= mkids sin of (r) .
LAW OF BIOT AND SAVART.
443
The force is moreover applied to the element, and is per-
pendicular to the plane Pds. Its direction is on the right of the
current that is, on the right of an observer placed in the element,
and looking at the pole, for the action of the element on the pole is
in the opposite direction.
VII. Law of Biot and Savart. The experiments of Biot and
Savart (444) demonstrate that the magnetic action of a rectilinear
current on a pole, is inversely as the distance of the current from
the pole.
According to a remark of Laplace, this law is satisfied if we
assume that the action of a pole on an element of current is inversely
as the square of the distance that is to say, if we have /(?) =
We may conversely prove that the law of the square is the only one
which satisfies Biot and Savart's experiments.
Fig. 107.
Consider, in fact, two parallel rectilinear currents, unlimited and
of the same strength AS and A'S', at distances a and a' from the
pole P (Fig. 107). For two elements ds and ds\ comprised between
the same two radius vectors drawn through the point P, and the
distances of which from this point are r and r', we have
ds _ r a
ds' r a '
and therefore
rds' = r'ds
444 ELEMENTARY ACTIONS.
The ratio of the actions d$ and d<$ of the pole on the elements
ds and ds' becomes then
ds sin a
d( r'" r .r ds r a
i r . rds' r a
ds sin a
The actions of the corresponding elements being inversely as the
distances a and ', this will also be the case with the resultants.
This is the law resulting from experiment. The action of a pole on
an element of current is then expressed by
dssina
d = r'd$, the point of application
of the partial resultant is the pole P. This is also the case for the
general resultant. The action of the whole circuit is sensibly equal
to that of the rectilinear part.
If the intensity is expressed by means of the electromagnetic unit
(460), the action of the unlimited current on the pole m, placed at
the distance a, is expressed by m , and the elementary formula
a
becomes
mlds sin a
(i) d$ = -
or, noting that is the magnetic action F of the mass m at the
point occupied by the element of current,
(2) d
dA denoting the surface of the parallelogram constructed on the
element and on the force F.
The action exerted on the current Ids, situate in a magnetic field,
only depends on the intensity of the field at this point, whatever be
the system from which the force proceeds (458) :
The action exerted on an element of current placed in a magnetic
field is equal to the product of the intensity of the current by the area of
the parallelogram constructed on the element of current, and on the
intensity of the field. This force is perpendicular to the plane of the
parallelogram, and directed to the left of the observer placed in the
current who is looking in the direction of the field.
The plane of the parallelogram to which the magnetic force is
perpendicular, was called by Ampere the directive plane.
446 ELEMENTARY ACTIONS.
Although we have given the name elementary to the force which
we have defined, it cannot so be considered in the strict sense of
the word ; thus, as Ampere observes, " we cannot apply the term
elementary either to a force which is manifested between two
elements which are not of the same kind, or to a force which does
not act along the straight line joining the two points between which
it is exerted."
464. RECIPROCAL ACTION OF A POLE AND OF A CURRENT.
Starting from this elementary law we shall prove as above (346)
that the components of the action of a unit pole, placed at the
origin of the co-ordinates, on a current element ds situated at a
point whose co-ordinates are x, y, and #, are
^ = -^
(3) dri =
It may be observed that the moment dM. z of this force, in
reference to the z axis, is
^M z = xdv) -yd% = \z (xdx +ydy) - (x 2 +/) dz .
The equation
gives
xdx +ydy + zdz = rdr.
It follows that
--\z rdr zdz - r* *' dz\ =- zc
2 ~r*[_ Z r r* ZL
- Id ( - )
r
But - is the cosine of the angle, which the right line r makes
with the z axis; we have then
*/M z = - \d cos y,
RECIPROCAL ACTION OF A POLE AND A CURRENT. 447
so that the moment M 2 of the actions exerted by the pole on any
arc AB has the value
(4) M^ = I(cosy a -cosy & ).
If the circuit is closed, this moment is null, and as the direction
of the z axis has been arbitrarily chosen, we see that the action of a
pole on a closed current passes through the pole. Conversely, the action
of a closed current on a pole also passes through the pole.
465. Instead of following the course taken, and of proving that
Biot and Savart's law is satisfied by an action which is inversely as
the square of the distance, we might have pursued a perhaps more
rigorous course, and have admitted as an experimental fact that the
action of a closed current on a pole passes through the pole.
The moment of the action of a pole on the element ds in
reference to the z axis will then be
and the moment relative to an arc AB
/B fA
M,= -I ry(r) Cydz zdy
(8)
c)co Czdx xdz
*y~ "J ^
<)a> Cxdy -ydx
^T J ^
468. ACTION OF Two ELEMENTS OF A CURRENT. The action
of two elements of a current may be established by an analogous
method by the aid of some principles and of facts taken from
experiment.
I. Equality of action and reaction. This principle does not
allow of experimental verification in the case of two elements of
currents. It must be regarded as a fundamental hypothesis ; it
carries with it the necessary consequence that the action of two
elements is along the right line which joins them. On the other
hand, the reciprocal action of two elements of current is obviously
G G
45 ELEMENTARY ACTIONS.
proportional to the length of each element, to the intensity of the
current in each of them, and to a function, which remains to be
determined, of the distance of the elements as well as of their
relative distances.
II. The action changes its direction when the direction of one of the
currents is changed ; it remains unaltered when the direction of the
two currents is simultaneously changed. This is a general property of
electrical currents.
III. Principle of symmetry. It follows from this principle of
symmetry that the reciprocal action of two elements a and b (Fig.
1 08), one of which a is in the plane perpendicular to the other in its
middle, is null.
X*
Fig. 108.
For consider a system a'V symmetrical with the first in reference
to a plane P parallel with the element a, and with the right line OC
joining the centres of the elements.
The actions of a on , and of a' on b' are respectively along
OC and O'C, and in the same direction from symmetry. But
the second is none other than the first, in which the direction
of the current has been changed in the element b; the force
should have changed its direction owing to this inversion, hence
it is null.
The force in particular is null, if the element a is perpendicular
to the right line OC, which joins the centres of the two elements, or
directed along this right line. These are two cases which will have
to be made use of.
IV. Principle of sinuous currents. The principle of sinuous
currents may be applied as above (463) and with the same limi-
tations ; we can always replace a current element by its projections
on three rectangular axes.
PRINCIPLE OF SINUOUS CURRENTS.
45 1
Let us consider two elements a and b (Fig. 109) in any position ;
let ds and ds' be their lengths, i and t' the intensities of the two
currents referred to a given unit, 6 and & the angles of their
directions with the right line OO' joining their centres, r the distance
OO', lastly co the angle of the planes drawn through the right line
OO' and the two elements.
Fig. 109.
Let us take the plane which passes through the element ds and
the right line OO' as plane of the figure, and replace each of these
elements by its projections on three rectangular axes ; one of these
axes is the right line OO', the other a right line in the plane of the
figure, and the third a perpendicular to this plane. The element a
has only two projections
a' = ds cos 0,
a" = ds sin B ;
the three projections of the element b are
b" ds' sin & cos w ,
b'" = ds' cos 0' sin w.
The total action consists of the actions of each of these elements
a' and a" on each of the elements b', b"> and b'".
Of these six actions, four are null from the principle of symmetry,
that of a' on b" and b'" t and that of a" on b' and b'".
There only remains to be examined the action of a' on &', and
that of a" on b".
The former is exerted between two elements directed along the
same right line; it might be represented by
ii'dsds cos cos 0'F(r).
G G 2
45 2 ELEMENTARY ACTIONS.
The second is exerted between two elements parallel to each
other, and perpendicular to the right line joining their centres ; we
might represent it by
ii'dsds' sin sin 0' cos wf(r) ,
the two functions of the distance being different since the conditions
are not the same.
The action d^ will then be expressed by the formula
( 9 ) d^ = ii'dsds' [cos cos & F (r) + sin 9 sin & cos (o/(r)] .
If e be the angle of the two elements, we have
cos e = cos cos & + sin sin & cos w,
and we may write
(i o) d*$ = ii'ds ds' [cos 6 cos & [F (r) -f(r)] + cos e/(r)] .
469. DETERMINATION OF THE FUNCTIONS F(r) and /(/). To
determine the functions F(r) and f(r\ we must have recourse to
experiment, and may employ very different methods, according to
the phenomenon to which we apply ourselves. We shall adopt the
following course, which is not perhaps the most rigorous from the
mathematical point of view, but which leads most rapidly to the final
formula.
We start from the two following experiments devised by Ampere.
V. When the homologous dimensions of three similar currents of
the same intensity are in geometrical progression that is to say, as i, m,
;/z 2 , and are moreover similarly placed the actions of the extreme
currents on the intermediate current are equal and of opposite sign. If
this latter is movable along a line passing through its centre of
similitude, and if it is disturbed from its position of equilibrium,
it returns to it of itself that is to say, that the equilibrium is
stable.
Ampere made this experiment with three circular currents situate
in the same plane, the intermediate circuit being movable about an
axis perpendicular to this plane.
DETERMINATION OF THE FUNCTIONS F(r) AND/(r). 453
VI. The action of a closed current on an element of current is
perpendicular to the element. The arrangement of this latter experi-
ment is the same as that for the action of magnets on currents
(463, IV.)
Consider the three similar currents of the first experiment (V.)
For the position of equilibrium, the distances to the centre of
Fig. no.
similitude of the three homologous points, A, B, C (Fig. no) of
the circles, satisfy the ratio
OA OB OC
from which is deduced
and therefore
OB - OC = B C = OA m (i - m),
AB
For three homologous elements of current #, b, and c, the lengths
will be ds t mdS) and m z ds ; the distance of the two former being r,
that of the second to the third will be mr.
If we assume that each intermediate element such as b is in
equilibrium between the two others a and c, which correspond to it,
the entire current S' will be in equilibrium between the two similarly
placed currents S and S". It does not seem that this condition is
always necessary, but it is evidently sufficient, and it enables us to
determine the form of the two functions ~F(r) and/(r).
454 ELEMENTARY ACTIONS.
It follows, in fact, that the action exerted on the element b ought
not to change when a is replaced by c that is to say, ds by m*ds t
and r by mr; we get then, from equation (9), suppressing the
common factor ii'ds ds', and observing that the angles 6 and 6' are
equal, and the angle w is zero,
cos 2 F (r) + sin 2 Of(r) = m* [cos 2 6 F (mr) + sin 2 Of(mr)~\.
