Maxwell's Treatise, part IV, Ch. 9
Premessa
Quella he segue e la tras rizione del Cap. 9 della parte II del Treatise di
Maxwell, presa dalla s ansione della primaedizione (1873) disponibile in rete.
Ho onservato on la massima fedelta he mi e stata possibile la stesura
originale, an he negli aspetti tipogra i. In parti olare ho las iato le derivate
omelehas ritteMaxwell,an hedoveavrebbedovutousareisimbolididerivata
parziale( heai tempidiMaxwellerano giain usoda tempo). Gli uni i ambia-
menti he ho apportato sono:
{ hosostituitoleletteregoti hemaius ole heMaxwellusoperindi arevettori
onla orrispondente maius ola nel tipo \ alligra o" di TeX
{ hoaggiuntounanumerazione(fraparentesiquadra) alleequazioni nelriepi-
logo nale, per poterle ri hiamare nel ommento he si trova appresso.
Il ommentoavevainizialmenteilsolos opodifa ilitarelalettura,mostran-
do la s rittura moderna della stesse equazioni e ri ordando a hi non l'avesse
immediatamente presente il posto he ias una equazione o upa nella teoria.
Tuttavia strada fa endo le ose si sono ompli ate, per he l'impostazione
di Maxwell in diversi punti di eris e pare hio da quella oggi tradizionale: mi
sono quindi trovato ostretto a un'interpretazione e un onfronto. Ho er ato
di essere piu stringato possibile, an he se forse ho dovuto sa ri are un po' la
omprensibilita. Purtroppo non era possibile fare diversamente, a meno di non
ambiare ompletamenteprospettiva,s rivendoun arti olostori o- riti o. Cosa
he non era e none nelle mieintenzioni.
Dopo il ommento ho inserito una \Appendi e sui quaternioni": sebbene
nonindispensabileper omprendereiltesto,puoaiutare hinonabbiafamiliarita
on questa struttura, oggi po o usata, e voglia vedere meglio la relazione tra la
formadata da Maxwell alle sue equazioni e quella poi a ermatasi no ai nostri
giorni.
GENERALEQUATIONS OFTHE ELECTROMAGNETIC FIELD.
604.℄ Inourtheoreti aldis ussionofele trodynami swebeganbyassumingthat
asystemof ir uits arryingele tri urrentsisadynami alsystem,inwhi hthe
urrentsmayberegardedasvelo ities,andinwhi hthe oordinates orrespond-
ingtothesevelo itiesdonotthemselvesappearintheequations. It followsfrom
thisthatthekineti energyofthesystem,sofarasitdependsonthe urrents,isa
homogeneousquadrati fun tionofthe urrents, inwhi hthe oeÆ ientsdepend
onlyontheformandrelativepositionofthe ir uits. Assumingthese oeÆ ients
to be known, by experiment or otherwise, we dedu ed, by purely dynami al
reasoning,the lawsof the indu tion of urrents, and of ele tromagneti attra t-
ion. In this investigation we introdu ed the on eptions of the ele trokineti
energy of a system of urrents, of the ele tromagneti momentum of a ir uit,
and of the mutual potential of two ir uits.
Wethenpro eeded toexplore the eldbymeansofvarious on gurationsof
these ondary ir uit,andwerethusledtothe on eptionofave torA,havinga
determinate magnitudeand dire tion at anygivenpoint of the eld. We alled
this ve tor the ele tromagneti momentum at that point. This quantity may
be onsidered as the time-integral of the ele tromotive for e whi h would be
produ ed atthat point bythe suddenremoval of all the urrents fromthe eld.
It is identi al with the quantity already investigated in Art. 405 as the ve tor-
potential of magneti indu tion. Its omponents parallel to x, y, and z are F,
G, and H. The ele tromagneti momentumof a ir uit is theline-integral of A
round the ir uit.
We then, by means of Theorem IV, Art. 24, transformed the line-integral
ofA intothesurfa e-integral ofanotherve tor,B, whose omponentsarea,b, ,
and we found that the phenomena of indu tion due to motion of a ondu tor,
andthoseof ele tromagneti for e anbe expressedinterms ofB. WegavetoB
thenameof theMagneti indu tion,sin eitspropertiesareidenti alwiththose
of the lines of magneti indu tion as investigated by Faraday.
