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* Philosophical Magazine, May 1846, or Expenmintal Researches, ill. p. 447.



536 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

proposed by him, is the same in substance as that which I have begun to
develope in this paper, except that in 1846 there were no data to calculate
the velocity of propagation.

(21) The general equations are then applied to the calculation of the coef-
ficients of mutual induction of two circular currents and the coefficient of self-
induction in a coil. The want of uniformity of the current in the different
parts of the section of a wire at the commencement of the current is investi-
gated, I believe for the first time, and the consequent correction of the coefficient
of self-induction is found.

These results are applied to the calculation of the self-induction of the coil
used in the experiments of the Committee of the British Association on Standards
of Electric Kesistance, and the value compared with that deduced from the
experiments.



PART 11.

ON ELECTROMAGNETIC INDUCTION.

Electromagnetic Momentum of a Current.

(22) We may begin by considering the state of the field in the neigh-
bourhood of an electric current. We know that magnetic forces are excited in
the field, their direction and magnitude depending according to known laws
upon the form of the conductor carrying the current. When the strength of
the current is increased, all the magnetic effects are increased in the same pro-
portion. Now, if the magnetic state of the field depends on motions of the
medium, a certain force must be exerted in order to increase or diminish these
motions, and when the motions are excited they continue, so that the effect
of the connexion between the current and the electromagnetic field surrounding
it, is to endow the current with a kind of momentum, just as the connexion
between the driving-point of a machine and a fly-wheel endows the driving-point
with an additional momentum, which may be called the momentum of the fly-
wheel reduced to the driving-point. The unbalanced force acting on the driving-
point increases this momentum, and is measured by the rate of its increase.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 537

In the case of electric currents, the resistance to sudden increase or dimi-
nution of strength produces effects exactly like those of momentum, but the
amount of this momentum depends on the shape of the conductor and the
relative position of its different parts.

Mutual Action of two Currents.

(23) If there are two electric currents in the field, the magnetic force at
any point is that compounded of the forces due to each current separately,
and since the two currents are in connexion with every point of the field,
they will be in connexion with each other, so that any increase or diminution
of the one will produce a force acting with or contrary to the other.

Dynamical Illustration of Reduced Momentum.

(24) As a dynamical illustration, let us suppose a body C so connected
with two independent driving-points A and B that its velocity is p times that
of A together with q times that of B. Let u be the velocity of A, v that
of B, and w that of C, and let hx, By, Sz be their simultaneous displacements,
then by the general equation of dynamics*;

C~Sz = XZx+YBy,

where A' and Y are the forces acting at A and B.
^ , dw du dv

and Zz=phx + qSy.

Substituting, and remembering that Bx and By are independent,



Y = j^{Cpq"+Cq'v)



0)-



We may call Cphi+Cpqv the momentum of C referred to A, and Cpqu + Cq'v
its momentum referred to B ; then we may say that the effect of the force
X is to increase the momentum of C referred to A, and that of Y to increase
its momentum referred to B.

* Lagrange, Mec. Anal. ii. 2, § 5.



538 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

If there are many bodies connected with A and i? in a similar way but
with diffierent values of p and q, we may treat the question in the same way
by assuming

L = %(Cp% M=t{Cpq), and N=t{Cq'),
where the summation is extended to all the bodies with their proper values of
C, p, and q. Then the momentum of the system referred to A is

Lu + Mv,
and referred to B, Mu + Nv,

and we shall have -^ = X (Lu + Mv)



dt
Y=^(Mu + Nv)



(2),



dt
where X and Y are the external forces acting on A and B.

(25) To make the illustration more complete we have only to suppose
that the motion of A is resisted by a force proportional to its velocity, which
we may call Ru, and that of ^ by a similar force, which we may call Sv^ R and
S being coefficients of resistance. Then if ^ and -q are the forces on A and B,



^=X + Ru = Ru + j^ (Lu + Mv)

rj= Y + Sv= Sv +j^{Mu + Nv)



(3).



If the velocity of A be increased at the rate -r- , then in order to prevent B
from moving a force, rj = -,- (Mu) must be applied to it.

This effect on B, due to an increase of the velocity of A, corresponds to
the electromotive force on one circuit arising from an increase in the strength
of a neighbouring circuit.