This condition should be satisfied, whatever be the particular
values of m, 0, and r; we must have separately
m*f(mr)=f(r).
Making r=i, and m = r, we get
r 2 F(r) = const = /*, ,
r' 88 const M
and, consequently,
Thus the functions F(r) and /(r) are both inversely as the
square of the distance.
The expression for the elementary action then becomes
(u) d^= U * S -\ k cos# cos^' + sin^ sin 0' cos w ,
or
(I2 )
J m >d ^ S> \(k - i) cos cos & + cos el .
470. DETERMINATION OF THE RATIO OF THE Two CON-
STANTS. The last experiment (VI.) enables us to determine the
ratio k of the two constants.
DETERMINATION OF THE RATIO OF THE CONSTANTS. 455
Let us place the origin of the co-ordinates at the centre of the
movable element ds', and the x axis in the direction of the element
itself. The action of an element ds of a closed circuit in which the
intensity is i is expressed, as we have seen, by
= m * S (k-i) cos cos 0' + cos e .
The co-ordinates of the element ds being x, y, z, and its distance
from the origin r, we have
-,
dr
-,
dx
COS = - .
ds
The elementary action may then be written in the form
hii 'dsds ' f x dr dx~\
= - (_!) +
r 2 \_ rds ds\
and the projection of this force on the x axis is
d*$ cos 6' = d^- = hii 'ds 1 \(k - i) + 1
r [_ r 4 r B J
The component parallel to the x axis of the action of the closed
circuit on the element ds' is expressed by
Integration by parts gives
456
ELEMENTARY ACTIONS.
For a closed circuit, the first term of the second member is null ;
we get then
~ hii'ds' ,
Since from experiment this component must be null, we get
0, or k= .
2
With this value of , the elementary action becomes
(i3)
or
ii'ds'ds I 3
cos e - - cos 6 cos | ,
7.--'7,r^ 3'awfrl
^hu 'ds'\ -T---T .
[_r 2 2 r 3 J
471. DETERMINATION OF THE CONSTANT A The components
of the action of the current parallel to the other axes are then
ydx $xydr
r* 2 r*
zdx 3 xzdr
~^~~2~^
di)'= \d^ = hu'ds'\
d= \d^ Z - = hu'ds'{
Integration by parts gives
xydr / i xy\ i Cxdy +ydx
" = V~3^/ 3J ^~
The former term of the second member being null for a closed
circuit, we have lastly
hii'ds' C
- J:
ydx - xdy
hii'ds' Czdx-xdz
DETERMINATION OF THE CONSTANT k. 457
The value of the action F, of the current on unit magnetic mass
placed at the origin, and therefore the intensity of the field which
this current produces at the point where is the element, is expressed
by IG, where I is the electromagnetic intensity of the current (466),
and its components are
z=-ic "'^-^
-I
The three components d%, dtj, d, of the action d$ of the
circuit on the element ds' may then be written
Mi'ds 1
from which is deduced
It follows from this that the two forces F and d$ are perpendicular
to each other.
As the x axis is the only one which is defined, we may choose
the two others in such a way that the magnetic action F of the
current is in the plane xz\ we have then-
Y = 0, B = 0,
X = Fcosa, A = Gcosa,
Z=Fsina, C=Gsina,
a being the angle which the force F on the straight line G makes
with the x axis.
458 ELEMENTARY ACTIONS.
It follows from this that
hii'ds' . hii'ds' .
dn -- G sm a = -- - Jb sin a = afVi,
we get
In this case the currents are parallel, of unit length, perpen-
dicular to the line which joins their centres, and at unit distance j
the strength of the current, which is equal for each of them, and
is taken at unity, is such that the reciprocal action is equal to the
unit of force.
Supposing the currents equal, equation (14) will give,
The electrodynamic intensity of a current is equal to its electro-
magnetic intensity multiplied by \/2.
In virtue of the ratio which connects the numerical expression
of a magnitude into the unit which serves to measure it, we see that
the electrodynamic unit of current is equal to the electromagnetic unit
divided by \/2.
474. The identity between the mutual action of currents and
that of the correlated magnetic systems has been confirmed in all
experiments as long, at least, as a steady condition has been estab-
lished in the circuits.
We may cite, for instance, the experiments of Weber on the
reciprocal action of the cylindrical coils with circular bases. This
action is proportional to the strength of the two currents ; it varies
with the relative distance and direction of the coils according to
the same law as that of two magnets whose axes are respectively
parallel to the axes of the coils.
460 ELEMENTARY ACTIONS.
475. FORMULA EQUIVALENT TO THAT OF AMPERE. We have
seen (349) that the action of two elements of the contour of two
magnetic shells, which is equivalent to the elementary electro-
dynamic action, may be expressed in an infinity of different ways,
with this condition that the resultant of the actions of a closed
circuit on an element has a determinate value.
476. (i.) Formula of M. Reynard. The first form which we
have met (348) for the action of ds upon ds' is, by supposing the
element ds' at the origin of co-ordinates and directed along the
x axis, a force whose components are
x r* y
The factor a in these equations represents the product Il'^y', and
x, y, z are the co-ordinates of the element ds.
The force itself is expressed by the formula
ll'ds'ds .
/= sin 6 cos IM,
in which is the angle of the element ds with the right line ?*, and
p! the angle of the element ds' with the plane rds.
If d$ is the angle under which the element ds is seen from the
element ds', an angle which is equal to - - , this formula may still
be written
/=
which is the formula of M. Reynard.
In order to determine the direction of this elementary force, we
observe, in the first place, that it is perpendicular to the element ds'
since/,. = 0.
It is in the plane rds. The equation of this plane, of which
X, Y, and Z are the co-ordinates, is
X (ydz - zdy) + Y (zdx -ydz) + Z (xdy -ydx) = 0.
FORMULAE EQUIVALENT TO THAT OF AMPERE. 461
The intersection with the yz plane is
Y (zdx - xdz) + Z (xdy -ydx) = ;
from which follows
Y_Z
We have further
r 2 ds 2 sin 2 = (ydz - zdyf + (zdx - xdz? + (xdy -ydx)*
which gives
_ a W sin 2 r (ydz -
r L
Now, the expression - : - is the cosine of the angle which
rds sin
the perpendicular to the plane rds makes with the x axis : hence the
quantity in brackets is the square of the sine of this angle, or the
square of the cosine of the angle /*' which the plane makes with
the x axis that is to say, with the element ds', and we have
ads sin cos M' \Vdsds' .
/ = - smtfcos/*.
Thus the action of ds upon ds l is in the plane rds t perpendicular
to the element ds', proportional to the sine of the angle which the
element ds makes with the distance r, and to the cosine of the angle
which the element ds' makes with the plane rds, and lastly inversely
as the square of the distance.
Let us take the plane rds as that of xz, and in this plane the line
OO', which joins the two elements, as the x axis. The force acting
on the element ds' placed at the origin of the co-ordinates is in the
xz plane and is perpendicular to ds'. To obtain its direction, we
must project trie element ds' on the xz plane ; a straight line in this
plane, perpendicular to the projection, will be the direction in ques-
tion ; it is perpendicular to the projecting plane, and therefore to the
element which passes through its foot in this plane.
462 ELEMENTARY ACTIONS.
The components of this force parallel to the axes are
II'dssmB, ll'dzdz'
f x =fcos/3 = ds cos/* cos/3 = 2 ,
A = 0>
> , .' ll'dssmO , . ll'dzdx'
f z =/sm p = ds cos fi! sin p = .
OC 00
If we still denote by & the angle which the right line r makes
with the element ds', and by w the angle of the two planes rds and
rds' , we have
dz' = ds' sin 0' cos w,
<&' = <&' cos 0',
which gives
f x = sin 6 sin & cos o> dsds' ,
f z = ^ sinO cos O'dsds'.
The action of two elements of consecutive currents is evidently null.
In fine, we have not here an equal and opposite action and
reaction, but there is a different action on each of the two elements,
directed perpendicularly to this element, and in the plane determined
by the other element.
The existence of a force perpendicular to the element is incom-
patible with the idea of an action at a distance; but if, on the
contrary, we view electrodynamic forces as resulting from a modifi-
cation in the elastic properties of the medium, we can easily see that
the reaction of this medium on an element of current may be
perpendicular.
477. (n.) General Formula. We may add an exact differential of
the co-ordinates to each of the components f x ,f y , and/ 2 without the
action of the closed circuit on the element ds' being modified. We
may then take as components of the elementary action the following
general expressions in which X, Y, and Z, are any given functions of
the co-ordinates :
GRASSMANN'S FORMULA. 463
478. (in.) Amplre's Formula. If we impose on the elementary
force the condition of being directed along the right line joining the
two elements, we get Ampere's formula ; this formula is the only one
which satisfies the general principle of action and reaction, and
consequently the essential conditions of a true elementary force.
For any other solution, the action of the element ds on the element
ds', will not be equal and directly opposite that of the element ds'
on the element ds.
479. (iv.) Grassmanris Formula. Let us replace the arbitrary
functions X, Y, and Z, respectively by xf t yf, and zf, f denoting a
function of the distance r. The components of the elementary force
will then be
This operation is the same as adding to the force given by
M. Reynard's formula, another force d 2 ^ the components of
which are
The force itself is given by the equation
[XViP = a*[f*ds* + r*(df)* + 2frd(fdr}},
or, taking into account the relation dr = ds cos 0,
~ 2 Wi) 2 = [flr + rdfj
This force makes with the straight line r an angle, the cosine
of which is
xd (xf) +yd (yf) + zd (zf) _ frdr + r*df_ d(rf)
'
464 ELEMENTARY ACTIONS.
Finally, the angle 8 which it makes with the element ds is
or
If we impose on this force the condition of being perpendicular
to the right line which joins the two elements we have then
'( an< ^ tne force given by M. Reynard's
formula is also null, the action of the two elements is null.
On this hypothesis, which is that of Grassmann, the true force
d^ will be the resultant of a force which is inversely as the square of
the distance, and of a force ^Vi which is inversely as the distance,
is perpendicular to the right line joining the elements, and whose
direction makes with the element ds an angle equal to - + 6.
Several other conditions might be imagined equally compatible
with experiment; but these few examples will suffice to show the
indeterminateness of the problem, and to point out the principal
solutions.
ACTION OF TWO PARALLEL CURRENTS. 465
CHAPTER III.