We also established three sets of equations: the rst set, (A), are those of
magneti indu tion, expressing it in terms of the ele tromagneti momentum.
The se ond set, (B), are those of ele tromotive for e, expressing it in terms of
the motion of the ondu tor a ross the lines of magneti indu tion, and of the
rate of variation of the ele tromagneti momentum. The third set, (C), are the
equations of ele tromagneti for e, expressing it in termsof the urrent andthe
magneti indu tion.
The urrentinallthese asesistobeunderstoodasthea tual urrent,whi h
in ludes notonly the urrent of ondu tion,but the urrent due tovariation of
the ele tri displa ement.
in Art. 400. In an unmagnetized body it is identi al with the for e on a unit
magneti pole,butifthebodyismagnetized,eitherpermanentlyorbyindu tion,
it is the for e whi h would be exerted on a unit pole, if pla ed in a narrow
revasse in the body, the walls of whi h are perpendi ular to the dire tion of
magnetization. The omponents of B area, b, .
It follows fromthe equations (A),by whi h a, b, are de ned, that
da
dx +
db
dy +
d
dz
=0:
This was shewn at Art.403 to bea property of the magneti indu tion.
605.℄ We have de ned the magneti for e within a magnet, as distinguished
from the magneti indu tion, to be the for e on a unit pole pla ed in a narrow
revasse utparallel tothe dire tionof magnetization. This quantityisdenoted
by H , and its omponents by , , . See Art.398.
If I is the intensity of magnetization, and A, B, C its omponents, then,
by Art.400,
a= +4A;
b= +4B;
= +4C:
9
=
;
(Equations of Magnetization.) (D)
We may all these the equations of magnetization, and they indi ate that
intheele tromagneti systemthe magneti indu tionB, onsideredas ave tor,
is thesum, in the Hamiltoniansense, of two ve tors, themagneti for e H , and
themagnetization I multipliedby 4, orB = H+4I. In ertainsubstan es,
the magnetization depends on the magneti for e, and this is expressed by the
systemof equations of indu ed magnetism given at Arts.426 and 435.
606.℄ Up to this point of our investigation we have dedu ed everything from
purely dynami al onsiderations, without any referen e to quantitative experi-
ments in ele tri ityor magnetism. The only use we have madeof experimental
knowledge is to re ognise, in the abstra t quantities dedu ed from the theory,
the on retequantities dis overedbyexperiment, andtodenote thembynames
whi h indi ate their physi al relations rather than their mathemati al genera-
tion.
In thisway we have pointedout theexisten e of theele tromagneti mom-
entumA asave torwhosedire tionandmagnitudevaryfromonepartofspa e
to another, and from this we have dedu ed, by a mathemati al pro ess, the
magneti indu tion, B, as a derived ve tor. We have not, however, obtained
any data for determining either A or B fromthe distribution of urrents in the
quantities and the urrents.
We begin by admitting the existen e of permanent magnets, the mutual
a tionof whi hsatis es theprin ipleof the onservation ofenergy. Wemakeno
assumptionwithrespe tto thelaws of magneti for eex ept that whi hfollows
from this prin iple, namely, that the for e a ting on a magneti pole must be
apableof being derived froma potential.
We then observe the a tion between urrents and magnets, and we nd
that a urrent a ts on a magnet in a manner apparently the same as another
magnetwoulda tifitsstrength,form,andpositionwereproperlyadjusted,and
thatthe magnet a tson the urrent in the sameway asanother urrent. These
observationsneednotbesupposedtobea ompaniedwitha tualmeasurements
of the for es. They are not therefore to be onsidered as furnishing numeri al
data, but areuseful only in suggesting questions for our onsideration.
The question these observations suggest is, whether the magneti eld
produ ed by ele tri urrents, as it is similar to that produ ed by permanent
magnets inmanyrespe ts, resembles it also in being related to a potential?
The eviden e that an ele tri ir uit produ es, in the spa e surrounding
it, magneti e e ts pre isely the same as those produ ed by a magneti shell
bounded by the ir uit, has been stated in Arts.482-485.
We know that in the ase of the magneti shell there is a potential, whi h
hasadeterminatevalueforallpointsoutsidethesubstan eof theshell,butthat
the values of the potential at two neighbouring points, on opposite sides of the
shell, di er by a nite quantity.