This dynamical illustration is to be considered merely as assisting the
reader to understand what is meant in mechanics by Reduced Momentum. The
facts of the induction of currents as depending on the variations of the quantity
called Electromagnetic Momentum^ or Electrotonic State, rest on the experiments
<^f Faraday", Felicif, &c.

* Experimental Besearclics, Series i., ix. t Annates de Chimie, ser. 3, xxxiv. (1852), p. 64.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 539

Coefficients of Induction for Two Circuits.

(26) In the electromagnetic field the values of Z, M, N depend on the
distribution of the magnetic effects due to the two circuits, and this distri-
bution depends only on the form and relative position of the circuits. Hence
L, M, N are quantities depending on the form and relative position of the
circuits, and are subject to variation with the motion of the conductors. It will
be presently seen that L, M, N are geometrical quantities of the nature of lines,
that is, of one dimension in space ; L depends on the form of the first conductor,
which we shall call A, N on that of the second, which we shall call B, and
M on the relative position of A and B.

(27) Let ^ be the electromotive force acting on A, x the strength of the
current, and R the resistance, then Ex will be the resisting force. In steady
currents the electromotive force just balances the resisting force, but in variable
currents the resultant force ^-Rx is expended in increasing the "electro-
magnetic momentum," using the word momentum merely to express that which
is generated by a force acting during a time, that is, a velocity existing in a
body.

In the case of electric currents, the force in action is not ordinary
mechanical force, at least we are not as yet able to measure it as common force,
but we call it electromotive force, and the body moved is not merely the
electricity in the conductor, but something outside the conductor, and capable
of being afiected by other conductors in the neighbourhood carrying currents.
In this it resembles rather the reduced momentum of a driving-point of a
machine as influenced by its mechanical connexions, than that of a simple
moving body like a cannon ball, or water in a tube.

Electromagnetic Relations of two Conducting Circuits.

(28) In the case of two conducting circuits, A and B, we shall assume
that the electromagnetic momentum belonging to A is

Lx + My,
and that belonging to B, Mx + Ny,

where L, M, N correspond to the same quantities in the dynamical illustration,
except that they are supposed to be capable of variation when the conductors
A or B are moved.

G8— 2



540 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

Then the equation of the current x in A will be

^=R^ + :jl{Lx-^My) (4),

and that of y in ^ r} = Sy+ -^-(Mx-j-Ny) (5)^

where ^ and tj are the electromotive forces, x and y the currents, and R and S
the resistances in A and B respectively.



Induction of one Current by another.

(29) Case 1st. Let there be no electromotive force on B, except that
which arises from the action of A, and let the current of A increase from
to the value x, then

Sy + ^^(Mx + Ny) = 0,



'hence 1



ft M
-^\ydt=.- — x, (6)



that is, a quantity of electricity Y, being the total induced current, will flow
through B when x rises from to x. This is induction by variation of the
current in the primary conductor. When M is positive, the induced current
due to increase of the primary current is negative.

Induction hy Motion of Conductor.
(30) Case 2nd. Let x remain constant, and let M change from M to M',



then

M'-M



(7)



so that if M is increased, which it will be by the primary and secondary
circuits approaching each other, there will be a negative induced current, the
total quantity of electricity passed through B being Y.

This is induction by the relative motion of the primary and secondary con-
ductors.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 541

Equation of Work and Energy.

(31) To form the equation between work done and energy produced,
multiply (1) by x and (2) by y, and add

^x + -ny=^Rx^^Sy' + Xj^(Lx + My) + yj^(Mx + Ny) (8).

Here ^x is the work done in unit of time by the electromotive force ^ actmg
on the current x and maintaining it, and r^y is the work done by the elec-
tromotive force 7). Hence the left-hand side of the equation represents the work
done by the electromotive forces in unit of time.

Heai produced by the Current.

(32) On the other side of the equation we have, first,

Rx? + Sf = H (9),

which represents the work done in overcoming the resistance of the circuits in
unit of time. This is converted into heat. The remaining terms represent
work not converted into heat. They may be written

Intrinsic Energy of the Currents.

(33) U L, M, N are constant, the whole work of the electromotive forces
which is not spent against resistance will be devoted to the development of
the currents. The whole intrinsic energy of the currents is therefore

^Laf + Mxy + iNy' = E (10).

