauss, I cannot
say. but I have not myself been able to construct a oonsieteDt
mental representation of Neumann's theory.
1864.] ProfesMT Betti *, of Pisa, has trented the Buhject in n
iffercnt way. He supposes the closed circuits in which the electric
orrents Sow to consist of elements each of which is polarized
periodically, that is, at equidistant inter%'nl8 of time. These polar
ized dements act on one another as tf they were little magnets
wbow sxoi are in the direction of the tang«nt to the circuits.
The )>eriodic time of this polarixulion is the same in all electric
«ircuit«. Betti aujipowK the action of one polarized oleineut on
* .Vnom Ctmmlo, civU (IMS).
448
AT A DIBTAHCB.
MiothOT at a distance to take place, not instantaneoosly, but ttfia
a time proportional to ihe distance l)etwe«n the elemeats. In (in
way he obtains espreaeioos for the action of one electric circuit ua
annthor, which coincido with thoee which ore knowD to be true.
ClauNintt, however, has, iu tliis caec also, criticized eouie parts of
tlie mnthcmaticiil calcuhitionti into which vrc Hhall not here enter.
865.] Tlicre nppmrs to he, in the minds of these eminent mo,
some prejudice, or i priori ottjectioa, againitt the hypotiieds of ■
medium in which the phenontena of radiation of light and bnt
and the eleotric actions at a distance take place. It is truo that at
one time those who speculated ox to tlic miMii of physical pheno
mena were in the hahit of accounting for each kind of action at ■
distance by moans of a Hpecia] lethereal fluid, wfaoM function and
projierty it was to produce these actions. They filled all spsw
three and four times over with tHhers of difTereut kinds, the pro
perties of which were invented merely to ' save apitearances,' so that
more rational enquirere were willing' rather to accept not only Nov
ton'si definite law of attractinn at a distance, hut even the dogma of
Cotes*, that action at a distance is one of the primary properties of
matter, and that no explanation can be more intelligible than this
fact. Hence the undulatory theory of li^ht has met with much
oppoeition, directed not againtit its failure to explain the pbeno
mena, but against its assumption of the existenoe of a medium in
which light is propagated.
866.] We have seen that the mathematical expresaionB for electro ^
dynamic action led, in the mind of Ganas, to the conviction that
theory of the propagation of electric action in time would be foundl
to be the very keystone of electrodynamics. Now we arc nnablflj
to conceive of propagation in time, except cither as the flight of
material substance through space, or as the propagation of a con"
dition of motion or stress in a medium already existing in apac«.
In the theory of Neumann, the mathcmatkal conception called
Potential, which we are unable to conceive as a material substance.
is supposed to be projected from one particle to another, in a manner
which is quite independent of a medium, and which, as Neumaaa
has hiniself jwintcd out, is extremely differcot from tliat of tlie pro.i
pagation of light. In the theories of Rieniunn and Betti it wonldl
appear that the action is Bup[iosed to h« propagated in ■ mannuj
aomewhut more similar to thai of light.
But in all of these theories the question naturally oocnrs : — I
• Vitlaoi to HewMn'* Primofia, &id AUiioa.
8663
MEDIUM NECESflART.
440
•otnethiriff is transmitted from one particle to another at n distance,
what is its condition an«r it has IcFt the one particle and bcrorv
it has reached the other ? If this Komitthing i>; the potential energy
of the Iwo partietes, as in Neumann's theory, how are wc to eon
ceivc tlii» cncffiy as existing in a point of upace, coinciding neither
with the one paHicK> nor with the other ? In fact, whenever energy
it tmnHniitl^ from one hody to another in time, there inuxt he
a laedium i>r Kub^tance in which the ener^ exists after it ttavef
one hody and before it reaches the other, for energry, as Torrieelli *
remarked, * is a quintessence of so suhtile a nature that it cannot he
contained in any vessel except the inmost sahstanoe of material
things.' Hence all these theories lead to the conception of a
medium in whiuh the propufjation takes place, and if we admit
this medium as an hypothesis, I think it ought to occupy a pro
minent place in our investifjationii, iind that wc ought to endeavour
to coattriict a mental representation of all the details of its action,
and this has been my constant aim in this treatise.
