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plane, their directions cannot intersect although they are not pa-
rallel. In this case a line may always be drawn, which is at the
same time perpendicular to both. To assist the imagination in
conceiving such a geometrical combination, let a vertical rod be
supposed to be erected, and from two different points of this
rod let lines be drawn horizontally, but in different directions,
one, for example, pointing to the north, and the other to the
east. If voltaic currents pass along two such lines, they will
mutually attract, when they flow both to or both from the ver-
tical rod ; they will mutually repel, when one flows to the ver-
tical rod and the other from it.

In either case the mutual action of such currents will have
a tendency to turn them into the same plane and to parallelism.
If they mutually attract, their lines of direction
turning round the vertical line will take
position parallel to each other, and at the same
side of that line. If they mutually repel, they
will turn on the vertical line in contrary di-
rections, and will take a position parallel to
Fig. 635. each other, but at opposite sides of it.
In fig. 635., AB and CD represent two currents which are not in


the same plane. Let PO be the line which intersects them both
at right angles, and let planes be supposed to pass through
their directions respectively, which are parallel to each other,
and at right angles to PO. If, in this case, CD be fixed and AB
moveable, the latter will be turned into the direction a b pa-
rallel to CD; or if CD were free and AB fixed, CD would take
the position cd ; if both were free they would take some po-
sition parallel to each other ; and if free to change their planes,
they would mutually approach and coalesce. It follows from
this, that if the direction of either of the two currents be re-
versed, the directions of the forces they exert on each other
will be also reversed ; but if the directions of both currents be
reversed, the forces they exert on each other will be unaltered.

2010. Mutual action of different parts of the same current.'
Different parts of the same current exercise on each other a re-
pulsive force. This will follow immediately as a consequence
of the general principle which has been just established. Since
a repulsive action takes place between oc and oc',jig. 632.,
and such action is independent of the magnitude of the angle
coc', it will still take place, however great that angle may be,
and will therefore obtain when the angle occ' becomes equal to
180 ; that is, when oc' forms the continuation of CO, or coalesces
with oc'. Hence, between oc and oc' there exists a mutually
repulsive action.

2011. Ampere's experimental verification of this. Inde-
pendently of this demonstration, M. Ampere has reduced the
repulsive action of different pai-ts of the same rectilinear current
to the following experimental proof:

Let A B c D, Jig. 636., be a glass or porcelain dish, separated
into two divisions by a partition AC, also of glass ; and let it be
filled with mercury on both sides
of AC. Let a wire, wrapped
with silk, be formed into two
parallel pieces united, by a se-
micircle whose plane is at right
Fig- 636. angles to that of the straight

parallel parts, and let these two parallel straight parts be placed
floating on the surface of the mercury at each side of the
partition AC, over which the semicircle passes. The mer-
cury in the divisions of the dish is in metallic communication
with the mercurial cups E and F placed in the direction of


the straight arms of the floating conductor. When the cups
E and F are put in connexion with the poles of a voltaic bat-
tery, a current will pass from the positive cup to the end of
the floating conductor, from that along the arm of the con-
ductor, then across the partition by the semicircle, then along
the other floating arm, and from thence through the mercury
to the negative cup. There is thus on each side of the par-
tition a rectilinear current, one part of which passes upon
the mercury, and the other part upon the straight arm of the
floating conductor. When the current is thus established, the
floating conductor will be repelled to the remote side of the
dish. This repulsion is effected by that part of the straight
current which passes upon the mercury acting on that part
which passes along the wire.

2012. Action of an indefinite rectilinear current on a finite
rectilinear current at right angles to
it. A finite rectilinear current a b,
fig. 637., which is perpendicular to an
indefinite rectilinear current cd lying
all at the same side of it, will be acted
on by a force tending to move it parallel
to itself, either in the direction of the
Y\f. 637. indefinite current, or in the contrary

direction, according to the relative di-
rections of the two currents.