This energy exists in a form imperceptible to our senses, probably as actual
motion, the seat of this motion being not merely the conducting circuits, but
the space surrounding them.

Mechanical Action between Conductors.



(34) The remaining terms,



dL dM dN ^y (11)



542 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

represent the work done in unit of time arising from the variations of L, M,
and N, or, what is the same thing, alterations in the form and position of the
conducting circuits A and B.

Now if work is done when a body is moved, it must arise from ordinary
mechanical force acting on the body while it is moved. Hence this part of
the expression shews that there is a mechanical force urging every part of the
conductors themselves in that direction in which L, M, and N will be most
increased.

The existence of the electromagnetic force between conductors carrying
currents is therefore a direct consequence of the joint and independent action
of each current on the electromagnetic field. If A and B are allowed to approach
a distance ds, so as to increase M from M to M' while the currents are x
and y, then the work done will be

{]\r-3i)xij,

and the force in the direction of ds will be

f^^ 02).

and this will be an attraction if x and y are of the same sign, and if 31 is
increased as A and B approach.

It appears, therefore, that if we admit that the unresisted part of electro-
motive force goes on as long as it acts, generating a self-persistent state of
the current, which we may call (from mechanical analogy) its electromagnetic
momentum, and that this momentum depends on circumstances external to the
conductor, then both induction of currents and electromagnetic attractions may
be proved by mechanical reasoning.

What I have called electromagnetic momentum is the same quantity which
is called by Faraday '" the electrotonic state of the circuit, every change of which
involves the action of an electromotive force, just as change of momentum
involves the action of mechanical force.

If, therefore, the phenomena described by Faraday in the Ninth Series of
his Experimental Researches were the only known facts about electric currents,
the laws of Ampere relating to the attraction of conductors carrying currents,

* Experiinental Researches, Series i. 60, &c.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 543

as well as those of Faraday about the mutual induction of currents, might be
deduced by mechanical reasoning.

In order to bring these results within the range of experimental verifica-
tion, I shall next investigate the case of a single current, of two currents, and
of the six currents in the electric balance, so as to enable the experimenter
to determine the values of L, M, N.

Case of a single Circuit.
(35) The equation of the current a: in a circuit whose resistance is 7i*,
and whose coefficient of self-induction is L, acted on by an external electro-
motive force ^, is

f-^=ai^- <")■

When i is constant, the solution is of the form

x = h + (a — h)e-L\
where a is the value of the current at the commencement, and h is its tinal
value.

The total quantity of electricity which passes in time t, where t is great, is

rxdt = ht + {a-h)~ (14).

The value of the integral of of with respect to the time is

jydt = lHHa-l)^{^) (15).

The actual current changes gradually from the initial value a to the final vatue
b, but the values of the integrals of x and af are the same as if a steady

current of intensity ^(a + h) were to flow for a time 2-^, and were then suc-
ceeded by the steady current h. The time 2 -^ is generally so minute a fraction

of a second, that the effects on the galvanometer and dynamometer may be
calculated as if the impulse were instantaneous.

If the circuit consists of a battery and a coil, then, when the circuit is
first completed, the eflfects are the same as if the current had only half its



544 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

filial strength during the time 2-^. This diminution of the current, due to
induction, is sometimes called the counter-current.

(36) If an additional resistance r is suddenly thrown into the circuit, as
by breaking contact, so as to force the current to pass through a thin wire

of resistance r, then the original current is ct=pj ^^^ ^^^ ^^^^ current is

The current of induction is then i^ p/p , .\ ' ^^^ continues for a time

2 . This current is greater than that which the battery can maintain in

the two wires R and r, and may be sufficient to ignite the thin wire r.

When contact is broken by separating the wires in air, this additional
resistance is given by the interposed air, and since the electromotive force across
the new resistance is very great, a spark will be forced across.

If the electromotive force is of the form Esinpt, as in the case of a coil
revolving in the magnetic field, then



X = — sin (pt — a),
P



where p' = R + L^2^\ and tan <^ = -& -



Case of two Circuits.

(37) Let R be the prunary circuit and S the secondary circuit, then we
have a case similar to that of the induction coil.