• L^tmi Aeeadmtcht (Fir«iu«, tTK), p. S9.
VOL. 11.
Qg
I
1
^fc'iBFtios. cl<ctric, 63. 227, 828.
— of liglit. 788.
AocumuUlun or onndcnnmi, M, S:2aSSS,
Action at ft diUanM, 103, 641640, 816
S6A.
Aayelio region, ID. US.
^ibet, T82 n.
Abr. SbG. B., 4S4, S30.
AmW«^ Aulnf Muis, 483, 90S5SS,
e»S, «8T, 833, 84S.
Aoion, 337.
Anodes 33".
Anson lUill, fiSil. 6M.
AaUde bkUnoii, SOI.
Atn»it[divric •■•ctririt]'. 221.
AUnution, elnotric, 27. 3». 103.
— ctpliJnod bj lUtB in a malluiii, I0&.
Bu^lsf and UilxHiii, 9!P, 789.
Batten, vcluuc, 331
lhcU.W,. 26S. aeS. 4)1
Battt. E., 173, KOI.
Bifilar uuMiMioii, tit.
BbmuUi, 415.
Borf*. J. C, S.
Bowl, fliheriial. i:*I8t.
BrLit. »Vi"*t*unM>», 84T, 7SB, 775, 7T8.
>— •Icotmtiilit, 2r>3,
BriEbt, (TlrC. and aark. SSI. U7.
Bio^e, Sir a (.'., 36U,
firoun, Jolm Allan, 182.
Bnub. !>a.
lufT, Hdnrtuh, 371, 368.
(dMtiMtMio), GO, S36.
VeMkdMiaM, 60, 87, 1D3. IQS, S37
771, 77471W.
GiiHtclty, calciilalliiri of, 10^ iSA.
— uiivuiirviiiPiit of, 22722S.
— In (itectnunoLiiDlLc Iitaa*B». 774,
775.
Cufmdty (eleotronugnotlc) <il k omi. fOt,
760, 778, 779.
OUh.>tl». S37.
Cation, 237.
Cauohy, A. L., S27.
Cantuliih. lloory, 88, 7*.
(^;1ef , A.. aS3.
Ceutrulmric, t>8.
UtruuitM. alKtric E7efiS4.
Ciroular «UTr«nla, 69470G.
— wlid Angle (ubt«id«d by. dSS.
dutrgo. (tcctric^ >l.
Ciarii. iMiautT. 8SS, 62P. 71i.
QwHllioLliiMi uf iilootrio*3 iluantitiw, 1130
esv.
Clauiiuf. R.. 70, 2SS, 8$3.
C«iH)«iont* of elsctrnaUttic cajiaclty and
(niluclton. 87. BO. 102.
— tif iKiUititial. H7, UO.
— nf r«ti*Unvi> ami oHniluvUrity, 267,
— of indund ma^fiiDtuation, 436.
— of «lcctnnna^otic inducUcpn, 7S6,
— »r Mlf.iniliK^Cion. TStf, 7i7.
Vorfomt ri)««, 421, 44(.
l^oiU ranittuiiw. 33G3II.
— «l«DtnHiugnetic. 691 706.
— mcvMuremont of. 708.
— cniuparliun af, 763767.
C<iiiii*ri«n iif eiinuille*, 229.
— ..fc"il>.75S JS7.
— ornlHrlniiiKiliia fomM, 3S8.
— . of rMUtADCC*. 3l(i3CS.
Connntntion. 26, 77.
Condtnter, fiO, 226SS8,
Sir CIiuIm WtiivutMno, in Iiit pHVr on 'N*w luatrnmont* and CivceBn,' I'iSl.