PARTICULAR CASES.
480. ACTION OF Two PARALLEL CURRENTS. According to
Ampere's formula, two elements of currents parallel to each other,
and perpendicular to the right line which joins their centres, will
attract or repel according as the currents are in the same, or in
opposite directions.
This result is usually verified by bringing a rectilinear current,
which we may suppose unlimited, near a portion of a rectilinear
current movable parallel to itself. The experiment is really more
complicated, for each of the currents in question forms part of a
closed circuit. In whatever manner we may suppose the planes of
the two currents placed in reference to each other, bringing the two
rectilinear portions near each other will increase, for each of them,
the flow of force which it will receive from the other by its negative
surface, and will diminish the relative energy if the currents are in
the same direction ; the converse takes place when they are in
opposite directions.
Let I be the intensity of the unlimited current, I' that of the finite
current, which is parallel to it, and b its length. If we vary the
distance a of the two currents (which we suppose in the same
direction) by da, the variation in the flow of force which enters the
circuit from the movable current is
= -bda = -2lb ;
a a
the force exerted upon the movable part of the circuit is expressed
by I'- = 211'-, it is therefore inversely as the distance a.
da a
481. ANGULAR CURRENTS. Two rectilinear currents placed
near each other tend to set parallel. This result is usually enunciated
H H
466 PARTICULAR CASES.
by saying, that two currents which form an angle with each other
attract if they both approach, or both recede from the apex of the
angle or the common perpendicular, and that in the contrary case
they repel.
The experiment is made by bringing an unlimited rectilinear
current near the bottom of a movable rectangular frame traversed by
the current. The movable frame turns so as to receive, on its
negative face, the maximum flow of force which proceeds from the
rectilinear current. There is no simple expression for the work of
any given displacement ; but the total work corresponding to the
displacement of the frame, from the position in which its plane is
perpendicular to the current, to that in which it becomes parallel, is
proportional to the flow of force which traverses the frame in the
second case. If a Q and 1 are the distances from the unlimited
current of the two sides of the frame parallel to it, and b the length
of one of these sides, we have
and the electromagnetic work is equal to 2 1 !'/. .
O-Q
These movements are easily accounted for by supposing the
currents replaced by equivalent magnetic shells, and considering the
actions of these shells.
We may arrive at the same object on Faraday's plan, by con-
sidering the lines of force and their distribution in the field. The
figured lines of force of the field resulting from the various systems
near each other, are closer in certain regions than in others. If we
represent these lines of force (105) as elastic threads exposed to a
strain in the direction of their length, and to a repulsion in the
direction perpendicular to this, we shall have a very definite idea
of the relative motion which they tend to produce.
482. APPARENT REPULSION OF Two CONSECUTIVE ELEMENTS
OF CURRENT. This important experiment of Ampere consists in
putting the two poles of a battery in connection with two rectangular
troughs containing mercury, and separated by an insulating division.
A copper wire is bent so as to form two horizontal legs floating
on the mercury, and a cross piece (in the form of a bridge) which
connects the two former. When the battery circuit is closed, the wire
is seen to glide along the surface of the mercury, and to recede from
the points by which the current enters.
ELECTROMAGNETIC ROTATION 7 ".
467
Ampere thought this a proof that the two elements of current
directed along the same right line, and in the same direction, repel
each other, as the elementary formula indicates; but it is easy to see
that the interpretation of the phenomena does not entail this conse-
quence.
In this experiment the current traverses a circuit, one of whose
portions is movable, and the surface of which tends to become a
maximum (455). This result may moreover be arrived at directly
by replacing the current by a flexible shell bent upon itself, as
shown in Fig. in. The three shells superposed in the space
ABB'A' do not give rise to any force among them parallel to the
plane of the current ; but their external action is equivalent to that
of a simple shell. The portion aDC& tends to recede, and the shell
extends so as to occupy the greatest surface.
Fig. in.
483. ELECTROMAGNETIC ROTATION. ist. Barlow's Wheel
A toothed metal wheel, movable about a horizontal axis, is arranged
so that one or more teeth plunge with their lower ends in a cup
containing mercury. If the system is traversed by a current
which enters by the axis and leaves by the mercury, the only
action of the current on itself will tend to move the bottom
teeth in a direction which displaces them from the rest of the
circuit so as to increase the total surface; but this action is
generally too weak to overcome friction. A stronger effect is ob-
tained by putting the trough between the limbs of a horseshpe
magnet arranged horizontally. The lines of magnetic force then
traverse the plane of the wheel ; if they are directed from back
to front that is, with the north pole in front the rotation will
be in the opposite direction of the hands of a watch.
In order to get a phenomenon easy to calculate, replace the
magnet by a uniform magnetic field of intensity F, parallel to the
axis of the rotation. Let a be the radius of the wheel, 6 the angle of
H H 2
4 68
PARTICULAR CASES.
two consecutive teeth, and suppose the surface of the mercury
placed so that one of the teeth touches the liquid at the moment the
preceding one quits it. The flow of force through the triangle
formed by the radius, which corresponds to these two teeth, is
F , or sensibly F ; that is to say, the product of the force
2 aW
by the surface of the sector, and the corresponding work is IF .
For an entire turn the work is IFS that is to say, proportional to
the whole surface S of the wheel.
484. 2nd. Ampere's Experiment. This experiment, in which
the rotation of a magnet is produced by a current, is the converse
of that, the theory of which has been given above (456). The
apparatus is arranged so that only one of the poles of the magnet
can traverse the current ; a continuous rotation is obtained in this
way. The magnet (Fig. 112), loaded by a counterpoise of platinum,
floats on the mercury, and can rotate about itself on its own axis ;
the current is brought to the surface of the liquid, traverses the
projecting part of the magnet, and emerges by a fixed conductor
Fig. 112.
which dips in a drop of mercury in the top N. If we suppose that
the current goes rigorously along the axis of the magnet, the work at
each turn for a magnetic mass outside the axis is 47r;;/I, and gives
rise to a couple the moment of which is 2ml. The phenomenon is
really more complicated, because the current traverses the whole
section of the magnet.
Faraday repeated the experiment by placing the magnet outside
the circuit. The magnet is brought to the centre of the vessel by a
metal rod, and the magnet floats in an eccentric position.
In both cases, if the current ascends by the axis, and the top of
ROTATION OF LIQUIDS. 469
the magnet is a north pole, the rotation is opposite to the motion of
the hands of a watch. Faraday's arrangement gives greater friction,
and the rotation is less rapid.
485. ROTATION OF LIQUIDS. When the current traverses a
liquid, the liquid filaments, which coincide with the lines of electrical
flow, may be considered as movable circuits, capable of obeying
electromagnetic actions, and experiment shows that the liquid is
moved along with the current which it carries.
i st. Davy's Experiment. Two platinum electrodes just pro-
ject a very little below the top of the mercury. If the N pole
of a magnet is placed over one of them the negative electrode,
for example a depression of the mercury is observed ; and, at
the same time, a rotation in the same direction as the hands of
a watch.
2nd. M. Jamiris Experiment. The two electrodes of a volta-
meter are placed in the same vertical line, and on the axis
of the poles of a horseshoe magnet. If the liquid molecules in
a filament of the current formed a rigid thread, we should be
in the same condition as in Faraday's experiment, in which
rotation is impossible. The electromagnetic forces really act inde-
pendently, and in the same manner as in Davy's experiment, on
the portions of the filaments which diverge as they start from
each of the electrodes. The liquid divides them into two super-
posed layers which rotate in contrary directions, and the rotation
is made visible by the bubbles of gas which result from the
decomposition of water.
3rd. M. Bertiris Experiment. In M. Bertin's experiments the
movement of the liquid is made visible by small pieces of cork
which float on the surface. The liquid is in an annular dish
containing two rings of metal, one inside the other. If these
circles are electrodes, a series ot either centripetal or centrifrigal
radiating currents is obtained in the liquid. When a magnet is
placed in the axis of the current, the liquid acquires a rotation
in a definite direction in agreement with theory. The direction
of the rotation is not altered, if the central magnet is replaced
by a magnetised tube encircling the dish. For if the north
pole is at the top in the two cases, the flow of magnetic force is
diverted downwards, both inside and outside the hollow magnet.
This is not the case when the magnet is replaced by a coil :
the rotation of the liquid changes its direction according as the
coil is inside or outside the dish, and each of the lines of force
constitutes a closed circuit.
47 PARTICULAR CASES.
486. ELECTRODYNAMIC ROTATION. Consider an unlimited
rectilinear current X'X of strength I (Fig. 113), and a finite recti-
linear current of length a, and strength I, perpendicular to the first
A
a.
Fig. 113-
and in the same plane. If we give the current a a displacement
doc parallel to the current I, then if r Q is the distance BC, the
corresponding work will be
f r o
\.'dx\
J''o
1 a
=
r
The force which acts on the movable current is perpendicular
to its direction, parallel to the unlimited current, and its value is
2117. ( i + ). This current will be carried parallel to itself by a
\ r o/
constant force, and will ascend or descend the unlimited current
according as it is ascending or descending in reference to this latter.
The experiment is ordinarily made by causing a circular current
to act on a portion of a current movable about an axis perpendicular
to its plane and passing through its centre. The movable current
then rotates in the opposite direction to the principal current.
If the movable current is closed, or if its ends are on the axis,
there is evidently no movement, for then each line of force meets
the edge twice.
487. ACTION OF A UNIFORM FIELD. Consider first two un-
limited rectilinear conductors AA', BB' (Fig. 114) parallel to each
other at the distance b, and let us suppose that, while the two ends
are in communication with the poles of a battery, the circuit is
closed by a cross bar CC', movable parallel to itself along the
conductors A A' and BB'.
Let Z be the component of the intensity of the field perpen-
dicular to the plane of the conductors ; for a displacement dx of
ACTION OF A UNIFORM FIELD. 471
the movable bar, the variation in the flow of magnetic force in the
circuit is bUx. In the case of the terrestrial field, and if the rails
are horizontal, the component Z is directed downwards in our hemi-
sphere, and with the direction of the current shown by the arrows,
the movable bridge CC' will recede from AB under the influence of
electromagnetic forces ; it will approach, if the current is in the
opposite direction.
In Ampere's experiment (482) the action of the current on itself
tends to increase the surface, and to repel the movable bridge.
This action and that of the Earth will add themselves or oppose
according to the direction of the current ; the motion of the wire is
more or less easy according to the case.