If the finite current do not meet the indefinite current, let its
line of direction be produced till it meets it at a. Take any
two points c and d on the indefinite current at equal distances
from a, and draw the lines cb and db to any point on the finite

First case. Let the finite current be directed towards the in-
definite current. Hence the point b will be attracted by d and
repelled by c (2007) ; and since db=cb, the attraction will be
equal to the repulsion. Let the equal lines be and bf represent
this attraction and repulsion. By completing the rectangle,
the diagonal bg will represent the resultant of these forces ;
and this line bg is parallel to cd, and the resultant is contrary
in direction to the indefinite current.

The same may be proved of the action of all points on the
indefinite current on the point b, and the sum of all these re-
sultants will be the total action of the indefinite current on b.


The same may be proved respecting the action of the de-
finite current on all the points of the indefinite current.

Hence the current a b will be urged by a system of forces
acting at all its points parallel to cd, and in a contrary direction.

Second case. Let the finite current be directed from the in-
definite current. The point b will then be attracted by c and
repelled by d, and the resultant bg' will be contrary to its
former direction.

Hence the current a b will be urged by a system of forces
parallel to c d, and in the same direction as the indefinite

Since the action of the two currents is reciprocal, the in
definite current will be urged by a force in its line of direction,
either according or contrary to its direction, as the finite current
runs from or towards it.

2013. Case in which the indefinite current is circular. If
the indefinite current cd be supposed to be bent into a circular
form so as to surround a cylinder, on the side of which is placed
the vertical current a b, it is evident that the same reciprocal
action will take place ; but in that case the motion imparted will
be one of rotation round the axis of the cylinder as a centre.

2014. Experimental verification of these principles. These
principles are experimentally verified by the apparatus,^. 638.,

where azsb repi'esents a rib-
bon of copper coated with
silk and carried round the
copper circular canal v. A
conductor connects the mer-
curial cup c with the central
metallic pillar which supports
a mercurial cup p. In this
cup the metallic point m is

Fio . 63g placed. The mercurial cup

d is in metallic communica-
tion with the acidulated water in the circular canal v. A hoop
of metal h is supported by the point m by means of the rect-
angular wire, and is so adjusted that its lower edge dips into
the liquid in the canal v.

Let the mercury in a be connected with the positive pole of
the battery, and the mercury in d with the negative pole. The
current entering at a will pass round the circular canal upon
R 5


the coated ribbon of copper, and, arriving at b, it will pass to c
by a metallic ribbon or wire connecting these cups. From c it
will pass to the central pillar, and thence to the cup p. It will
then pass from m as a centre in both directions on the wire,
and will descend to the hoop k, from which it will pass into the
liquid in the canal v, and thence to the cup d, with which the
liquid is in metallic communication, and, in fine, from d it will
pass to the negative pole of the battery.

By this arrangement, therefore, a circular current flows
round the exterior surface of the vase v, while two descending
currents constantly flow upon the wire at right angles to this
circular current. The circular current being fixed, and the
vertical currents being moveable, the latter will receive a
motion of continued rotation by the action of the former ; and
in the case here supposed, this rotation will be in a direction
contrary to the direction of the circular current. If the con-
nexions be reversed by the reotrope, the direction of the cir-
cular current will be reversed, but at the same time that of the
vertical currents on the wire will be also reversed ; and, con-
sequently, no change will take place in the direction of the
rotation. These changes of direction of the two currents neu-
tralize each other. But if, while d is still connected with the
negative pole, b be connected with the positive pole, the con-
nexion between b and c being removed, and a connexion be-
tween a and c being established, then the direction of the
circular current being from * to z will be reversed ; while that
of the vertical cm-rents remains still the same, the direction of
the rotation will be reversed.