PM, 1813. btmiijbt tbit att*n^inonl intb I'libtic nolioa^ villi ilua acknonlfit^ment
ot Um orijjiul icvrnl'ir. >lr. S, II iinter Cbri<ti<^ nbu liad dMsribwI it i» I'U ltprr mi
■ladsMd CuiHDia.' Pkii. Tmn:, 1833. no'ler ILq name of a IKfffirTiiIisl Arnuig*
Bicnl. Sm tlw rnuarlu of Ur. tMirant Clark In the Sttittf <^ Tttegmfk t'a^latm,
May », U:i
ag2
i
IXDrJT.
HlN.
«rtl««nfcdlU N.
fcffciM. U71M, US.
»t
3SX.MT.
USSOC
pri^d. f » Ma. 7M, Tsa,
I «( CMIT, »^ 3U; 3«X M3.
C«iUatt fara^ tic
CMi*«tM. ss, ssai u*.
Cotvv, ftl, S«h S«S, ;«i.
Codnodv C. A, Sa, 71. 315, S3. 373.
GMlMib1*lMr, ;».W.
Cryul. oMdMte li, WT.
— WH***^ iniMHkB at. 435, 43(, 4S3.
■ T W f^fWioa <f %)it ia Ik 7>i;S7.
jJmm,3it.
CniSS.
CwnM, ikctricv ISO.
— bM MMkod «(a{^fln(, T4L
— iadacwiSSl
— •t«dy, S»2.
— tkcnu*l«(rtri«, U»3H.
— MMenl, 331. UO. KM, U7, SS3.
748. 748. 780, 771, 77».
Camatatajghar, 7M.
C^Ma npoa, U. 113, 481.
Cjlind^, deetriftMtlao o( IM.
— iMgMtiMUaa oT. tM^ 43^ 43>.
— tmnwBMM ik. <H2«»0.
CjtlBdrle oculi, <76<3l.
Itaa>p«l TibntiMv 783713, 7*1
OwiMf', 78*.
I>miMI'>mU,1S%S71
Dc^ bcu piT— cM«l«r, 74 1.
Dtcnmoit, loguiUiaic, 79C.
MUilo^ 4S3, 743.
MUanln, J. IL J.. 3.
I)«lluuuii>. I'., SSI.
UEiwity, cbxtrio. Bl,
— 111 A cuncnt, 3sy
— nouaraaunt << 313.
IHaMsMtwD, tV. 440. S3S.
IHMMiric, a& 103, HI, 339, S35 381,
HHi^.37<t, 7SI.
t»«a.
Willi ■111. •teAw. M. ;s. 71. lU.
»»«, «oe. 733, 7*1.
S,ll<.
IteA, 1 i^iiiiw ■< iC347l.
Brwk,M.
31.
1».
Xll. X3S, 3«e, 95, »*.l
33*.
»67.
•0, 73, 7«, til,
SS4, MS; 7<3, 7)1.
— M>K7 M.
— I»«,6S.
— !LlacttB«.S«.
— tMidM«.M7.
iinHMfal 70
— .faA,B7.
— waiba, 4S. B9, 107. 108, ni.
— •w.ss
ghafrwiy—fe ijatf of i
BiBB ttl yna ww l T. 73S.
nMtDlp&i, 330, SS5373.
BaBtraljtav SS7. 3SS.
Klac«nMM anHhiCFiMn, 3S327^
7»».
— pobrbuk^ Se7. W4373.
Bkemtaacaaiw fotn^ 475, &S0. S83.
— OMMOnauBl. 40S.
— ■■■u^lBm. »S5.
— Bta mati aai^ 730780,
4
' unI Jntti iwu tk anllB MauM«4
7Ba.
— raUlLa, 491.
Ei«em«aaEthin, djmnieal (1im*t «C
S38577.
Elaetranwtcn, 311320.
ElMtfDOMiiv run*. 49, S3. IJl,
34«3S4. at. SOS, t».