A"
y i
Fig. 114.
488. Suppose now that the movable conductor forms a closed
circuit ; let S be the surface of this circuit, if it is plane, or the
maximum projection of its surface on a plane, which we will call the
plane of the current. When such a circuit is movable in a uniform
magnetic field, like that of the earth, stable equilibrium corresponds
to the case in which the flow of force across the negative face of the
circuit is a maximum ; the plane of the circuit tends then to set at
right angles to the force. Under the influence of the earth this plane
will be perpendicular to the dipping needle ; the current will move
from east to west in the lower part. If F is the strength of the field,
I that of the current, the potential energy of the current in the
position of equilibrium is
Wj = - ISF ;
when the face of the current is turned upside down, it becomes
W mF
o JLO J7
472 PARTICULAR CASES.
The work done against electromagnetic forces in this operation
is then
If the current is made to turn about a vertical axis we need only
consider the horizontal component H of the field. The work of the
rotation of 180 about the axis from the position of equilibrium
is then
W'-alSH.
If the current turns about a horizontal axis parallel to the
magnetic meridian, the vertical component Z of the field alone
comes into play. For a rotation of 180 from the position of
equilibrium the work is still
W" = 2 ISZ.
The ratio of the works in the two latter cases,
W" Z
is equal to the tangent of the Inclination. Hence, if we could
measure these works, we could determine the elements of terrestrial
magnetism without having recourse to magnets.
If the axis of rotation is in the magnetic meridian, the total
work is null, when the current, which at first was in this plane,
comes back to it after having been turned through 180; the works,
which correspond to the two halves of the rotation, are therefore
equal and of contrary signs.
Lastly, the work would be null for any given rotation if the axis
of rotation were parallel to the direction of the field.
489. ASTATIC CIRCUITS. The work of any given displacement
is still null when the circuit comprises two closed curves, such that
their projections on any plane give two equal surfaces S and S',
surrounded by currents moving' in opposite directions. This is the
condition which movable currents must satisfy, which are arranged
so as not to be under the action of the earth ; what are called astatic
ASTATIC CIRCUITS.
473
currents. Figures 115, 116, and 117 give examples of currents
which realise these conditions.
If the two surfaces S and S' were not equal, the action of the
field would be proportional to their difference S - S'.
H- S'
Fig. 115-
B
Fig. 1 1 6.
Fig. 117.
490. ROTATION OF A CURRENT UNDER THE ACTION OF THE
EARTH. A portion of a current not closed, and movable about
an axis, takes in general a continuous rotatory motion under the
influence of terrestrial magnetism.
We observe, in the first place, that in a uniform magnetic field,
like the terrestrial field, we may always replace a current by its
projection on three rectangular planes ; this amounts, in fact, to
replacing the strength of the field by its three rectangular
components.
Consider any given current movable about an axis and determine
its projections on three planes, one perpendicular to its axis of
rotation, the two others passing through this axis, and such that
one of them is parallel to the direction of the field ; let S, S', and S"
be these three projections. The projection S" perpendicular to the
field will not produce any action. The action on the projection S'
will be purely directive ; the circuit of the current will be carried
along in such a manner that the surface S" is a maximum, and
presents its negative face to the force ; in our hemisphere the current
must be descending in the part turned towards the east. There
remains to be considered the projection S on the plane perpendicular
to the axis. If it is closed, and of a fixed shape, it undergoes no
action ; if part of it is movable, the component of the field parallel
to the axis will have a constant moment relatively to this axis, and
will produce a continuous rotation.
474 PARTICULAR CASES.
491. For instance, let the system be formed of a current OP of
length a (Fig. 118) movable about a vertical axis, one of whose ends
is on the axis of rotation and the other dips in a mercury cup.
The current enters the mercury at A, traverses the two parts ABP
and ACP in opposite directions, and regains the axis by the movable
part PO. Let I be the total strength of the current, x the strength
in the arc B, y in the arc C ; the current will evidently be equal to
1 in the movable portion PO,
Fig. i i 8.
The surface comprised by the horizontal projection S consists
of two parts, one ABPO presenting its negative face to the com-
ponent Z of the terrestrial action; the other, ACPO, its positive face.
The former tends to increase, the second to diminish, and for an
angular displacement of the radius PO, the total work is
-a*(x + y)ZO = -
2 2
This work is independent of the position of the conductor OP;
the force then is constant. The work corresponding to an entire
turn will be
If the current has a vertical projection S', the motion of rotation will
be modified by the directive action corresponding to this projection.
It is easy to see that according to the ratio of the two surfaces S and
S', the initial velocity, and the value of friction, the moment of the
directive action might preponderate over the moment of rotation,
ACTION OF TWO RECTANGULAR CIRCUITS. 475
and keep the apparatus in equilibrium in a position perpendicular to
the magnetic meridian. In the apparatus used for this experiment,
we take a movable current symmetrical in reference to the axis of
rotation. The projection S' is then null, and the couple of rotation,
which would impart to the system a uniformly accelerated rotation if
there were no friction, ultimately makes it rotate uniformly.
492. ACTION OF Two RECTANGULAR CIRCUITS. We may cal-
culate the action of two rectangular frames AC and A'C', the sides
of which are parallel. Suppose, for the sake of simplicity, that
the frames are equal (Fig. 119), and their corresponding summits
Fig. 119.
A and A' on a perpendicular to their plane. The mutual energy
of the two circuits, with currents equal to unity, is expressed on
Neumann's formula (352) by
W= 1 I -dsds'.
flf
The value of cos e is equal to unity for two parallel sides, and null
for two perpendicular sides such as AB and B'C'. The energy thus
becomes
This expression only contains terms relating to parallel wires.
Consider, in the first place, the two sides AB and A'B', of length
a, and at the distance //. Let ds and ds' be two elements, placed
respectively at M and M' and r their distance ; lastly, suppose that
we measure the lengths s and s f from the points A and A'. From
the ratio
47^ PARTICULAR CASES.
we get for the first integration, in which the distance s is taken
constant,
or
/
The second integration relative to ds is easily effected, for we
have in general
l>(-u + ^WT^ 2 )du = ul\-u + ^W^ 2
we get then
p *-*W(^g+g
Jo -*+^ 2 +^ 2
Changing the sign of this expression and replacing /$ by the
distance # of the sides AB and CD', we get in like manner the term
relating to this last side. If the rectangle is a square, h' = ,Ja 2 + h 2 ,
and the two terms of the energy corresponding to the side AB
give
The total energy is then
_ + ,
W J I n-f , o TTi 7
When the distance of the frame is altered by dh, the variation
of the energy ^W, is equal to the work -Mfc of the force F,
PROPERTIES OF CIRCULAR CURRENTS. 477
considered as attractive, which is exerted between the two circuits,
and we have
dh '
We thus obtain, all reductions being made,
If the strengths of the currents in the two frames are respectively
I and I', the expression for the mutual action is
where P is the sum of the terms in the brackets.
493. PROPERTIES OF CIRCULAR CURRENTS. The potential of
a circular current is equal, within a constant, to that of a shell of the
same strength and the same contour. We have given above (368)
the expression of this potential for any given point. If the point is
on an axis at a distance x from the centre, it is sufficient in equation
(16) to make p = 0; replacing < by I, we get
V=27Tl
from which is deduced
a 2 IS
denoting by S the surface of the circle. For a point on the axis the
force is inversely as the cube of the distance to the contour.
This force is a maximum at the centre of the circle ; we
have then
IS I IL
L being the length of the circumference.
47$ PARTICULAR CASES.
This latter result would follow directly from a consideration of
the equivalent shell. Let 2h be the thickness of the shell supposed
to be plane, then denoting by l a the intensity of magnetisation,
The value of the action of the two terminal layers on a point
at the centre is (322)
and the magnetic induction is
We may now reject the shell outside the point in question,
without changing the value of the force (451).
494. ELECTROMAGNETIC SOLENOID. Ampere gave the name
solenoid to a system of equal circular currents, infinitely near, and
infinitely close, equidistant and perpendicular to any given curve
passing through their centre, which is called the directrix.
Let dS be the common surface of the elementary currents, h
their distance, and I the strength of the current. Each elementary
current may be replaced by a magnetic shell of the same magnitude,
of thickness h, and surface density a-, such that we have
As the surfaces in contact of all these shells have equal and
opposite charges, they neutralise each other except at the ends, and
the system is identical with that of a solenoidal filament The
external action reduces then to that of two magnetic masses M
placed at the ends. If / be the length of the solenoid, n the total
number of elementary currents, and n : the number of these currents
in unit length, we have
495. CYLINDRICAL COIL. Let us suppose that a cylinder is
covered with equidistant currents perpendicular to the axis. The
system of these currents forms a kind of cylindrical solenoid, of
CYLINDRICAL COIL. 479
finite transversal dimensions ; it is approximately realised by winding
a wire in the form of a helix on the surface of the cylinder. Each
element of the helix may be replaced by its projections on the
axis and on a plane perpendicular to the axis. If the section of
the cylinder is small, we sensibly destroy the effect of the former by
bending the wire back in a contrary direction parallel to the axis.
Whatever be the diameter, if the individual turns are sufficiently
near, and the coil consists of an equal number of layers in which the
inclination of the windings is alternately in opposite directions, the effect
of the projections on the axis is still sensibly zero, and the external
action differs very little from that of the perpendicular projections.
The system of the currents perpendicular to the axis is equivalent
to a solenoidal magnet of the same form ; we may, in fact, replace
each of them by a shell, and decompose the system into an infinity
of parallel solenoids, each of which is equivalent to a solenoidal
filament.
The action of the system on points outside the cylinder, reduces
then to that of two equal and opposite layers spread uniformly on
the bases, and the density cr of which is n-J..
For internal points the force is equal to the induction of the
equivalent magnetic system. If the cylinder is so long that the
action of the ends may be neglected in part of its extent, the lines of
force are parallel to the axis of the cylinder ; the field is uniform and
its strength is
The flow of induction across the section of the cylinder is
this flow is in the opposite direction to the internal flow from the
bases of the equivalent magnet.
It is, moreover, evident that a coil is not equivalent to a hollow
magnet ; in the hollow magnet all the lines of force, both internal and
external, start from the positive surface, and are absorbed at the
negative surface ; in coils, on the contrary, the internal lines of force
are the continuation of the external lines of force, and form closed
curves which never terminate at magnetic masses.