The equations of currents are those marked A and B, and we may here
assume L, M, N as constant because there is no motion of the conductors.
The equations then become






(13*).



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 545

To find the total quantity of electricity which passes, we have only to
integrate these equations with respect to t', then if x„ y, be the strengths of
the currents at time 0, and x„ y, at time t, and if A', Y be the quantities
of electricity passed through each circuit during time t,

(in

Y=\[M(x,-x,) + N{y,-y,)} J

When the circuit R is completed, then the total currents up to time t,
when t is great, are found by making

P



then X = x,{t - ^'j, Y= - ^x, (15*).

The value of the total counter-current in R is therefore independent of the
secondary circuit, and the induction current in the secondary circuit depends only
on M, the coefficient of induction between the coils, S the resistance of the
secondary coil, and x^ the final strength of the current in R.

When the electromotive force ^ ceases to act, there is an extra current
in the primary circuit, and a positive induced current in the secondary circuit,
whose values are equal and opposite to those produced on making contact.

(38) All questions relating to the total quantity of transient currents, as
measured by the impulse given to the magnet of the galvanometer, may be
solved in this way without the necessity of a complete solution of the equa-
tions. The heating effect of the current, and the impulse it gives to the
suspended coil of Weber's dynamometer, depend on the square of the current
at every instant during the short time it lasts. Hence we must obtain the
solution of the equations, and from the solution we may find the effects both
on the galvanometer and dynamometer ; and we may then make use of the
method of Weber for estimating the intensity and duration of a current uniform
while it lasts which would produce the same effects.



546 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

(39) Let n„ lu be the roots of the equation

(LN-M')n'-\r(RN+LS)n-\-RS=-0 (16),

and let the primary coil be acted on by a constant electromotive force Re, so
that c is the constant current it could maintain; then the complete solution of
the equations for making contact is

^-i^.£-^y-^.-^y-''^] (^^)'

y-'-^-^r-^^ <^^)-

From these we obtain for calculating the impulse on the dynamometer,
Ix-* = o'{<-|^-i^^} (19),

m = c'l ^^^l^^ (20).

The effects of the current in the secondary coil on the galvanometer and
dynamometer are the same as those of a uniform current

MR

~^ RN+LS

for a time 2 ("d + "^) •

(40) The equation between work and energy may be easily verified. The
work done by the electromotive force is

^lxdt = c'{Rt-L).

Work done in overcoming resistance and producing heat,

R]oedt + Sly'dt = & {Rt - |Z).

Energy remaining in the system, =^-L.

(41) If the circuit R is suddenly and completely interrupted while carrying
a current c, then the equation of the current in the secondary coil would be



M



M

This current begins with a value c-^, and gradually disappears.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.



547



M M*

Tlie total quantity of electricity is c -^ , and the value of jy'dt is c' -^ .

The effects on the galvanometer and dynamometer are equal to those of a

M . N

uniform current ^c ^r for a time 2 -^ .

The heating effect is therefore greater than that of the current on making
contact.

(42) If an electromotive force of the form $=E cos pt acts on the circuit
R, then if the circuit S is removed, the value of x will be

x = -j sin (pt — a),



where
and



A' = R' + Ly,



tan a =



-^



R



The effect of the presence of the circuit S in the neighbourhood is to
alt€r the value of A and a, to that which they would be if R became



R+p'



MS



and L became



X-p'



S'+p'N''

MN



S'+p'N''



Hence the effect of the presence of the circuit >S is to increase the apparent
resistance and diminish the apparent self-induction of the circuit R.



On the Determination of Coefficients of Induction hy the Electric Balance.

(43) The electric balance consists of six con-
ductors joining four points, A, C, D, E, two
and two. One pair, AC, of these points is con-
nected through the battery B. The opposite pair,
DE, is connected through the galvanometer G.
Then if the resistances of the four remaining
conductors are represented by P, Q, R, S, and
the currents in them by x, x — z, y, and y + z,

69—2




548



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.



the current through G will be z. Let the potentials at the four points be ^, C,
D, E. Then the conditions of steady currents may be found from the equations
Px = A~D, Q{x-z) = D-C ^

Ry = A-E, S(y + z) = E-C i (21).