2015. To determine in general the action of an indefinite
rectilinear current on a finite rectilinear current. First. Let
, n ^ it be supposed that the finite current A B, Jig. 639.,

has a length so limited that all its points may be
considered as equally distant from the indefinite
current, and therefore equally acted on by it.
In this case the current AB may be replaced by
two currents, AD perpendicular and AC parallel
to the indefinite current, and the action of the

Fig. 639. indefinite current on AB will be equivalent to its

combined actions on AD and AC.

If A be supposed to be the positive end of the finite current,
it will also be the positive end of the component currents AD


and AC. Supposing the indefinite current parallel to AC to
run in the same direction as AC, then AD will be urged in the
direction AC (2012), and AC in the direction AC', by forces pro-
portional to AD and AC. Hence, if AD'=AD, and AC'=AC, AD'
and AC' will express in magnitude and direction the two forces
which act on the component currents. The resultant of these
two forces AD' and AC' will be the diagonal AB', which is evi-
dently perpendicular to AB and equal to it.

Secondly. Let the finite current have any proposed length,
and from its positive end A, Jig. 640., let a line AO be drawn

perpendicular to the indefi-
nite current x'x, this cur-
rent being supposed to run
from x' to x.

If the distance OA be
greater than AB, that cur-
rent AB, whatever be its
position, will lie on the same

^ side of x'x, and the action

~o X of x'x on every small ele-

Fig. 64O. ment of AB will be perpen-

dicular to AB, as has been just demonstrated. The current AB
will therefore be acted on by a system of parallel forces perpen-
dicular to its direction. The resultant of these forces will be a
single force equal to their sum, and parallel to their common
direction. Hence the indefinite current x'x will act on the
finite current AB by a single force R in the direction CD.

If the current AB be supposed to assume successively different
positions, B a , B 2 , B 3 , &c., around its positive end A, the line CD
will represent in each position the direction of the action of
the current x'x upon it.

It is evident that when the indefinite current runs from x'
to x, the action on the finite current is such as would cause it
to turn round its positive end A with a direct, or round its ne-
gative end B with a retrograde rotation.

If the indefinite current run from x to x', the direction of its
action on AB, and the consequent motions of A B, would be reversed.
The point c of the current AB at which the resultant R acts
will vary with the position of the current AB, approaching more
towards x'x as AB approaches the position AB 3 ; but in every
position this resultant must be between A and B. The force

R 6


producing the rotation therefore having a varying moment, the
rotation will not be uniform.

If the distance o A be very great compared with AB, the resultant
R will be sensibly constant, and will act at the middle point of AB.

In this case, if the middle point of AB be fixed, no rotation
can take place.

If the distance OA be less than AB, the current AB will in
certain positions intersect x'x, Jig. 641., and a part will be at

Fig. 641.

one side and a part at the other. In this case the action on AB,
in all positions in which it lies altogether above x'x, is the same
as in the former case.

When it crosses x'x, as in the positions AB 2 , AB 3 , AB 4 , the
action is different. In that case the forces which act on Am,
and those which act on WB, are in contrary directions, and their
resultant is in the one direction or in the other, according as
the sum of the forces acting on one part is greater or less than
the sum of the forces acting on the other part. If Am be in
every position of AB greater than mB, then the resultant will
be in every position in the same direction as if the current AB
did not cross x'x; and if the point A were fixed, a motion of
continued rotation would take place, in the same manner as in
the former case, except that the impelling force would be di-
minished as the line AB would approach the position AB S .

^ But if AO be less than

half AB, the circum-
stances will be different.
In that case there will

^ be two positions AB O and

B AE 4 , Jig. 642., at equal

distances from AB. 5 , at
^ which the line AB will be

Fig. 642. bisected by x'x.

In all positions of AB not included between AB 2 and AB 4 ,


the action of the indefinite current upon it takes place in the
same direction as in the former cases.

But in the positions AB' and AB", where WE' and mv" are
greater than m A, the forces acting on m B' and m B" exceed
those acting in the contrary direction on m A, and consequently
the resultant of the forces on A B in all positions between A B 2
and AB 4 is contrary to its direction in every other position of
the line A B.