If the magneti eld in the neighbourhood of an ele tri urrent resembles
thatin the neighbourhood of a magneti shell, themagneti potential,as found
by a line-integration of the magneti for e, will be the same for any two lines
of integration, provided one of these lines an be transformed into the other by
ontinuous motionwithout utting the ele tri urrent.
If, however, one line of integration annot be transformed into the other
without utting the urrent, the line-integral of the magneti for e along the
one line will di er from that along the other by a quantity depending on the
strength of the urrent. The magneti potential due to an ele tri urrent is
thereforeafun tion having anin niteseriesofvalueswitha ommondi eren e,
the parti ular value depending on the ourse of the line of integration. Within
the substan e of the ondu tor, thereis no su h thing as a magneti potential.
607.℄ Assuming that the magneti a tion of a urrent has a magneti potential
of this kind, we pro eed to express this result mathemati ally.
In the rst pla e, the line-integral of the magneti for e round any losed
urve is zero, provided the losed urve does not surround the ele tri urrent.
losed urve in the positive dire tion, the line integral has a determinate value,
whi hmaybe usedasa measureof thestrengthof the urrent. Forif the losed
urvealters its formin any ontinuous mannerwithout utting the urrent, the
line-integral will remain thesame.
In ele tromagneti measure, the line-integral of the magneti for e round
a losed urve is numeri ally equal to the urrent through the losed urve
multiplied by 4.
Ifwe takefor the losed urve theparallelogramwhose sidesaredy anddz,
the line-integral of the magneti for e round the parallelogram is
d
dy d
dz
dydz;
and if u, v, w are the omponents of the ow of ele tri ity, the urrent through
the parallelogramis
udydz:
Multiplying this by 4, and equating the result to the line-integral, we obtain
the equation
4u= d
dy d
dz
;
with the similar equations
4v= d
dz d
dx
;
4w= d
dx d
dy
; 9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
(Equationsof
Ele tri Currents.) (E)
whi h determine the magnitude and dire tion of the ele tri urrents when the
magneti for e at every point is given.
When there is no urrent, these equations are equivalent to the ondition
that
dx+ dy+ dz = D;
orthatthe magneti for e isderivablefromamagneti potential inallpoints of
the eld wherethere are no urrents.
Bydi erentiatingtheequations (E)withrespe ttox,y,andz respe tively,
and addingthe results, we obtainthe equation
du
dx +
dv
dy +
dw
dz
=0;
whi h indi atesthatthe urrent whose omponents areu, v,w is subje t tothe
onditionof motionof an in ompressible uid, andthatit mustne essarily ow
in losed ir uits.
ele tri ow whi h is due to the variation of ele tri displa ement as well as to
true ondu tion.
We have very little experimental eviden e relating to the dire t ele tro-
magneti a tion of urrents due to the variation of ele tri displa ement in
diele tri s,buttheextreme di i ultyofre on ilingthelawsof ele tromagnetism
withthe existen e of ele tri urrents whi h arenot losed is one reasonamong
manywhywemustadmittheexisten eoftransient urrentsduetothevariation
of displa ement. Their importan e will be seen when we ome to the ele tro-
magneti theory of light.
608.℄ Wehavenowdeterminedtherelationsoftheprin ipalquantities on erned
in the phenomena dis overed by
Orsted, Ampere, and Faraday. To onne t
these with the phenomena des ribed in the former parts of this treatise, some
additionalrelations are ne essary.
When ele tromotive for e a ts on a material body, it produ es in it two
ele tri al e e ts, alled by Faraday Indu tion and Condu tion, the rst being
most onspi uousin diele tri s, and the se ond in ondu tors.
Inthistreatise, stati ele tri indu tionismeasuredbywhatwehave alled
the ele tri displa ement, a dire ted quantity or ve tor whi h we have denoted
by D, and its omponentsby f, g, h.
In isotropi substan es, the displa ement is in the same dire tion as the
ele tromotive for e whi h produ es it, and is proportional to it, at least for
small values of this for e. This may be expressed bythe equation
D= 1
4
KE;
(Equation of Ele tri
Displa ement.)
(F)
where K is the diele tri apa ity of the substan e. SeeArt. 69.