Gz=D-E, B{x + y)=-A + C+F\

Solving these equations for z, we find



{i



1 1 1 j,[l 1



^+i^+^






R'^ S



BG , j-j ^



R^S^ = Fi



PS



_1

QR.



.(22).



In this expression F is the electromotive force of the battery, z the current
through the galvanometer when it has become steady. P, Q, R, S the resistances
in the four arms. B that of the battery and electrodes, and G that of the
galvanometer.

(44) If PS=QR, then 2 = 0, and there will be no steady current, but a
transient current through the galvanometer may be produced on making or
breaking circuit on account of induction, and the indications of the galvano-
meter may be used to determine the coeflScients of induction, provided we
understand the actions which take place.

We shall suppose PS=QR, so that the current z vanishes when sufiicient
time is allowed, and



x{P + Q) = y(R + S)



F{P+Q){R + S)



(P + Q){R + S) + B{P+Q){R + S)

Let the induction coeflScients between P, Q, R, S
be given by the following Table, the coefficient of in-
duction of P on itself being p, between P and Q, h,
and so on.

Let g be the coefficient of induction of the gal-
vanometer on itself, and let it be out of the reach of
the inductive influence of P, Q, R, S (as it must be
in order to avoid direct action 'of P, Q, R, S on the
needle) Let .Y, Y, Z be . the integi-als of x, ~y, z



(23).





P


Q


R


S


1 p


P


h


k


I


1 Q


h


?


m


I
n i


' R


k


771


r


1


S


I


n





s 1



with respect to t. At



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.



549



making contact x, y, z are zero. After a time z disappears, and x and y reach
constant values. The equations for each conductor will therefore be



PX +{p + h)x + (k+l)y = jAJt-jDdt^

Q(X-Z) + (h'+q)x-{-{m + n)y = jDdt-jCdt
RY +{k+m)x + (r+o)y = jAdt-jEdt

S{Y+Z) -h(l +n)x + (o +s)y=SEdt-jCdt
GZ=\Ddt-\Edt.
Solving these equations for Z, we find



(24).



,(25).



(45) Now let the deflection of the galvanometer by the instantaneous
current whose intensity is Z be a.

'"Let the permanent deflection produced by making the ratio of PS to QR,
p instead of unity, be 6.

Also let the time of vibration of the galvanometer needle from rest to rest
be T.

Then calling the quantity
p q T s , /I 1'



r 5 J /I
Q~R'^ S'^'^P'Q,



S-fl-Ki-i)-&4)




+"($-!)+'' (5-i)=^-


...(26)


Z 2 sin ia r t

2 tan 'n I— p


...(27)



we find



* [In those circumstances the values of x and y found in Art. 44 require modification before
being inserted in equation (24). This has been pointed out by Lord Rayleigh, who employed the
method described in the text in his second determination of the British unit of resistance in
absolute measure. See the Philosophical Transactions, 1882, Part ii. pp. 677, 678.]



550 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

In determining t by experiment, it is best to make the alteration of resist-
ance in one of the arms by means of the arrangement described by Mr Jenkin
in the Report of the British Association for 1863, by which any value of p
from 1 to rOl can be accurately measured.

We observe (a) the greatest deflection due to the impulse of induction
when the galvanometer is in circuit, when the connexions are made, and when
the resistances are so adjusted as to give no permanent current.

We then observe (/3) the greatest deflection produced by the permanent
current when the resistance of one of the arms is increased in the ratio of
1 to p, the galvanometer not being in circuit till a little while after the con-
nexion is made with the battery.

In order to eliminate the effects of resistance of the air, it is best to vary
p till /3 = 2a nearly; then

^=^^(-^)^'' (^«)-

If all the arms of the balance except P consist of resistance coils of very
line wire of no great length and doubled before being coiled, the induction
coefticients belonging to these coils will be insensible, and r will be reduced

to 'p. The electric balance therefore afibrds the means of measuring the self-
induction of any circuit whose resistance is known.

(46) It may also be used to determine the coefficient of induction between
two circuits, as for instance, that between P and S which we have called m;
but it would be more convenient to measure this by directly measuring the
current, as in (37), without using the balance. We may also ascertain the

equality of ^ and — by there being no current of induction, and thus, when
P V

we know the value of jp, we may determine that of g by a more perfect method
than the comparison of deflections.



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