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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

RECIPROCAL INFLUENCE OF CURRENTS. 367

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

368 VOLTAIC ELECTRICITY.

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

current.

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.

RECIPROCAL INFLUENCE OF CURRENTS. 369

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

current.

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

370 VOLTAIC ELECTRICITY.

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

RECIPROCAL INFLUENCE OF CURRENTS. 371

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

372 VOLTAIC ELECTRICITY.

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 ,

RECIPROCAL INFLUENCE OF CURRENTS. 373

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

affected.

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.

374

VOLTAIC ELECTRICITY.

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

wire.

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.

RECIPROCAL INFLUENCE OF CURRENTS. 375

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 ;

376

VOLTAIC ELECTRICITY.

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

repulsion.

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

directed.

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

RECIPEOCAL INFLUENCE OF CURRENTS. 377

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

ensue.

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.

378 VOLTAIC ELECTRICITY.

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

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

RECIPROCAL INFLUENCE OF CURRENTS. 367

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

368 VOLTAIC ELECTRICITY.

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

current.

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.

RECIPROCAL INFLUENCE OF CURRENTS. 369

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

current.

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

370 VOLTAIC ELECTRICITY.

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

RECIPROCAL INFLUENCE OF CURRENTS. 371

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

372 VOLTAIC ELECTRICITY.

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 ,

RECIPROCAL INFLUENCE OF CURRENTS. 373

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

affected.

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.

374

VOLTAIC ELECTRICITY.

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

wire.

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.

RECIPROCAL INFLUENCE OF CURRENTS. 375

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 ;

376

VOLTAIC ELECTRICITY.

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

repulsion.

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

directed.

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

RECIPEOCAL INFLUENCE OF CURRENTS. 377

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

ensue.

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.

378 VOLTAIC ELECTRICITY.

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

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