Insubstan eswhi harenotisotropi ,the omponentsf,g,h oftheele tri
displa ement D are linear fun tions of the omponents P, Q, R of the ele tro-
motive for e E.
The formof the equations of ele tri displa ement is similar to that of the
equations of ondu tion as given in Art.298.
Theserelationsmaybe expressedbysaying thatK is, inisotropi bodies,a
s alar quantity, but in other bodies it is a linear and ve tor fun tion, operating
on theve tor E.
609.℄ Theothere e t ofele tromotivefor e is ondu tion. Thelawsof ondu t-
ionastheresultofele tromotivefor ewereestablishedbyOhm,andareexplain-
edin the se ondpart of thistreatise, Art.241. They maybe summedup inthe
equation
K=CE; (Equation of Condu tivity.) (G)
of the urrent of ondu tion, the omponents of whi h are p, q, r, and C is the
ondu tivity of the substan e, whi h, in the ase of isotropi substan es, is a
simple s alar quantity, but in other substan es be omes a linear and ve tor
fun tion operating on the ve tor E. The form of this fun tion is given in
Cartesian oordinates in Art.298.
610.℄ Oneofthe hief pe uliaritiesof thistreatiseisthedo trinewhi hitasserts,
that the true ele tri urrent C, that on whi h the ele tromagneti phenomena
depend, is not the same thing as K , the urrent of ondu tion, but that the
time-variation of D, the ele tri displa ement, must be taken into a ount in
estimating thetotal movement of ele tri ity, so thatwe must write,
C =K+ _
D (Equation of True Currents.) (H)
or, in terms of the omponents,
u =p+ df
dt
;
v =q+ dg
dt
;
w=r+ dh
dt :
9
>
>
>
>
>
=
>
>
>
>
>
;
(H)
611.℄ Sin ebothK and D depend on theele tromotive for e E, we mayexpress
the true urrent C in terms of the ele tromotivefor e, thus
C =
C+ 1
4
K d
dt
E; (I)
or, in the ase inwhi h C and K are onstants,
u=CP + 1
4
K dP
dt
;
v=CQ+ 1
4
K dQ
dt
;
w=CR+ 1
4
K dR
dt :
9
>
>
>
>
>
=
>
>
>
>
>
;
(I)
612.℄ The volume-density of the free ele tri ity at any point is found from the
omponents of ele tri displa ement by the equation
%= df
dx +
dg
dy +
dh
dz
: (J)
=lf +mg+nh+l 0
f 0
+m 0
g 0
+n 0
h 0
; (K)
where l, m, n are the dire tion- osines of the normal drawn from the surfa e
into the medium in whi h f, g, h are the omponents of the displa ement,
and l 0
, m 0
, n 0
are those of the normal drawn from the surfa e into the medium
in whi h they are f 0
, g 0
, h 0
.
614.℄ Whenthemagnetizationofthemediumisentirelyindu edbythemagneti
for e a ting on it, we maywrite the equation of indu ed magnetization,
B=H ; (L)
where is the oeÆ ient of magneti permeability, whi h maybe onsidered a
s alarquantity,oralinearand ve torfun tionoperatingonH , a ordingasthe
mediumis isotropi or not.
615.℄ These may be regarded as the prin ipal relations among the quantities
we have been onsidering. They may be ombined so as to eliminate some
of these quantities, but our obje t at present is not to obtain ompa tness in
the mathemati al formulae, but to express every relation of whi h we have any
knowledge. Toeliminateaquantitywhi hexpressesausefulideawouldberather
a loss than a gainin this stage of our enquiry.
There is one result, however, whi h we may obtain by ombining equat-
ions (A)and (E), and whi h is of very great importan e.
Ifwesupposethatnomagnetsexistinthe eldex eptintheformofele tri
ir uits,thedistin tionwhi hwehavehithertomaintainedbetweenthemagneti
for e and the magneti indu tion vanishes, be ause it is only in magnetized
matter thatthese quantities di er fromea h other.