EUcuuthom, 309.
eUc<i<>«»p«b 33, 814.
EbotnalalH lamnnaiiwto. 21iB
~ pofarisiaon. B&. 111.
— sUncUon. 103111.
rjtuai of iBiUiv ^, A«>
Eloctrutouk *UI<^ 540.
I£ttas>i>u<>. 734.
Blii««l. IM, 303. 137, ISO.
KUtptio inltrnk 140, 137. 70I_
BMfgj', 0, &, 03O ««, 7'*^ 7ine.
^^^^^^^^^^^^^^^^^^^^^^^^^46^^^^B
HSqiia^uua of ooniluctiTity, 39$, 609.
Glut, 51, ^^H
^t of coiiibuil;, 85.
Glov, oleotric, 55. ^^^^H
^K~ of cbctrio ouRoiU, 007.
Grawuuum. H., 526, 687. ^^^1
^^ ot (oUl eumnlii, 910.
Grating, electric clTMit of, 903. ^^^H
^m — 0r«l«lr<Ha«4[n'^>o fweo, 603.
Gmri, UeorEiv 70, 84, 318, 439. ^^H
^ — of aUobQiDoliia tator, 693.
Gnoti'a function, 9S. ^^^^
■^arLaplMr.77.
~ tiuanm, 96. ^^^^
^L^of nagnctiMtloD, 41)0, dOB.
Gtoovo, elwtdo «a*ot of. 199. ^H
^^~ of mnatiello Induction, G9t.
■ — of PoJMn, 77.
Gran, Sir W. R.. 372. 778. __^M
Glinnlring. SOI. 217. 2'J6. ^^^H
V— of r«ii«taii(w, 397.
GulUpen'hH, 51, 3U7. ^^^H
EijuUilriuni, poinUoT, 11311*.
^^H
lluiiiluai. Sir W. Rowui, 10. 561. ^^^
VunilHy, Al„ liiit diionvnric^ 53, Sn, 23G,
Hani imii, (34, ^^^H
Harri». .Sir W. Sitow, 36, 218. ^^H
2S6, S30, SSI, S3<, 646. 668, SOS.
Uent, ootnluotiiQ of, SOI. ^^^^1
— hii experimmu, 3S. 4i!9, 930, S68.
— gcnenlcd by the ourrviit, 342, 2SS, ^^^^
— tai tnnbodi. »7, !3. I±i, i03, 528,
^^^M
&2», fill. £»'i, r.0(. 604.
— «pMdfltvofolMtridt;, 213. ^^^H
— Li* •juuUliuM, S4, 60, S8, 107, 109,
US, 439, £03, £10, S<7, 569, 615, 762.
H«iliO, I3R, 140. ^^H
Helix, &1S. ^^^1
F»d, 6'ifl.
Helmholti. U., 30^ 431. 543, 713, 833, ^^H
Foctuur. G, T, 3S1, S74, 6(8.
^H
roltd, R., S36&>g, 669,
Hnlcriotallc electrnmiolen, 818. ^H
teirwnn^ 138, HO.
KTcnoaugnatfc, i^, 139. S44.
^ IVW, olectriti, ■*).
Hix^kin, C'liarlct, 353, 36u. 800, ^^^1
ItctU. W,, olotOiicail mMhiDO, 313. ^^^H
Ilotnitein, Kairi, 471n. ^^^^
— deolmiiiagiictic', 5SS61I>.
Huygoni. Chritian, 783, ^^^^
— of onirarm fmea, U7'i.
KyilntiiHe ram, 5S0. ^^^^
Pirn (wjng. 7(5.
HyjKjwliie, 151. ^^^^H
iriBoa. H. L., 787.
^^^^H
^L flaiiL aleoUio, 36, 3*.
H>^ inociiiit>T«OTi1<lc^ 61, 111, 2E>5, S29, 331.
^ — m«<Mllc, 3S0.
FlM/ia.