496. ANNULAR COIL. Suppose a ring to be covered by equal
currents equidistant from each other, and each situate in a plane
passing through the axis ; the system may be decomposed into a
series of solenoids, and it is equivalent to a solenoidal magnet of the
same form (411).
480 PARTICULAR CASES.
All the elementary solenoids comprise then the same number of
currents with the same intensity I, but of different lengths. If n^
is the number of windings comprised between two meridian planes
which make with each other an angle equal to unity, and if x is the
radius of an elementary solenoid, the distance of the successive turns
will be ; the intensity of magnetisation of the equivalent magnetic
filament is then 2L , and the induction, or the magnetic force, ^ HI .
x x
The value of the flow of induction across a surface S, taken in the
/ 7Q
meridional section of the ring, is 473-^1 .
J x
In the case in which the ring is a circular torus (372), we have
--WR
x
The total flow across the section is then
497. CASE OF ANY GIVEN SURFACE. Let us now consider the
general case in which any surface J5f is covered by plane currents of
the same strength, parallel to each other, and at such a distance that
these are n^ in unit length. These currents may be replaced by a
series of parallel solenoids terminating in the surface, and these
solenoids themselves by equivalent magnetic filaments ; in this way
a uniform magnet will be formed, the intensity of magnetisation
of which is n^, and the density at each point of the surface, has
the value n-J. cos 0, where 6 is the angle which the perpendicular to
the surface, at the point in question, makes with a perpendicular to
the plane of the currents.
The internal action of these currents is equal to the induction of
the equivalent magnetic system. In the case of the sphere (355) it
o
is constant and equal to -irn-^L ; the value of the flow of induction
across the great circle perpendicular to the line of- the poles on the
common axis of the currents is
where L is the circumference of the great circle.
AMPERE'S THEORY OF MAGNETISM. 481
The internal field would also be uniform in the case of an
ellipsoid (356).
From this we arrive at a new way of regarding terrestrial
magnetism ; the magnetic action of the earth is equivalent to that
of a series of circular currents situate in equidistant planes perpen-
dicular to the magnetic axis, these currents circulating from east
to west.
498. AMPERE'S THEORY OF MAGNETISM. We see that it is
possible, by means of currents situate in parallel planes, to realise
a system equivalent to a uniform magnet, which has the same
external surface ; the two systems are equivalent for all external
points, and produce the same induction in the interior.
Any given magnet may, in like manner, be replaced by a system
of superficial currents, in so far, at least, as the external action is
concerned.
This action, in fact (315), is equivalent to that of a layer of
total mass null, distributed on the surface. If o- be the density
of the layer at a point, F H and ' n the perpendicular components,
measured from the surface, of the actions which it exerts outwards
as well as inwards, we have (38)
Let us consider the internal potential V of the layer, and the
equipotential surfaces to which the force F' is perpendicular, and
suppose that on each of these surfaces we place equal and opposite
magnetic layers, the density of which, at each point, is determined
by the condition
The external action of this system of surfaces is null. We
observe now that the product cr'dri = -- dV is constant between
4 7T
two equipotential surfaces. If then we connect the negative layer
of the surface Jjf, where the potential is V, with the positive layer of
the following surface ^ 5 at the potential V + dV, we form a shell, the
magnetic power of which, o-'dn', is constant. A current, of the same
strength, which followed on the surface, the curve formed by the in-
tersection of the shell, would have the same action on the outside ;
we could proceed in the same way with all other shells. But in
i i
482 PARTICULAR CASES.
forming this shell a negative corona has been left corresponding to
the difference 2? - J5f' of the two surfaces, and the sum of the external
actions of these corona is equal and of opposite sign to that of the
superficial currents. If d^> and dS are the two corresponding
elements of this corona, and of the surface of the magnet deter-
mined by a tube of force, we have
which gives
= -- *,!,
^ ]
If the sphere is placed in a cylindrical coil which is so long that
the effect of the ends may be neglected, it will also acquire a uniform
magnetisation.
This would also be the case with an ellipsoid, and also with a
sufficient approximation for a cylinder, the axis of which would
coincide with that of the coil.
501. In the case of a cylindrical coil of great length (495), the
strength of the field in the interior is 47^! ; the value of the intensity
of magnetisation of a long cylinder parallel to the field would be
k$ (292) or 47rvfc# 1 I, and the internal induction F l will be 47n<, or
I 6w 2 # 1 I.
If S is the section of the bar, the flow of magnetic induction
across it is
F 1 S = 1 67^18,
and the total flow, including in it the induction 471-^18 of the current,
has the value
Q = 471-;^ (i + 47r/) IS.
This value can be experimentally investigated, and we might deduce
from it the coefficient of magnetisation k.
502. The determination of this coefficient is still more accurate
by means of a piece of soft iron in the form of a torus, which is
surrounded by equidistant currents (496). In this case the
1
strength of the field produced by the currents is - - at each
DETERMINATION OF THE COEFFICIENT OF MAGNETISATION. 485
point. This being the only effective force, the value of the intensity
of magnetisation at the point in question is
The induction is equal to ^-rrkfa so that the total flow of induction
across the section S of the soft iron, comprising still that which arises
from the currents, is
Q =
/d S
*'
If the soft iron does not occupy the whole of the space bounded
by the currents, but merely a portion S' of the section S, the total
flow of induction, across the surface of the currents, is
T f (VS , (VS'1
= 4 7r 1 I J + 4**J
Suppose, for instance, that the section of the iron is a circle of
radius a! concentric with the circular section of a torus ; the total
flow of induction in the torus will be
Q' = 4*^1 [~R ~
-* +
If the section of the coil were a rectangle of height b parallel to
the axis of revolution, and of the thickness 20, the mean radius
being R, we should have
x R-a
IR-a
The iron ring, in like manner, might have a rectangular section
of height b\ of thickness 20', and of mean radius R'; the total flow
of induction would then be
We shall see further on (559) the use which can be made of
these various formulae.
486 PARTICULAR CASES.
503. MEASUREMENT OF CURRENTS. GALVANOMETERS. The
strength of currents is usually measured by the electromagnetic or
electrodynamic actions which they exert, and the instruments which
are used for this purpose are called galvanometers, or electrodyna-
mometers, according as they depend on one or the other of the two
actions.
A galvanometer consists of a magnetised needle, or of any
magnetic system on which a conductor traversed by a current is
made to act; the effect produced is measured by means of an
antagonistic force, such as the torsion of a metal wire, or of a bifilar
suspension, or by the action of an external magnetic field.
Let us consider the simple case of a horizontal magnetic needle
suspended by a wire without appreciable torsion, and placed in the
centre of a frame on which is coiled a wire forming a series of
parallel turns.
If the turns are parallel to the magnetic meridian, and they are
traversed by a current, they produce a magnetic field, the strength of
which is proportional to the strength of the current, and which
may be represented by GI. The horizontal component of the
terrestrial field at this point being H, the horizontal component of
the field is ^/G' 2 ! 2 + H 2 , and its direction makes an angle 8 with the
C*T
magnetic meridian, the tangent of which is equal to .
An infinitely small needle placed at this point, and which at
first was in equilibrium in the plane of the needle, will be deflected
through an angle <5, and from it we may deduce the strength of the
current by the expression
TT
This formula is only exact provided the magnetic field is uniform
throughout the whole space which the needle occupies. When the
needle has a length which is considerable in reference to the
dimensions of the frame, the intensity of the field is not constant,
and the formula for the deflection is less simple. In that case, by
an empirical graduation, we could determine the ratio which exists
between the strength of the current and the deflection produced.
The magnetic moment of the needle has no influence on its
position of equilibrium ; it has no other effect than that of modifying
the strength of the forces, and therefore the duration of the
oscillations of the needle.
TANGENT GALVANOMETER. 487
In order to increase the sensitiveness of the galvanometer that is,
the deflection 8 for a given current we must increase the value of G,
and diminish that of H. The value of G is increased by increasing
the number of turns by Schweigger's method, and by placing them as
near the needle as possible. In order to diminish H, a magnet is
placed at a certain distance, which produces at the centre of the
frame a magnetic field parallel, and in the opposite direction to, that
of the Earth.
Use is sometimes made of a quasi-astatic system of two needles
(299), one of which is inside the frame and the other is outside ;
the action of the Earth on the movable system is then far feebler
without there being any appreciable modification in the action of the
current, which is exerted more particularly on the inner needle. We
may also use two frames, each having one of its needles in the centre,
and pass the current in opposite directions, so that the actions exerted
on the two needles are concordant.
504. TANGENT GALVANOMETER. In order to determine the
absolute value of the strength of a current, besides knowing the
component H of the terrestrial magnetism, we must also know the
constant G of the galvanometer. The name tangent galvanometer \s>
given to a galvanometer, the wire of which has been coiled in such a
manner that this coefficient may be calculated from the dimensions
of the wire and the shape of the frame.
If on the frame a wire L is coiled on a circle of radius a in such
a manner as to make n turns, and if the needle, which is supposed to
be infinitely small, is placed at a point of the axis at a distance u
from the circumference, we shall have (493)
2TTa 2 La A7T 2 ?
G = n = = -
u* fc 3 L
which gives
I = --r- tan 8 = / - } tan 8 .
La
The distance u is equal to a, when the needle is at the centre
of the circle.
If the length is to be taken into account, we must estimate the
strength of the field outside the axis of the currents by the formulas
of (368).
The formula of the tangent galvanometer would be exact and
independent of the length of the needle if the field of the current
were uniform. This would be the case, for instance, with a
488 PARTICULAR CASES.
cylindrical coil (495) or a spherical coil with equidistant currents
(497). If n^ is the number of turns for unit length, we have
o
G = 47r^j in the first case, and in the second G = - Tm v
O
505. ELECTRODYNAMOMETERS. In an electrodynamometer we
measure directly the action exerted between two circuits, one
fixed and the other movable, traversed by the same or by different
currents. Suppose, for instance, that the magnet of a tangent
galvanometer is replaced by a small coil, through which a current
could be passed by a bifilar suspension, and which is in equilibrium
when the axis of the coil is in the magnetic meridian. If a current
I is passed through the wire on the frame of the galvanometer and
a current I' in the coil, the latter is displaced, and by a suitable
torsion a of the suspension, it is restored to its original position.