A ording to Ampere's hypothesis, whi h will be explained in Art. 833,
the properties of what we all magnetized matter are due to mole ular ele tri
ir uits, so that it is only when we regard the substan e in large masses that
our theory of magnetization is appli able, and if our mathemati al methods
are supposed apable of taking a ount of what goes on within the individual
mole ules, they will dis over nothing but ele tri ir uits, and we shall nd
the magneti for e and the magneti indu tion everywhere identi al. In order,
however, to be able to make use of the ele trostati or of the ele tromagneti
systemofmeasurementatpleasureweshallretainthe oe i ient,remembering
that its value is unity in the ele tromagneti system.
a= dH
dy
dG
dz
;
b= dF
dz
dH
dx
;
= dG
dx
dH
dy :
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
The omponents of the ele tri urrent are by equations (E), Art.607,
4u= d
dy d
dz
;
4v= d
dz d
dx
;
4w= d
dx d
dy :
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
A ording to our hypothesis a,b, are identi al with , , respe t-
ively. We therefore obtain
4u= d
2
G
dxdy d
2
F
dy 2
d 2
F
dz 2
+ d
2
H
dzdx
: (1)
If we write
J = dF
dx +
dG
dy +
dH
dz
; (2)
and*
r 2
=
d 2
dx 2
+ d
2
dy 2
+ d
2
dz 2
: (3)
we maywrite equation (1),
4u= dJ
dx +r
2
F:
Similarly, 4v=
dJ
dy +r
2
G;
4w= dJ
dz +r
2
H:
9
>
>
>
>
>
=
>
>
>
>
>
;
(4)
*The negative sign is employed here in order to make our expressions onsi-
stent with those in whi h Quaternions are employed.
F 0
= 1
u
r
dxdydz;
G 0
= 1
ZZZ
v
r
dxdydz;
H 0
= 1
ZZZ
w
r
dxdydz;
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
(5)
= 4
ZZZ
J
r
dxdydz; (6)
where r is the distan e of the given point from the element x, y, z, and the
integrations are to be extended over all spa e, then
F =F 0
+ d
dx
;
G=G 0
+ d
dy
;
H =H 0
+ d
dz :
9
>
>
>
>
>
=
>
>
>
>
>
;
(7)
The quantity disappears from the equations (A),and it is not related to
any physi al phenomenon. If we suppose it to be zero everywhere, J will also
be zero everywhere, and equations (5), omitting the a ents, will give the true
values of the omponents of A.
617.℄ Wemaythereforeadopt,asade nitionofA,thatitistheve tor-potential
of the ele tri urrent, standing in thesame relationto the ele tri urrent that
thes alarpotentialstandstothematterofwhi hitisthepotential,andobtained
by a similar pro ess of integration, whi h may be thus des rihed.
From a given point let a ve tor be drawn, representing in magnitude and
dire tion a given element of an ele tri urrent, divided by the numeri al value
of the distan e of the element from the given point. Let this be done for every
element of the ele tri urrent. The resultant of all the ve tors thus found is
the potential of the whole urrent. Sin e the urrent is a ve tor quantity, its
potential is also a ve tor. See Art.422.
Whenthedistributionofele tri urrentsisgiven,thereisone,andonlyone,
distributionof thevaluesof A, su hthat Ais everywhere niteand ontinuous,
and satis es the equations
r 2
A=4C; SrA=0;
and vanishes at an in nite distan efrom the ele tri system. This value is that
givenby equations (5), whi h maybe written
A= 1
ZZZ
C
r
dxdydz:
618.℄ In thistreatisewehave endeavouredtoavoidanypro essdemandingfrom
the reader a knowledge of the Cal ulus of Quaternions. At the same time we
have not s rupled to introdu ethe idea of a ve tor when it was ne essary to do
so. When we have had o asion to denote a ve tor by a symbol, we have used
aGerman letter, thenumber of di erent ve tors being so greatthat Hamilton's
favourite symbols would have been exhausted at on e. Whenever therefore,
aGerman letter is usedit denotes aHamiltonianve tor, and indi atesnot only
its magnitude but its dire tion. The onstituents of a ve tor are denoted by
Roman or Greek letters.
Theprin ipal ve tors whi h we have to onsider are :|
Symbolof
Ve tor.
Constituents.
The radius ve tor of a point:::::::::::::: % x y z
The ele tromagneti momentum of a point A F G H
The magneti indu tion:::::::::::::::::: B a b
The (total) ele tri urrent::::::::::::::: C u v w
The ele tri displa ement::::::::::::::::: D f g h
The ele tromotive for e:::::::::::::::::: E P Q R
The me hani alfor e::::::::::::::::::::: F X Y Z
The velo ityof a point::::::::::::::::::: G or %_ x_ y_ z_
The magneti for e::::::::::::::::::::::: H
The intensityof magnetization::::::::::: I A B C
The urrent of ondu tion:::::::::::::::: K p q r
Wehave also the followings alar fun tions :{
Theele tri potential .