^^^^H
IdlnnUitlc cIrctroiDobin. 318. ^^^^H
\ai»tt^ elmtdc, llll, 155181, 189. ^^^H
— uin^Mic. 318. ^^^H
Farc«. cUatmtni^cticv ITS, 580, GS3.
— niovinj;, 663. ^^^^H
~ dKlnimolivo, t». Ill, 333, S4J, iMO
Inmgitikrjr magnctlo mBtbr, 380. ^^^^
364, ZiS. 569, 379. 5uA, 608.
loduool cnrrtaita, 528' 553. ^^^H
^•^ nechuiical. 69, US, KlStll, 174, SSO,
 In a plana ebecl, fiSOdtfO. ^^^H
■ 602.
— WoMr'n tboorjr of, ^56. ^^^^
^H — luiaiiiiruiiiont </, (1,
IqiIocoI ■na{n»tia>tiDii, 4S44I8. ^^^^
^K — acting >1 * iliiUniv, 103.
IndaoUon, vleclnvlaUr, 36, 75, 76, tlL ^^^H
m _ Bnwtt 83, 117 123, *0(. 
— nu^eticv 400. ^^^^
Foomuli, L., 767.
luertll^ olcolrio. 550. ^^^^
Foumr, J. B. J,, £., S4L 332, 333, SOI
 mmnunU and iirodiicti <£, 56S. ^^^^
H W5.
IniiUUn, 31>. ^^H
InvrreioD, 'loctric, 16^161, IM, 314. ^^^H
Ion, 337. 255. ^^^
GitraMOMUv, 210. 707.
^^^1
^ • diOnUkl. 3(6.
K  Muiilinv 717.
■ — •Imk1u<1,708l
poicliloTld* of, 800. ^^^H
IrroauiuiUablo outvc% SO, 421. ^^^^
^^^^H
— ohMTvMfon oC 7(I~7SL
^^^^H
Gasov otMrln* iliictiarge in, 3677, 370.
Juobi. Kl. H., 336. ^^H
— iwiiMOM of, 36S.
Jonklni, Winiam, 546. Sto Phil. Uoj., ^^H
Churfe*. J. P, C7.
1834, III. SI, 9l 351. ^M
Jcnkhi. rlwiiiini/, 763, 77*. ^H
OmHStla, J. M., 3<6, 712.
OBIItc* «Ue4naH4«r, 318.
Jcichnuinii, K, iAS, ^H
OaoHt C. >'., 16, 70, 131, HO, U«. 400,
Joiilo. J. p., 344 363. 448, 4fi7, 468, 796, __^M
421. 45(. 45», 470, 706, 733, 744, 851.
^^H
Gcomcttic nioan di«*ate^ 691693.
^^^^H
G«o)nob[7 of iHxilluD. 431.
^^^^H
Olbwa aod IluvUi, S39, 76».
■K«}rien«ufdMt(oajraaii>lM,'Ml ^^^M
IT"
KInetiw, 969565. ^^^1
454
iwwby:
KtTcliliolT, CxtUv, 282. Sie, 439. 7M.
KoblnuKh. Ituaolpb. 3a&. 965, TU, 7f 1.
Urnf. G.. 17. Ml.
LunGllaf rMAot, 412.
[jiplaco. P. a. 70.
LopUon'i conlKciunti, lSft'146,
— «<ioatiOD, Se, J7, SOI.
— MpuiBon, I3S.
LaBsndn's cMlItoient. 13&.
L^nlu. G. W.. Id, 4!t.
L«iu. E.. :iijG. a;<<>. fits.
Uglit, «lrt;lr.iiu»ftnrjc tIit>Di7 of, 7Sl SOB.
—  and iiu>L:t»'Li.<ir>. 80^^31.
IiiniMlQiuiiv. 04, SI.
— •intognl. lA30.
^ oT electric tbieo. tH, tli.
— of iiuicnalc rnn'*, 401. 4S1, 4D8. 1U».
5»0, 800, 60T, 633.