The magnetic moment of the movable coil is proportional to I',
and may be represented by S* ; the couple produced by the action
of the frame is then GSII'. As the couple of torsion of the bifilar
is proportional to the sine of the angle, if T is the moment of the
couple which corresponds to an angle of torsion equal to ,
GSII' = Tsina.
If the two wires are traversed by the same current I, the expression
becomes
GS
Hence we might determine the strength of the current in
absolute measure if we knew the constants T, S, and G, or we might
leave these constants undetermined, and use the apparatus as an
instrument of comparison. This is the principle of Weber's
experiments.
If we suppose that the current traverses the parallel rectangular
frames (492), as in Cabin's experiments, the intensity might be
deduced from the attractive or repulsive action exerted between the
two circuits.
506. MEASUREMENT OF DISCHARGES. When the duration
of the current is so short that the needle has no time to undergo
an appreciable displacement before the current stops, it has
nevertheleless received an impulse or throw, and acquired a
certain velocity; it is impelled from its position of equilibrium,
and returns to it after a series of oscillations. This is the case,
MEASUREMENT OF DISCHARGES. 489
for instance, of the discharge of a condenser through a conducting
wire in which is a galvanometer; the total quantity of electricity
may be deduced from the angle of throw imparted to the needle.
The strength of a permanent current in the galvanometer in
question* is given by an expression of the form
in which f (B) reduces to the angle 8 when the deflections are very
small. If p, be the magnetic moment of the needle, the action
of the current on the needle produces a couple, the moment of
which is /xGI.
We know, on the other hand, that when a body is movable
about an axis, the product of the moment of inertia K, by the
angular velocity is equal to the moment of the resultant couple
in reference to the axis of rotation. Hence, since the deflection
during the discharge is so small, that the action of the Earth can
be neglected, we have for the needle in question,
v d{ * <-T
K =/xGI.
dt
If dm is the quantity of electricity which flows in unit time
dt, this equation becomes
da) dm
K = MG __
dt dt
From which, if w o is the initial angular velocity, and m the total
discharge, we get
Kw o = /xGw.
The needle, once impelled with this velocity co o , has a vis
K.CO*,
viva equal to - L and it stops at an angle 0, when it has done
work of the same value against the action of the terrestial field.
We have then
TT- a f\
= H/x(i - cos 6) = 2H/x sin 2 - ,
or
n 2 -,
i /HK
m = L \ 2 sm - .
G \ /A 2
49 PARTICULAR CASES.
If the deflections are small enough, we may simply take
i /HK H /~K~
= cV^ eW^'
=
It appears thus that the angle of throw 6 is proportional to
the quantity of electricity which flows during the discharge, and
this law of proportionality will be sufficient for all comparative
experiments.
To determine m in absolute value we must know the constant
G of the galvanometer, and the quantities which come under
the root.
It may be observed that if the needle is left to itself under
the influence of terrestrial magnetism, the time T of infinitely
small oscillations is
from which follows
HT
m = - .
g 7T
As a matter of fact, the true angle of throw is diminished
by the resistance of the medium, and by the induction currents
which the motion of the needle produces in the wire. But if
the oscillations do not diminish very rapidly, this effect is allowed
for by adding to the angle 0, a quarter of the excess of this
deflection over the deflection produced on the same side by the
succeeding oscillation. We shall have, finally,
m =
FARADAY'S DISCOVERY. 491
CHAPTER IV.
INDUCTION.
507. FARADAY'S DISCOVERY. The electromagnetic actions
studied in the preceding chapters are purely mechanical ; they are
exerted on conductors traversed by currents, and correspond to a
permanent condition of currents, and of the magnets near them. In
all cases in which the systems experienced relative displacements, we
have implicitly assumed that those displacements had no influence
on the electric condition of the conductors. Faraday discovered
in 1831 a class of phenomena of a totally different kind which
corresponds to the variable condition of the system ; these
phenomena, which he comprised under the term induction^ are
of an electrical character, and are manifested by the production of
temporary currents in conductors.
The currents which are formed are called induced currents ; the
induced circuit is that submitted to induction ; the term inductor is
applied to the current, the variation in which has been the cause of
the induced current.
508. The phenomena discovered by Faraday may be classed
under several heads :
i st. A closed circuit becomes the seat of a temporary current
whenever a magnet is displaced near it ; or if the magnetisation is
varied ; or still more generally when the magnetic field is modified
in which the circuit is placed. This is magnetoelectrical induction.
2nd. Analogous effects are obtained by substituting a system of
currents for the magnetic system. The circuit in question is
traversed by an induced current whenever the distance, strength, or
form of the external current is altered. The effect is the same as
that which would produce the corresponding modification of the
equivalent magnetic system or the current. This is electro dynamic
or voltaic induction.
3rd. The change of form or of relative position of a closed
49 2 INDUCTION.
circuit, in reference to the magnetic field of a system of magnets or
of currents, is ordinarily sufficient to give rise to an induced current
in this circuit, which comes under one of the preceding heads.
4th. Finally, the mere fact of altering in any way the strength of
the current in a circuit, even when it is withdrawn from any external
action, produces an induction current in this circuit which adds itself
to the principal current, and always tends to counteract the change
of strength which it experiences ; it is a current of self-induction
or an extra-current.
509. Experiment has established the following general facts in
reference to induction currents :
i st. Whatever be the kind of variation which gives rise to an
induction current, two equal variations in opposite directions always
give rise to equal and opposite currents.
2nd. The duration of the induced current is equal to that of the
variation of the inducing system.
3rd. The quantity of electricity set in motion in the induced
current by any operation is independent of the duration of the
variation, and therefore of that of the induced current itself.
4th. Lastly, the nature of the conductor in which the induction
currents are transmitted is only of importance in so far as it affects
the resistance which it brings into the circuit.
510. Examining the various circumstances in which induction
currents are produced, it is easily seen that their common charac-
teristic is that of corresponding to a variation in the flow of magnetic
force which traverses the induced circuit. This is evident for all
the phenomena of the relative displacement of currents or of
magnets; experiment shows, moreover, that any displacement, or
any deformation of the induced circuit which does not modify the
value of the flow which traverses it, never produces induced currents.
This is also the case with the extra current. For a current gives
rise to a magnetic field, and therefore to a flow of force in the
surface of the circuit which it traverses. It is easily seen that
any change of intensity, or of shape, which modifies this flow, may
produce an effect analogous to that which would be produced by the
displacement of an external magnet, giving rise to the same variation.
We are thus led to define the phenomena of induction in the
following manner :
When the flow of magnetic force which traverses a closed circuit is
in any way modified, this circuit becomes the seat of a temporary
current, the duration of which is equal to that of the variation of
the flow.
LENZ'S LAW. 493
This enunciation defines the conditions in which induced
currents are produced. It remains for us to establish the direc-
tion and the magnitude.
511. LENZ'S LAW. A short time after Faraday's discovery,
Lenz enunciated the following law, which establishes a connection
between the induction produced by the displacement of the inducing
system, and the electromagnetic work as defined by Ampere's
formula :
Any displacement of the relative positions of a dosed circuit, and of
a current or magnet, develops an induced current, the direction of which
is such as would tend to oppose the motion.
512. NEUMANN'S THEOREM. Lenz's law, which is of great
practical utility, merely gives the direction of the induced current,
but not the intensity. Assuming, as an experimental fact, that the
induction produced in a very short time is proportional to the velocity
with which the conductor moves, Neumann has given a complete
theory of the induction currents produced in a movable linear
conductor in the presence of any magnetic system. He has thus
demonstrated this theorem, which we shall afterwards meet with
under a more general form :
The electromotive force of induction is equal to the work which
would be done in unit time by the magnetic system, if the intensity of
the current in the induced circuit was equal to unity.
513. THEORY OF HELMHOLTZ AND THOMSON. The existence
of phenomena of induction may be considered as a necessary
consequence of the conservation of energy combined with the
electromagnetic law of Ampere and the law of Joule. This
proposition was first put forth in 1847 by Prof. Helmholtz in his
celebrated memoir on the Conservation of force. Sir W. Thomson
arrived independently at the same conclusions.
Consider an invariable magnetic system, in the vicinity of a fixed
conductor S, in communication with a battery. If the magnet is
stationary, the strength I of the permanent current is determined by
Ohm's law, and if E is the electromotive force of the battery and R
the resistance of the circuit,
(1) E-I.R.
Multiplying both sides by I <#, we get
(2) EI <#=
494 INDUCTION.
This equation expresses that during the time *#, the energy due
to the chemical actions is equal to the thermal energy expended in
the circuit on Joule's law.
Suppose now that instead of being stationary, the magnetic
system moves in accordance with electromagnetic actions. The
external work resulting from this displacement, can only be borrowed
from the sole source of energy in the system (that is, the chemical
action), and the preceding equation must be in default. On the
other hand, there is no reason for supposing that the laws of Faraday
and Joule cease to hold ; in other words, the weights of the bodies
combined in the different couples must still be proportional to the
strength of the current, and the thermal energy disengaged in the
circuit is equal to the product of the resistance into the square of
the strength. Hence, the strength of the current could not retain
its original value.
514. Suppose now that the magnet is displaced in such a way
that the new value of the strength remains constant. So long as
this condition is fulfilled, the excess of chemical work over the
thermal energy expended in the circuit in the time dt, serves to
produce the external work dT corresponding to the electromagnetic
forces. Hence, if I is the strength of the current,
(3) EL#=I 2 R, gives
(12) (E
Suppose that while the magnetic power <3> is constant, the shell is
brought from an infinite distance to a determinate position in
presence of the induced circuit which we suppose fixed ; we have
then
ELECTROMAGNETIC INDUCTION. 499
If both sides of this equation are multiplied by I, and we
integrate from /=0 to the time t when the shell takes up its final
position, we get
ft ft / , \ t
(13)
The first member of this equation represents the excess of the
chemical energy, furnished by the battery during the time /, over the
energy which appears as heat in the circuit during the same time.
The first term of the second member is the total work of the
electromagnetic actions ; this work depends on the law of the
motion. We may imagine, for instance, that the shell may have
been approached very slowly, so that the induction is very feeble,
and the principal current differs very little from its initial value I ;
in this case, if M is the flow of force corresponding to the final
position of the shell, the electromagnetic work will be equal
to 3>MI .
This latter term represents the change in the potential energy of
the current ; it is zero if the two limiting values of the current are
the same that is, if the magnetic shell is in a state of rest in its
final position.