In the positions AB O and AB 4 the resultant of the forces in
one direction on Am is equal and contrary to the resultant of
the forces on B m. There will in these positions be no tendency
of the current AB to move except round its middle point.

If the indefinite current x' x pass through A, fig. 643., the
resultants of its action on A B will
be in contrary directions above and
-jj- below x'x, and will in each case
/\ ~ tend to turn the current A B round

' / \. i the point A so as to make it coin-

cide in direction with the indefinite
Fig. 643. current x'x.

2016. Experimental illustration of these principles. These
effects may be illustrated experimentally by means of the
apparatus, fig. 638., already described. The circular current
surrounding the canal v being removed, and the currents on the
wire m being continued, let an indefinite rectilinear current be
conducted under the apparatus at different distances from the
vertical line passing through the pivot, and the effects above
described will be exhibited.

2017. Effect of a straight indefinite current on a system of
diverging or converging currents. If any number of finite
rectilinear currents diverge from or converge to a common
centre, the system will be affected by an indefinite current near
it, in the same manner as a single radiating current would be

Thus if a number of straight and equal wires have a common
extremity, and are traversed by currents flowing between that
extremity, and the circumference of the circle in which their
other extremities lie, an indefinite current x'x placed in the
plane of the circle, as represented in^. 644., will cause the ra-
diating system of currents to revolve in the one direction or the
other, as indicated by the arrows in the figures.



2018. Experimental illustration of this action. These
actions may be shown experimentally, by putting a vertical

wire, fig. 645., in communication with the centre of a shallow
circular metallic vessel of mercury v, and another wire N, com-
municating with the outside of the vessel,
into communication with the poles of a
battery: diverging currents will be trans-
mitted through the mercury in the one di-
rection or the other, according to the con-
nexion ; and if a straight conducting wire
CD, conveying a powerful electric current,
is brought near the vessel, a rotation will
Fig. 645. ^6 imparted to the mercury, the direction

of which will be in conformity with the principles just ex-
plained. Davy used a powerful magnet instead of the straight

2019. Consequences deducible from this action. The follow-
ing consequences respecting the action of finite and indefinite
rectilinear currents will readily follow from the principles which
have been established.

When a finite vertical conductor AB, moveable round an axis
oo', is subjected to the action of an indefinite horizontal current
MN, the plane ABo'owill place itself in the position O'OB'A',
when the vertical current descends, and the horizontal current
runs from N to t&,Jig. 646.


Fig. 646.

If the direction of the
vertical or horizontal cur-
rent be reversed, the po-
sition of equilibrium of
the former will be OO'AB ;
but if the direction of
both be reversed, the po-
sition of equilibrium will
remain unaltered.

When two vertical conductors AB and A'B' are moveable
round a vertical axis oo', and connected together, they will
remain in equilibrium, whatever be their position, if they are
both traversed by currents of the same intensity in the same
direction, provided that the indefinite rectilinear current which
acts upon them be at such a distance and in such a position
that its distances from the points B and B' may be considered
always equal. When the wires AB and A'B' are traversed by
currents in opposite directions, one ascending and the other de-
scending, the system will then turn on its axis oo' until the
vertical plane through AB and A'B' becomes parallel to MN, the
descending current being on that side from which the inde-
finite current flows.

2020. Action of an indefinite straight current on a circulating
current, The circulating current, &.,fig. 647 v is affected by the

Fig. 647.

indefinite current PN in the same manner as would be affected
the rectangular current B. The current PN affects the de-
scending side a by a force contrary to, and the ascending side b
by an equal force according with, its own direction (2012).
In the same manner it affects the sides c and d with forces in
contrary directions, one towards, and the other from, PN. But
the side c, being nearer to PN than d, is more strongly affected ;



and consequently the attraction, in the case represented in
fig. 646., will prevail over the repulsion. If the direction of
either the rectilinear or circulating current be reversed, the
repulsion will prevail over the attraction.