Linn orequOiliriutn. 112.
~ or flow. 33.203.
— of •loctrio IndndlaD. 82, 117123.
— of niMipiMic lailucUun, 4(14, ifV. It'i,
641L5S7, 702.
LituiBUB, C, 23,
liouvillc. J.. KS. 17S.
LiKlntr. J. B.. 18, 23, 421.
IiOmii, L., SOG n. •
LoKhliiidl. J,i.
Maitiwci7>it«Ulc (ihenDQieiui, 4'JS, 43G.
Ma((iiv(, lU inip<atliM, 371.
— (lir*clH>ii vt BiU, 872 31*0.
— nt^vlio iniiniriil ^f. 384, 390.
— MOtrv Bad priDcip*! mlm, 353.
— pcrteniiril anergj of, 380.
Mutictic kctton uf llsht, fiOO,
— itiaturbniiDC^ 473.
— (orw, law of, 374
(Itnolian of, S7t, 493.
Inlcnrity of. 4U.
~ Iniliicliuii. I<i(>.
M^^tio ' nialtur,' 350,
— mvMurBiuciili^ 44(i4<14
— p«lc>. 4SS.
— wjircy. 488.
— varUll'ino. 472.
HafiMiiHii ofiiliipfv 441.
— (vrrwLrial, KJA174.
Mwwtintkii, coiuiiDaeolB oC 881.
— UidiMed. 424'43U.
— Aiop^'« iLeory of, 883, 333.
— P<4wnn'ii IhoiBy of. 4SR,
— Wawr". Ofirj of. 442, 338.
Uairna*' ((>^ 1*«, 251
Uanco'i, IlcBry. nut'i'l. SG7.
MaltblMBii. Aui.. 3.V2. SBO.
MoMiiiviHvnl, Xinxtrj of, 1.
— (if riistria l(«otv is.
— orolMtraaUdo etfmMn, SSBSSO.
MeanmmMit nf DteotonnotiviB fona
potcBlul, 818, S&S.
— of naiiilanci!, X3G3S7.
— i)f oiinHtMil vurnaU. 748.
— «( tniwlfinl oiiirtBU. 748.
— ofooii.. 708, 75»7a7.
» magnecic, 449 VH.
Mnliuni, eltctfomafiictic, S98.
I uiiiiiiifcr(,iiiii. 6CI9.
Mlireiiry, rntiiUnot of, ML
.M rula. KwtUooe of, 383.
MjohoU. John. 3«.
MlUoT, W. n.. 28.
Mirror OMthod, 430.
MoWulu' cliar)(« «r tloclricily, S59.
— (lunnilii, 633,
— ■Unilarda. 9.
— vortlMf. 832.
MolaeuU*. liaa <£, h.
~ alMlrie. 300.
— magnvlU, 430. 893 34K.
Moraont. ma^tUo, 334.
— of liwnia. 589.
Moiaaiitliin. 3.
— alvcinikiiiDllc. 578, S8S.
MottcUi. <i. V^ 6±
Motion, «if iiatiiHU of, M3945.
Movins kin, 300.
— ooadu«(«n, COS.
— tRu«ai. ees.
MnltI>lo cniiducKoi^ 3T8. S14.
— FuiHTtiiaif, P.
Mullipliotiaii, MMMI oC 747. 7M.
Naatuann. F. %, eodBotcat of aa^aUiia
UoD, 4,10,
— niagnatimtion id «Ilfp«rid. 4SB.
— tiiODrjr of indund concaiU, 943.
NamuMui. C. G . lUO^ S30. 383.
Ntehoban'i [t«n>lTlnf Dovbkr. 20ft.
Niokal, 429.
Una lottho!'^ 114. 348, SOS.
f)iwud. a C «*», 47S.
OI><i>.l^.?t. 241.338.
Ohm. Uw, ■.iii, 340, 9S».
OpMilr. 788,
OTatj sllitwiia. 153.