521. If the power of the magnetic shell, while still at rest,
was variable, the equation would be quite analogous :
(El - RI 2 X/= M I - dt+ ( L
and would lead to the same conclusions.
522. The most general case is that in which the three functions
M, < and L vary simultaneously that is to say, when the magnetic
shell changes its strength, its form, and relative position, and that
the circuit itself is deformed. Equation (12) gives therefore the
electromotive force of induction
LI) d& ,dU dl d"L
(14) e = --^ = M + $--- + L-- + I .
at at at dt dt
If the induced circuit contains no electromotive force inde-
pendent of induction, we need only make E = in the preceding
equations.
K K 2
500 INDUCTION.
523. ELECTRODYNAMIC INDUCTION. If the inducing system
were a constant current, we might replace it by the equivalent
magnetic shell, and thus bring it within the preceding case ; but in
consequence of reactions, the inducing circuit itself will be under
induction, and the strength of the current will no longer be constant.
If R' and L' are the resistance and the coefficient of self-induction
of the inducing circuit, and E' the electromotive force which it
contains, the strength of the current in the two circuits will be
determined at each instant by two simultaneous equations
OiLi/CQRY. (E
+LT).
The complete solution of these equations generally presents
great difficulties, and in the next chapter we shall investigate the
simplest cases in which it can be obtained; but the differential
equations already suggest some important remarks.
If we add these equations, after having multiplied the first by I
and the second by I', we get
(16) (El + ET - RI 2 - RT 2 )<# = L/(MI' + LI) + IV(MI + LT).
The left hand side represents the excess of the energy furnished
by the sources in the two circuits over the thermal energy expended
in the conductor.
The right hand side may be written as follows :
(17) -
m' denoting a magnetic mass of suitable magnitude, and h and /'
lengths. We shall have then
As the three first fractions are abstract numbers, it will be
seen that the resistance expressed in electromagnetic units is equal
to the quotient of a length by a time that is to say, a quantity
of the same order as a velocity.
We may, indeed, easily discover in the experiment itself, a
physical representation of this velocity. Suppose, in fact, that
the cross-bar moved uniformly with a velocity u, and that the
intensity I of the current is measured by the action which it
exerts on a needle placed in the centre of a tangent galvanometer
(504) ; we shall have
On the other hand, MR = HS, or
it follows from this that
tan 8 = -
CLOSED CIRCUIT IN A UNIFORM FIELD. 505
If the velocity u is so great that the action of the current is
equal to that of the Earth that is, that the deflection of the
needle in the galvanometer is 45 we shall have
and if we take fl 2 = L, R = #.
Hence the resistance of the circuit in question is equal to the
velocity with which the bridge must be uniformly moved under
the given conditions, in order that the action of the current
induced in a galvanometer of suitable dimensions may produce a
deflection of 45.
527. The following experiment, suggested by Faraday, may be
considered as an application of the same problem.
Suppose that two electrodes, A and B, are immersed in water
on the opposite edges of a river, of a canal, or of a current in
the sea, and are connected by a metal conductor. If u is the
velocity of the current, and a the distance of the electrodes,
under the influence of the Earth's magnetism, an electromotive
force equal to iSLa will be established between them, which, in a
circuit of resistance R, will develop a current of intensity - .
R
The experiment is not impracticable; but, unless we could work
with very great values of u and a, the polarization of the electrodes
would no doubt make it very difficult to verify the conclusion.
528. CLOSED CIRCUIT IN A UNIFORM FIELD. Consider a
closed circuit, which may be supposed plane (487), and let S be
its surface. Let us suppose it placed in a uniform field of in-
tensity F the terrestrial field, for instance and perpendicular to the
direction of the field. If the frame be made to turn through an
angle a, the variation of the flow of magnetic force is equal to
FS (i-cosa), and the quantity M of electricity put in motion in
the circuit, which we suppose has the resistance R, is given by
the ratio
FS(i -cos a)
-
If the frame turns through 180, face for face, we have
506 PARTICULAR CASES OF INDUCTION.
This quantity of electricity may be measured in absolute value
(506) by the throw of the needle of the galvanometer, which
would enable us to determine the intensity of the magnetic field.
The direction of the current on Lenz's law, is that which ought
to traverse the current in its original position, in order that it may
be in stable equilibrium.
529. DETERMINATION OF THE INCLINATION BY INDUCTION
CURRENTS. We have seen (487) that the tangent of the magnetic
inclination is equal to the ratio of the works which for the same
strength of current correspond to a rotation of 180 of the circuit,
starting from a position at right angles to the magnetic meridian :
ist, about a horizontal axis perpendicular to the meridian; 2nd,
about a vertical axis. This ratio is that of the electromotive
forces of induction for the same displacements, and therefore that
of the corresponding quantities of induced electricity (506) ; a
measurement of this latter ratio will therefore give the inclination.
530. FARADAY'S Disc. A metal disc, movable in a uniform
field about an axis parallel to the direction of the field, forms part
of a circuit which communicates on the one hand with the axis of
rotation, and on the other with a spring which presses on a point
of the circumference. When the disc is put in uniform rotation,
a uniform current is also produced in the circuit. It will be seen
that the arrangement of this experiment is, as it were, the inverse
of that of Barlow (483). -If the plane of the disc is vertical, and
the force of the field F traverses it from front to back, and if
the direction of the rotation is that of the hands of a watch, the
induced current traverses the disc from the centre to the edge.
If a is the radius of the disc, and w the angular velocity, the
electromotive force is
,-^-- 2 F
~ dt 2
An analogous result is obtained, though with a less simple
calculation, by placing the disc between the poles of a horse-shoe
magnet or between the armatures of two electromagnets. In this
latter form, M. Le Roux obtained currents so strong that bright
sparks passed between the disc and the spring.
Moreover, all the experiments, particularly those which were
examined in Chap. III., in which the motion of a conductor is
produced by electromagnetic or electrodynamic actions, would pro-
duce an inverse induction current if the motion was kept up by
an extraneous cause.
TERRESTRIAL CURRENTS. 507
531. TERRESTRIAL CURRENTS. Let us consider, for example,
a sphere magnetised uniformly. Let us suppose that a conducting
arc, resting with one end at the pole and the other on the
equator, turns about the axis with a uniform motion; this arc
will cut the same flow of force as if it were applied on the sur-
face along a meridian. An element ds, the velocity of which is v,
cuts in each unit of time, a flow of force equal to Zvds, Z
being the perpendicular component of the magnetic force on the
corresponding parallel. If V be the velocity at the equator, F^
the magnetic force at the pole, a the radius of the sphere, and A
the latitude of the element ds, we have
z> = VcosA, Z = FpsinA, ds = ad\.
The flow of force cut in each unit of time by the whole arc
is equal to the electromotive force of induction *?, which gives
e VFtf sin A cos \d\ - F v Va.
Jo 2
If the arc is insulated, this value of e represents the difference
of potentials at the two ends.
If the ends of the arc were connected to two bodies of the
same capacity C, these bodies would acquire, after a longer or
shorter time, equal and opposite statical charges, the absolute value
of which would be -Ce.
2
Finally, if the arc were closed by a fixed conductor on the
inside or outside of the sphere, the circuit would be traversed by
a continuous uniform current, from the equator to the pole or
conversely.
As the Earth may be compared to a sphere magnetised uni-
formly, it will be seen that an external arc, which does not share
the rotatory motion, should be traversed by an induction current
from the equator to the pole, for the direction and magnitude of
the induced currents only depend on the relative motion of the arc
and of the magnetic system. It is probable that this induction
plays an important part in certain natural phenomena, such as
the aurora borealis (which seem to be electrical discharges in the
upper regions of the atmosphere), the currents observed on the
surface of the Earth in telegraphic wires, and the perturbations of
the magnetic elements.
508 PARTICULAR CASES OF INDUCTION.
532. VARIABLE STATE OF A CURRENT. The establishment of
a current in a circuit represents a certain amount of work which
is the potential energy of the current; this energy is absorbed at
the starting of the current, and is restored when the electromotive
forces disappear. In all cases, the effects of self-induction, which
are the consequence, determine the law of intensity during the
variable period, whether at the closing or opening of the circuit.
Consider a single circuit. Let R be the total resistance, L its
coefficient of self-induction, and E the electromotive force which
it contains. We have the equation
dt
If we suppose L constant as well as E, the strength at each
moment is given by the formula
(2) l-^ + ^-l^-r,
I being the initial value and I x that which corresponds to the
permanent state.
The total quantity of electricity which passes in time /, is
f' -d g-)
J' l ^
If the time / is sufficiently great, we have simply
C \ L
\o) i \ JR.
Jo
We have also, for a sufficiently long time,
this expression is proportional to the calorific energy expended in
the circuit.
VARIABLE STATE OF A CURRENT. 509
It may be observed that these values of the two integrals are
the same as if there had been a current of strength - during
T
the time - , which had been succeeded by a current of the
R
normal intensity I v during the rest of the time.
Suppose that the electromotive force is constant, and that we
measure the time from the closing of the circuit ; we have then
i.-o,
and therefore
'-
533. The expression ^T represents the strength of the extra
R
current obtained after the moment /. It will be seen that the current
only attains its normal strength after an infinitely long time ; but if
-p
the ratio - is very great, which it is in most cases, the exponential
J_/
tends rapidly towards zero, and after a very short time, the real
strength only differs from the final strength, by a quantity which
may be neglected. In order to calculate after what time this
difference would be below a given quantity, - for instance, we
may put
from which we deduce
The total quantity of electricity which corresponds to the extra
current is
(6)
R 2
it is the same as if the current had had half the intensity of the
11 lE - 2L
normal value - -- m the time .
2 R R
510 PARTICULAR CASES OF INDUCTION.
534. EXTRA CURRENT ON OPENING. Suppose that the per-
manent regime being established, we suddenly introduce a resistance
r in the circuit ; we shall have at the two limits
IO ~R'
E
E
and the strength at any instant is given by the equation
(7) I "
The value of the total quantity of electricity which corresponds
to the extra current is
(8) ' EL
This case has some analogy with that in which the circuit
is broken in air, the resistance r being that of the layer of gas
traversed by the spark on breaking; but in reality this resistance
is far from being constant 'while the phenomenon lasts.
Suppose that instead of breaking the circuit we had separated
it from the battery, by replacing the latter by a wire of the same
resistance, so that the total resistance of the circuit is still repre-
sented by R, which simply amounts to suppressing the electromotive
force. Equation (i) reduces to
(9) L
Determining the constant by the conditions that for /=0 we
p
have I = , it follows that
R
E _R
(10) ! = -* L.