Thus it appears, that au indefinite current flowing from
right to left, under a circulating current having direct rotation,
or one moving from left to right under a circulating current
having retrograde rotation, will produce attraction; and two
currents moving in the contrary directions will produce

If the current A be fixed upon an horizontal axis a b on which
it is capable of revolving, that side c at which the current moves
in the same direction as PN will be attracted downwards, and
the plane of the current will take a position passing through PN,
the side c being nearest to that line.

If the current A be fixed upon the line cd as an axis, it will
turn into the same position, the side b on which the current
ascends being on the side towards which the current PN is

2021. Case in which the indefinite straight current is perpen-
dicular to the plane of the circulating current. If the rectilinear
current &'&,fig. 648., be perpendicular to the
circular current QNN, and within it, and be
moveable round the central line oo', a motion
of rotation will be impressed upon it con-
trary to that of the circular current. This
may be experimentally verified by an appa-
ratus constructed on the principles repre-
sented infig. 649., consisting of a wire frame
supported and balanced on a central point
in a mercurial cup. The current passing
between this point and the liquid in a circular canal will ascend
or descend on the vertical wires according to the arrangement
of the connexions. The circular current may
be produced by surrounding the circular
canal with a metallic wire, or ribbon coated
with a non-conductor, upon which the current
may be transmitted in the usual way. The
wire frame will revolve upon the central point
with direct or retrograde rotation, according to
Fig. 649. fhg directions of the currents. If the current


ascend on the wires, they will revolve in the same direction as
the circular current ; if it descend, in the contrary direction.

The circular current may also be produced by a spiral current
placed under the circular canal, and the wire frame may be re-
placed by a light hollow cylinder, supported on a central point.
The spiral in this case may be moveable and the cylinder fixed,
or vice versa, and the reciprocal actions will be manifested.

2022. Case in which the straight current is oblique to the
plane of the circulating current. Like effects will be produced

when the rectilinear current, in-
stead of being perpendicular to
the plane of the circular current,
is oblique to it.

Let the rectilinear current a c,
fig. 650., be parallel to the plane
of the circular current NQ. If
the current flow from a to c, the
part a b which is within the circle
will be affected by force opposite
to the direction of the nearest
part of the current NQ, and the
part be outside the circle will be affected by a force in the same
direction. If the current flow from c to a, contrary effects will

If in this case the straight current be limited to ab, and be
capable of revolving round a in a plane parallel to that of the
circle, it will receive a motion of rotation in the same or in a
contrary direction to that of the circulating current, accordingly
as it flows from b to a, or from a to b. If the straight current
be limited to b c, it will, under the circumstances, receive rota-
tion in the contrary direction. If, in fine, it extend on both
sides of the circle, it will rotate in the one direction or the
other, according as the internal or external part predominates.

2023. Reciprocal effects of curvilinear currents. The
mutual influence of rectilinear and curvilinear currents being
understood, the reciprocal effects of curvilinear currents may be
easily traced. Each small part of such current may be regarded
as a short rectilinear current, and the separate effects of such
elementary parts being ascertained, the effects of the entire
extent of the curvilinear currents will be the resultants of these
partial forces.


2024. Mutual action of curvilinear currents in general. An
endless variety of problems arise from the various forms that
curvilinear currents may assume, the various positions they may
have in relation to each other, and the various conditions which
may restrain their motions. The solution of all such problems,
however, presents no other difficulties than those which attend
the due application of the geometrical and mechanical prin-
ciples already explained in each particular case.

To take as an example one of the most simple of the infinite
variety of forms under which such problems are presented, let
the centres of two circular currents be fixed ; the planes of the
currents being free to assume any direction whatever, they will
turn upon their centres until they come into the same plane,
the parts of the currents which intersect the line joining their

Online LibraryDionysius LardnerHand-book of natural philosophy and astronomy (Volume 2) → online text (page 36 of 45)