Puling, A.. 304.
Parabulaiiit, oonGical, 154.
Panmusnxfc (lanie aa FnrannvMilo);,
439. 428, 844.
PdUor. A., 248.
Poriiidj,! niBcUue). 8.
Poiplanrtio Tttfim, ti, 118.
Pnmu*l>ilitt, iiBii{wtk>, 428. ni.
pbimp*. a L, 342.
Phn of Ui* Ttmiiw, 99.
Piano curTMt.alw(4, 894««l.
llaKUiy oUlraoU, 131.
^^^^^B^^^^^/jr^j^^^^^^^^4s^^^^
PUtymetfff, clcetrcp, 22P.
Rloiniuiii. BemhitriJ. 431, SII3. ^^^^
Pincl[<T.J"llu». S3l'.
itijflil uiJ 1nfthaiiil*d ■ytt««iu of SM* ^^^^H
PdIuIh of ti]ulUbriiim. 112.
^^^H
P«i»on. S, J>, !.'..'.. 431, 487. «74.
— ciTa<iiiulr'><ilaria«d nyi, 613. ^^^^
FauaoD*! equKtioD, 77, 1 49.
Rilohie, W.. Hi. ^^H
Paiawn's tiEiMrjr of laagiivUim, 137, 430,
431, 441, S3S.
BiiCot'i ^J. W.) SctDDFlary Pile, S71. ^^^H
DntaUoii of pliuiu of pDlarimCian. 808. ^^^^H
■ — iheary of vavoprnpayatjon. 7SI.
— ma^utlinii, a iiliiiiiuiunuan of, 631. ^^^^H
PoUr diUuiliiin uf uutEiielio (urou, 3I>S.
Klltilniaiiii. K.. 37i>. ^^^^
Poliuil;. 381.
RiiIp ol clooljomiigiiatio dlraoUtm, 477, ^H
PoLiriattiDti. dwtTiMtttCio. M. III.
li>4, 196. ^^M
•~ •kotfolvtio. 3:>T, ^42;2.
— mMtwdo, 3»I.
^fl
— f IlKhl, SSI. 791.
ficaiar. ^^^M
— oiiwilw, SI 3,
Sc^ln f<tr HLirror iilTHHrrati^rni, 450. ^^^^^
Pol«* «f a magnst, 373.
Sntorial li.iimoDic, 133, ISS. • ^^^H
■ — magiwiie of Uib urth, 469.
Hcebeck. T, J . 350. ^^^H
PodliTd and motive. liiiDTflntiani about.
Selenium, CI. 363. ^^^M
K 2S. ST, an, 37. 63, t!H'81. 231, 374. 'i^i.
■ 417. 489. 498.
~ Piiieotial. 16.
ti<.'K[ii<tucti»u. 7. ^^^^H
— miuftimiiniii (if, 75S, T7A, TTtt. ^^^^
— coil ofiuuiiuuin, 700. ^^^^H
— vkMrio. «5, TO. SiO.
SLniJti« gntoiiuiiieter. 717. ^^^^H
Sixic* of otHorratitnidi, 74U, 750. ^^^^H
^ — aasMUc, 3«3. 391.
^r— oCtwu ciivuita. 4'J^).
KImU. mntpcilc, 409, 4S4. 485. 806. tH, ^H
070, 094. 0W«. ^^M
— of tun <M«*. a9S.
Sliri«ni(, (;. W.. 33fi. 361.  ^^^H
Potential, voctw, 40B, 48!, 590, 617,
Sinn, nit^tliinl uf, (55, 710l ^^^^H
667.
Sin^tar poinla, 139. ^^^^H
Prini^<«l axca. 29S>. 303.
Slope. 17. ^^^B
Prublnnu, •IwUflntadc, ir>tSOS.
H— •iMArekiotiiliiLi':, 3<9H.;1SS.
^P^nu^stie. 131441.