In this case the law of the extra current of opening is the
same as that of the current of closing (533), and the quantities
of electricity put in motion are the same.
VARIABLE ELECTROMOTIVE FORCE. 511
535. VARIABLE ELECTROMOTIVE FORCE. Suppose that the
electromotive force, instead of being constant as with ordinary
batteries, undergoes periodical variations, and is represented by an
expression of the form
(n) E = E sin 27T .
When the steady condition is obtained, the intensity of the
current evidently follows the same period, and may be represented
by the expression
(12)
/ -- (j>\
Substituting this value in equation (i), and determining the
constants A and , by the condition that it shall be satisfied for
any given time, we find
(13)
^2
and
277-L
(14) tan 27Td> = .
TR
It will be seen from this that the effect of the coefficient of
self-induction is to increase the apparent resistance of the circuit.
T
The strength of the current is zero whenever /- is - , that is to say, that the difference of phase
is equal to -. The maximum retardation of the induced current
4
is therefore equal to a quarter of the whole period, or to half
the semi-period.
512 PARTICULAR CASES OF INDUCTION.
During a semi-period, the quantity of electricity which passes
through the circuit is
[~$ f AT
(15) Q = A " sin27r-^= ,
Jo 7T
and the corresponding calorific work
(16) W = RA 2 f 2 sin 2 27r-
gives the equation
C~ ~dt
Observing that I = , this may be written in the form
dt
(19)
dt* -Ldt CL
OSCILLATING DISCHARGES. 513
The general integral of this equation is
(20) Q = A^ + Ay ;/ ,
p and p' being the roots of the quadratic equation,
which gives
= - + / R2 _L
"~~
According as L^ -- , these roots are real or imaginary.
4
The constants A and A' are determined by the condition that
for /=0, we have Q = Q , and 1 = 0, which gives Q = Ap + A'p.
If the roots of the equation (21) are real, then representing
the radical
Q = Q ^ 2L
(22)
/"R* ~
ical^/- ,
I R\ a, /x R\ -a,"]
-+ r ) e + ( --- r ) e h
2 4 aLy \2 4 aLy
at -at\
'-' )
When the roots are imaginary, we may still take the integral
in the same form, and replace the constants by their imaginary
V~i R2~
--- , we get then
r R i
cosaV + sinaV
(2$
O -
^fLC e '
L L
514 PARTICULAR CASES OF INDUCTION.
We get from these equations, in the two cases,
L//=Q ,
f
J
These results were evident a priori, for the discharge is com-
plete, and the work it produces is reduced to a disengagement of
/e
heat ; the calorific work RIV/ should be equal to the electrical
energy (89) which the conductor had before the discharge, that
iQ
is to say, - .
537. The nature of the discharge is very different according
as the solution is given by equations (22) or equations (23).
In the first case the discharge is continuous. The strength
of the current begins by being zero, passes through a maximum,
and then decreases to zero. The maximum is at the time deter-
dl
mined by the condition -r = ^t r
R \ 2a* R
a ) e = 1-a,
2 L J 2 L
which gives
538. In the second case, the values of Q and of I are given
by the periodic functions; the conductor takes alternately charges
in contrary directions, and the wire is the seat of alternating
currents.
The times of maxima and minima of charge correspond to
1 = 0, that is to say, to
sin a'/=0, or a!t = nir.
OSCILLATING DISCHARGES. 515
The oscillations of the discharge are regular, therefore, and
the value of the complete period T is
27T
The values of the alternate maxima are
+ Qo>
RTT
-Qo* 2La/ >
2R7T
+ Qo* 2La ',
_3Rir
-Qo' 2L *', etc.;
they decrease, therefore, like terms of a geometrical progression,
RTF
the ratio of which is e ^LT'.
The maxima of the intensity of the current in the two directions
d\
correspond to = 0, which gives
2La'
tana/= - ,
R
or
sina7= a'x/CL;
7T T
they are still equidistant, separated by a semi-period = - , and
a 2
succeed the periods at which the current is zero, by the time 0,
defined by the smallest angle which satisfies the condition
The values of the maxima of intensity are successively
2R:r
2a'L, etc.;
they also decrease in geometrical progression.
L L 2
516 PARTICULAR CASES OF INDUCTION.
Disregarding the sign, the total quantity of electricity put in
motion is
_Rir
/ __ R JL _2R!T \ !+<> sL7'
(2 5 ) Q (l + 2* 2La' +2 , 3La' + \ = Q Q _ _.
\ / -**.
i-e 2La'
RTT
This total mass is the greater the nearer the quantity e zLa? is
to unity, that is, the greater the factor - , or the greater is
R
. L
mtl
539. We pass to the case of a continuous current by making
Q
Q o = oo, C = oo, and = 1&. The roots of equation (21) are then
real, and the discharge is continuous ; we arrive thus at the formula
already found (532)
E
i e
If we suppose the coefficient L of self-induction to be very small,
we always fall into the case of real roots ; the coefficient a then
R / 2 L\ R i
becomes equal to ( i-^r^s ) or -z---^ t an d the equations
2Li \ CK. 4 / 2JL CR
(22) become
The current accordingly starts at once with its maximum value,
and then diminishes indefinitely.
It must, however, be remarked that all the preceding discussion
rests on the implied hypothesis that electricity does not possess
inertia, and that the intensity is the same throughout the whole
extent of the wire. These hypotheses appear justified in all cases
of permanent currents, or of slow variations, but we can no longer
assume them in the case of sudden discharges ; the results at
which we have arrived can only then be considered as a first
approximation.
CASE OF TWO CIRCUITS. 517
540. CASE OF Two CIRCUITS. Let us consider two adjacent
circuits C and C', whose relative position is fixed, and which contain
electromotive forces ; we shall represent by E, R, and L the electro-
motive force, the resistance, and the coefficient of self-induction of
the first circuit ; by the same quantities accentuated, the analogous
quantities of the second ; and by M the coefficient of mutual
induction. All the coefficients being supposed constant, we shall
have the two simultaneous equations
M + L + RI-E = 0,
dt dt
(27) dl dl ,
M + L' + RT-E' = 0,
dt dt
the solution of which is of the form
RI -E=
'
The coefficients A, B, A', B' are constants to be determined from the
conditions in respect to the limits.
Expressing the condition that these values of the strength satisfy
the differential equations (27), whatever be the time, we find that the
exponents p and p are roots of the equation of the second degree.
/ M 2 \ /R R'\ RR'
(29) ( i -- U2 + / _ + _ \p+ -- = o.
\ LL'/ \L L'/^ LL'
These roots are always real ; again they must be negative, seeing
that, under no circumstances, can the strength increase indefinitely
with the time. Hence we must have LL/>M 2 , which moreover is
evident (519); we could only have LL' = M 2 if the two circuits
coincided.
It follows also from this condition that if the coefficient of self-
induction L of a wire is very small, the coefficient of mutual
induction M of this wire with any other wire is also very small.
541. Consider the case in which E' = 0, that is to say in which
the second circuit contains no electromotive force.
If we only wish to determine the quantities of electricity m and
m' which traverse the two circuits in time /, we may calculate the
518 PARTICULAR CASES OF INDUCTION.
integrals \dt and I'dt by equations (27). If I and I' are the
intensities of the two currents at the beginning of the time in
question, we shall have
Suppose that we close the circuit, and that we consider the
phenomenon after a sufficiently long time, we shall have
i =o, i;=o,
i-f, r-oi
and therefore,
(-D-K'-i)-
<"> M EM V
m : = 1= .
R' RR'
If, on the contrary, we open the circuit C, after the steady con-
dition has been attained, we shall have at the two limits,
i =0, r = o.
The quantities of electricity induced in the two circuits are then
equal, and of opposite signs to those of the preceding case which
was evident, for the variation of the flow of force was the same in
both cases.
It is to be observed that the extra current of C is independent
of C', and, on the other hand, that the induction on the circuit C'
only depends on its resistance R', on the coefficient of mutual
induction of the two currents, and on the strength I of the permanent
current in the circuit C. The direct consideration of the flow
of force still enables us to foresee these results.
If we wish to know the strength of the currents at each instant,
the solution of the problem must be completed by determining
the constants.
CURRENT ON OPENING. 519
542. CURRENT ON OPENING. Let us first consider the case
in which, having closed the current in the principal circuit C, con-
nection with the battery is broken by leaving the circuit open;
an induced current is formed in the secondary circuit C', but the
principal current is entirely broken. The first of the equations (27)
no longer holds, and the second reduces to
L'
dt
it is identical with equation (9), and therefore the law of the extra
current will be the same in both cases.
In like manner, on the hypothesis that the opening of the
circuit C was instantaneous, and that the current is not prolonged
with a variable resistance, as in the case in which a spark is
produced, we may determine the initial value of the current pro-
duced in C'.
Let us integrate the second of these equations (27), taking E' = 0,
from /=0, to a time T which is infinitely small in comparison with
the duration of the induction current in C' ; denoting by l\ the
intensity of the current induced at the time r, we shall have the
equation
_M-+L'I;+R' f T r<#=o.
R Jo
If we make T diminish towards zero, the last term tends itself to
zero, seeing that T has an infinitely small value, and that the intensity
I' of the current retains a finate value ; we have then, in the limit,
ME M
(33) ' 1 "
Thus the initial intensity of the induced current is to the
intensity of the inducing current I in the ratio of the coefficients
M and L'.
Hence, at any given instant, we shall have, by equation (10),
ME _RL<
(34) 1' = --'.
52O PARTICULAR CASES OF INDUCTION.
From this is deduced, for the quantity of electricity put in
motion,
L'RR' RR''
and, for the calorific work expended in the circuit,
543. CURRENT ON CLOSING. The moment the inducing
current is closed, the two circuits react on each other, and we must
take into account the two simultaneous equations (27).
If we put
2 R'
(35)
the roots of the equation are
/ + R / L)-2RMa
2(LL'-M 2 )
2(LL'-M 2 )
The coefficients will be determined by the aid of equations (27)
and (28) by the condition that, for /=0, we have 1 = and I' = 0,
which gives
M L
Jx JK
M L'
- (A P + Ep) + ( A> + B >') = ,
K. K.
We get then
_ E( if/ R'L-RL'\ ft / R'L-RIA P 't~]
! =