Themagneti potential (where it exists).
Theele tri densitye.
Thedensityof magneti `matter'm.
Besidesthesewehavethefollowingquantities,indi atingphysi alproperties
of the medium at ea h point :|
C, the ondu tivity for ele tri urrents.
K, the diele tri indu tive apa ity.
,the magneti indu tive apa ity.
These quantities are, in isotropi media, mere s alar fun tions of %, but
ingeneral they are linear andve tor operators on theve tor fun tions to whi h
theyareapplied. K andare ertainlyalwaysself- onjugate,andC isprobably
so also.
a= dH
dy
dG
dz
;
may now be written
B=VrA; [1℄
where r is theoperator
i d
dx +j
d
dy +k
d
dz
;
andV indi atesthattheve torpartoftheresultofthisoperationistobetaken.
Sin eA is subje t to the ondition SrA =0, rA is a pure ve tor, andthe
symbol V is unne essary.
Theequations (B)of ele tromotivefor e, of whi h the rstis
P = y_ bz_ dF
dt
d
dx
;
be ome
E =VGB _
A r : [2℄
Theequations (C) of me hani alfor e, of whi h the rst is
X = v bw e d
dx m
d
dx
;
be ome
F =VCB er mr: [3℄
Theequations (D)of magnetization,of whi h the rst is
a= +4A;
be ome
B=H+4I: [4℄
Theequations (E) of ele tri urrents, of whi h the rst is
4u= d
dy d
dz
;
be ome
4C =VrH : [5℄
Theequation of the urrent of ondu tion is, byOhm's Law,
K=CE: [6℄
D= 1
4
KE: [7℄
The equation of the total urrent, arising fromthe variation of the ele tri
displa ement as well as from ondu tion,is
C =K+ _
D: [8℄
When the magnetizationarises from magneti indu tion,
B=H : [9℄
Wehave also, to determinethe ele tri volume-density,
e =SrD: [10℄
To determine the magneti volume-density,
m=SrI: [11℄
When the magneti for e an be derived froma potential
H= r: [12℄
Qui di seguito elen o le equazioni he si trovano nella pagina nale del
Cap. 9, indi andone il signi ato moderno. Comin io on due delle \eq. di
Maxwell" propriamente dette:
e =SrD rD=% [10℄
4C =VrH rH=4
j+
D
t
: [5℄
Man ano, ome si vede, due equazioni: quella per la divergenza di B, e quella
per il rotore di E. La prima in realta si trova s ritta, in omponenti, ome
eq. (18) all'Art.403:
da
dx +
db
dy +
d
dz
=0:
Mi sentoper io autorizzato ad aggiungerla:
SrB=0 rB=0: [13℄
Si noti he la [13℄e onseguenza della [1℄:
B=VrA B=rA: [1℄
Quanto al ampo elettri o, sembrerebbe de nitodalla [2℄
(1)
:
E =VGB _
A r E =vB
A
t
r : [2℄
Ma o orre osservare he quello he M. indi a on E non e io he oggi viene
inteso on questo nome. Basta seguire il ragionamento dell'Art. 599 per apire
heM.indi a onE il ampola ui ir uitazione, al olataaunistante ssatosu
un ir uito in moto, fornis e la f.e.m.indotta. Nel detto arti olo E viene diviso
in tre parti:
{ E
1
dipendedalmotodell'elementodi ir uito: lasuaespressione( onsimboli
moderni)e vB (forza di Lorentz).
{ E
2
dipende dalla variazione del ampo magneti o: la sua espressione
e A=t.
{ E
3
dipende dalla variazionespaziale del potenziales alare , e vale r .
(1)
M. usasistemati amenteilpunto, omein _
A oin _
D, dovenoioggis riverem-
mounaderivataparziale. An helederivaterispettoalle oordinate, heandreb-
bero s ritte ome parziali, le s rive ome derivate ordinarie: si veda ad es. la
divergenzadi B s ritta sopra.
ma non il primo. Se lo hiamo E 0
, dalla [2℄ prendendo il rotore e usando la [1℄
ottengo
VrE 0
= _
B rE
0
=
B
t
[14℄
he e la quarta eq. di Maxwell (legge dell'induzione).