Silvio. A.. 372. ^^^M
Stiiilh. AnhitnU. 441. ^^^H
Smith, Vi'. K.. 1S3. 316. ^^^H
— (iMtromagnttio. <J4T706.
>kvp bubble. 125. ^^^H
Proof of tha law of tlw lavatw Squiin,
Sa1«ioiii. iragncllo, 407. ^^^^^H
H 74.
~ aUcttiolnn<rn''l'c. 0706^1. 727. ^^^^
■ PnK>rpEanv,233.
ttolMlDldal dirt n Initios. 21. $S. 407, ^^^H
Kl
Solid MfI*. 400. 4174S3. 4S5. 69K. ^^^H
K
SpawvariatioQ, 17. 71. 639. ^^^H
Spatk. 57, 170 ^^H
Qoailric uirfKOT. 1471G4.
SpcfiiAc induetlra oapadtj. 53, 59; 94, ^^^H
Quaiitit)'. axpcwdMa Iwrm phjrukal, 1.
111, 32V. 32S, 334. 637. 763 ^^H
QuaatitiM. «l*iinliuMi«a of elcatnmu^
— oonductiTiiy, 27^ 637. ^^^H
Brticeao «29.
— midanot^ 277. 637. ^^^^
Qaatembma, 11. 303, 490. S2S, HIS.
^ baat of eloctririt;, 2Si. ^^^^H
QuJuke,G., 316 1..
Spherv, lis. ^^^H
Sphcdnal oanij union, 145. 146. ^^^^
SplnricHl Lanuimioi. 128146. 391. 481. ^^H
Badialian. faroM anovmed In, 799
Spin], lomriihiiitc. 781. ^^^^M
IUiildoe.W.J.M.. 115. CSI.
tJtMliliifl vlectruniftn, 317. ^^^^H
BayordoctruiiwfftinEicdldurbanoa, 791.
— gftlvanomotiT. 708. ^^^^^H
Bacipneal propuliL o, vlKtmalktic. 86.
StokM. G. G.. 24. 115, 761. ^^^H
— BlMlTDhilWInntio. set, SIS.
filuoar. Q. J.. ^^^H
— DtoKDacKv 4'il, 423.
Stntiflvd cniMliiDinn. 319. ^^^^
•— dtcirnuui^Detie. 538,
iStrMi. rlKtrvoUtio. li>5 HI. ^^^H
— kinollc Mi.
— dwlrokiuetic, «i 1. «I5. 016. ^^^H
ItaoAX iiiolliiid «f. 750.
Suult, Hon. J. W.. 103, SOS. ^^^H
Rfaianal cUiyK 3S743I.
SaiUee InloHnl. 15. 21, 75, 402. ^^^M
— ilu^atix*li"D, 444.
d«n>ity, 64, 223. ^^^H
Bcfiltaiitfaer. SIO.
Hiirfnur, iKiutiiDUvlial, 48. ^^^^
BoiiUtuw of cuDiluctani, SI, 4TB.
— •leotriHnl. 78. ^^^H
~ t^lo* of, 362300.
Suipoidod coil, 721';29. ^^^H
— MDnlioiin of Sn7.
Supciudan. brtiLar, IS9. ^^^^
— .laalc'i, ^^^H
— unit «f, 758767.
— «lM<iiM(atia mcaaura of. 353. 780.
— TbonuDD'a, 721. ^^^H
r""
— untfilu, ^^H
^^^45^^^^^^^^^^^?^^^^^^^^^^^
^^H 1W)lM«feaefB(iMitoarfto«IL 700l
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fVol. I. The Upanishads. Translated by F. Max Mulkr.
Part 1. ThrATindogii.upaniiliB'l.The Talavitini'iipiiaiihid. The Aitareya
iunnyaka. The Kauililtakibiihcnajiianiianitkid, aadTbc VSfiuancyi4aiiihlu
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^m IW. C«or][ UuUa. Pail 1. Ajmnamba and Gaaiuna. i<w. dt.
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1
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