Passiamo a questioni piusempli i. La [4℄
B=H+4I B=H+4M [4℄
e in uso an or oggi on la stessa forma, ed esprime la relazione tra B, H
e l'intensita di magnetizzazioneM. Le due eq. [7℄ e [9℄
D= 1
4
KE D=
"
4
E [7℄
B=H B=H [9℄
note ome\eq.di ollegamento"o\eq.distruttura"nonhannovaliditagenerale,
mavalgono solonei asilineari: quando lapolarizzazioneelettri a orisp.lama-
gnetizzazione dipendono linearmente dai ampi. Inoltre e assunta l'isotropia:
questoM. lodi eespli itamenteeri orda healtrimenti" ediventanotensori.
NaturalmenteM.nonusaquestotermine, heavrebbepresoilsigni atoattuale
solo anni dopo [?℄.
Un'equazione on arattere simile alle pre edenti e la [6℄:
K=CE j=E [6℄
heesprimeinformadi erenzialelaleggediOhm: ladensitadi orrente(di on-
duzione)eproporzionaleal ampoelettri o. Sinoti he ompareE,nonE 0
,ede
giusto, nel senso he se il mezzo onduttore e in moto, a muovere le ari he
ontribuis e an he la forza di Lorentz.
Non ho tradotto la [8℄
C =K+ _
D [8℄
in forma moderna, per he un equivalente della \ orrente totale" C oggi non e
in uso. Infatti nella [5℄, dove M. usa C, vi ho sostituito j+D=t, servendomi
proprio della de nizione [8℄ di C.
Vediamoora altre due equazioni. La prima e
m=SrI rM=%
m
[11℄
dove%
m
ela\densitadi ari adimagnetizzazione." Sebbenele ari hemagneti-
henonesistano,M. introdu equestotermineinanalogiaaquellodi\densitadi
quello di polarizzazione elettri a P, analogo alla magnetizzazione M, non mi
pare he siano presenti nel Treatise. Anti ipo he %
m
onparira nella [3℄ he
vedremo piu oltre.
La se onda equazionee
H= r H= r [12℄
Questaintrodu eil on ettodi\potenzialemagneti o", hehasensonei asiin
ui,annullandosiilse ondomembrodella[5℄,il ampomagneti oe onservativo.
Ho las iato per ultima la [3℄, he e un aso spinoso:
F =VCB er mr f =jB %r %
m
r: [3℄
Questa e hiamata da M. \equazione della forza me ani a." Fa io anzitutto
notare omeho tras rittolenotazioni. DoveM. usa C, hosostituitoj, henone
la stessa osa; spiegazione tra po o. Dove s rive F ho preferito la minus ola f:
questo per he si trattadella densita diforza (forza per unita divolume) agente
suunsistema ontinuo hepossiedeunadensitadi ari aelettri a%, unadensita
di ari a magneti a%
m
e porta una densita di orrente elettri a j.
Se ondo M. la (densita di) forza in presenza di ampo magneti o e dovuta
nona j ma aj+D=t. La giusti azione sitrova all'Art.604, dove sidi e he
\la orrente in tutti questi asi va intesa ome la orrente e ettiva, he in lude
non solo la orrente di onduzione, ma an he la orrente dovuta alla variazione
dell'induzione elettri a (D)." Ede tutto, quanto a spiegazione.
I asi di ui si tratta sono appunto la forza elettromotri e [5℄ e la forza
me ani a [3℄. Della [5℄ ho gia detto; quanto alla [3℄, non vedo ome si possa
motivare l'intervento di D=t. Si dovrebbe supporre he per es. undielettri o
polarizzabile inpresenza diun ampo elettri ovariabile( he generauna D=t
non nulla) e di un ampo magneti o B risenta una forza. A me questo non
sembra vero,opermeglio direloa ettereise alpostodi D=t i fosse P=t.
Ma las iamo la questione in sospeso, almenoper ora.
Nell'Art.607 e an he dedottal'equazione di ontinuita, s ritta
du
dx +
dv
dy +
dw
dz
=0:
Dato he u, v, w sono le omponenti della orrente totale C, questa si tradu e
SrK+Sr _
D=0 rj+r
D
t
=0
e usando la [10℄:
SrK+e_ =0 rj+
%
t
=0: [13℄
Eq. he non omparenel Treatise, mae onseguenza immediata di eq. he sono
presenti. Per questo motivo l'ho aggiunta alla lista.
Si possono seguire diversi appro i per introdurre i quaternioni. Qui, dato
lo s opo di questo s ritto, la via migliore e forse quella piu vi ina all'originaria
di Hamilton,ossia ome generalizzazione del ampo omplesso.
Chiamerodunque quaternione una s rittura del tipo
Q=a+bi+ j +dk
dove a,b, , d sononumerireali, e i, j, k sono treunita immaginarie. Hamilton
de nis e la\parte s alare" e la \parte vettoriale" di un quaternione:
SQ def
= a VQ
def
= bi+ j +dk:
SeVQ=0, Qedetto \s alare." Se SQ=0,edetto \vettoriale." Si noti hela
parte s alare di un quaternione puo essere identi ata on un reale, mentre la
parte vettorialepuo essere interpretata ome un vettore (b; ;d) di IR 3
.
Comenel asodei omplessi,traquaternionisonode nitisomma eprodotto.
La sommae ovvia: se
Q
1
=a
1 +b
1 i+
1 j +d
1
k Q
2
=a
2 +b
2 i+
2 j +d
2 k
abbiamo
Q
1 +Q
2 def
= (a
1 +a
2 )+(b
1 +b
2
)j+(
1 +
2
)j+(d
1 +d
2 )k:
Per il prodotto si parte de nendo i prodotti delle unita immaginarie, ome
segue:
i 2
=j 2
=k 2
= 1
ij = ji=k
jk = kj =i
ki= ik =j:
(A:1)
Dopo di he il prodottotra due quaternioni generi i si estende perlinearita:
Q
1 Q
2
=(a
1 a
2 b
1 b
2
1
2 d
1 d
2 )+
(a
1 b
2 +a
2 b
1 +
1 d
2
2 d
1 )i+
(a
1
2 +a
2
1 +d
1 b
2 d
2 b
1 )j +
(a
1 d
2 +a
2 d
1 +b
1
2 b
2
1 )k:
(A:2)
Si vede bene dalle (A.1), ma an he delle (A.2), he la moltipli azione non e
ommutativa.
L'espressione (A.2) del prodotto, piuttosto ompli ata, si sempli a molto
se Q
1 , Q
2
sono vettoriali. In tal aso
Q
1 Q
2
= (b
1 b
2 +
1
2 +d
1 d
2 )+(
1 d
2
2 d
1
)i+(d
1 b
2
d
2 b
1
)j +(b
1
2 b
2
1 )k:
1 1 1 1
e analogamente v
2
, si vede subito he
S(Q
1 Q
2
)= v
1
v
2
V(Q
1 Q
2 )=v
1
v
2 :
Si de nis e oniugato di un quaternione Q il
Q tale he
S
Q=SQ V
Q = VQ:
Osserviamo he
Q
Q=a 2
+b 2
+ 2
+d 2
(A:3)
Per nire: a Hamilton e an he dovuta l'invenzione dell'operatore vetto-
riale r:
r def
= i
x +i
y +k
z :
Appli ando r a un ampo quaternioni o puramente vettoriale, on la regola
formale del prodotto, si ottiene
SrQ= rv VrQ=rv:
Si ottengonoquindi in un olpo solo divergenzae rotore del ampo vettoriale v
(attenzionepero al segno \ " nellaparte s alare).
Poniamo iun problema: esiste sempre l'inverso diun quaternione? Questo
vuol dire, dato Q, se esiste Q 0
tale he QQ 0
= 1 (inverso destro), oppure
heQ 0
Q=1(inversosinistro). Sebbenelamoltipli azionenonsia ommutativa,
si dimostra fa ilmente he i due inversi, se esistono, debbono oin idere. Dal-
la (A.3)segue he se Q 6=0 l'inverso esiste (ede uni o):
Q 0
=
Q
a 2
+b 2
+ 2
+d 2
:
Pertanto i quaternioni formano un ampo non ommutativo.
