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A Treatise on Electricity and Magnetism
by James Clerk Maxwell

Electromagnetic Force

Ampère's Investigation of the Mutual Action of Electromagnetic Currents

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92287A Treatise on Electricity and Magnetism — Electromagnetic ForceJames Clerk Maxwell

PART IV.

ELECTROMAGNETISM.

CHAPTER I.

ELECTROMAGNETIC FORCE.

475.] It had been noticed by many different observers that in certain cases magnetism is produced or destroyed in needles by electric discharges through them or near them, and conjectures of various kinds had been made as to the relation between magnetism and electricity, but the laws of these phenomena, and the form of these relations, remained entirely unknown till Hans Christian Örsted^[1], at a private lecture to a few advanced students at Copenhagen, observed that a wire connecting the ends of a voltaic battery affected a magnet in its vicinity. This discovery he published in a tract entitled Experimenta circa effectum Conflictus Electrici in Acum Magneticam, dated July 21, 1820.

Experiments on the relation of the magnet to bodies charged with electricity had been tried without any result till Örsted endeavoured to ascertain the effect of a wire heated by an electric current. He discovered, however, that the current itself, and not the heat of the wire, was the cause of the action, and that the 'electric conflict acts in a revolving manner', that is, that a magnet placed near a wire transmitting an electric current tends to set itself perpendicular to the wire, and with the same end always pointing forwards as the magnet is moved round the wire.

476.] It appears therefore that in the space surrounding a wire transmitting an electric current a magnet is acted on by forces depending on the position of the wire and on the strength of the current. The space in which these forces act may therefore be considered as a magnetic field, and we may study it in the same way as we have already studied the field in the neighbourhood of ordinary magnets, by tracing the course of the lines of magnetic force, and measuring the intensity of the force at every point.

477.] Let us begin with the case of an indefinitely long straight wire carrying an electric current. If a man were to place himself in imagination in the position of the wire, so that the current should flow from his head to his feet, then a magnet suspended freely before him would set itself so that the end which points north would, under the action of the current, point to his right hand.

The lines of magnetic force are everywhere at right angles to planes drawn through the wire, and are there fore circles each in a plane perpendicular to the wire, which passes through its centre. The pole of a magnet which points north, if carried round one of these circles from left to right, would experience a force acting always in the direction of its motion. The other pole of the same magnet would experience a force in the opposite direction.

478.] To compare these forces let the wire be supposed vertical, and the current a descending one, and let a magnet be placed on an apparatus which is free to rotate about a vertical axis coinciding with the wire. It is found that under these circumstances the current has no effect in causing the rotation of the apparatus as a whole about itself as an axis. Hence the action of the vertical current on the two poles of the magnet is such that the statical moments of the two forces about the current as an axis are equal and opposite. Let m₁ and m₂ be the strengths of the two poles, r₁ and r₂ their distances from the axis of the wire, T₁ and T₂ the intensities of the magnetic force due to the current at the two poles respectively, then the force on m₁ is m₁T₁, and since it is at right angles to the axis its moment is m₁T₁r₁. Similarly that of the force on the other pole is m₂T₂r₂, and since there is no motion observed,

$m_{1}T_{1}r_{1}+m_{2}T_{2}r_{2}=0.$

But we know that in all magnets

m_{1}+m_{2}=0

.

Hence

T_{1}r_{1}=T_{2}r_{2}

,

or the electromagnetic force due to a straight current of infinite length is perpendicular to the current, and varies inversely as the distance from it.

479.] Since the product $Tr$ depends on the strength of the current it may be employed as a measure of the current. This method of measurement is different from that founded upon electrostatic phenomena, and as it depends on the magnetic phenomena produced by electric currents it is called the Electromagnetic system of measurement. In the electromagnetic system if $i$ is the current,

Tr=2i

.

480.] If the wire be taken for the axis of $z$ , then the rectangular components of $T$ are

X=-2i{\frac {y}{r^{2}}}

, ⁠

Y=2i{\frac {x}{r^{2}}}

, ⁠

Z=0

.

Here $X\,dx+Y\,dy+Z\,dz$ is a complete differential, being that of

2i\tan ^{-1}{\frac {y}{x}}+C

.

Hence the magnetic force in the field can be deduced from a potential function, as in several former instances, but the potential is in this case a function having an infinite series of values whose common difference is $4\pi i$ . The differential coefficients of the potential with respect to the coordinates have, however, definite and single values at every point.

The existence of a potential function in the field near an electric current is not a self-evident result of the principle of the conservation of energy, for in all actual currents there is a continual expenditure of the electric energy of the battery in overcoming the resistance of the wire, so that unless the amount of this expenditure were accurately known, it might be suspected that part of the energy of the battery may be employed in causing work to be done on a magnet moving in a cycle. In fact, if a magnetic pole, $m$ , moves round a closed curve which embraces the wire, work is actually done to the amount of $4\pi mi$ . It is only for closed paths which do not embrace the wire that the line-integral of the force vanishes. We must therefore for the present consider the law of force and the existence of a potential as resting on the evidence of the experiment already described.

481.] If we consider the space surrounding an infinite straight line we shall see that it is a cyclic space, because it returns into itself. If we now conceive a plane, or any other surface, commencing at the straight line and extending on one side of it to infinity, this surface may be regarded as a diaphragm which reduces the cyclic space to an acyclic one. If from any fixed point lines be drawn to any other point without cutting the diaphragm, and the potential be defined as the line-integral of the force taken along one of these lines, the potential at any point will then have a single definite value.

The magnetic field is now identical in all respects with that due to a magnetic shell coinciding with this surface, the strength of the shell being $i$ . This shell is bounded on one edge by the infinite straight line. The other parts of its boundary are at an infinite distance from the part of the field under consideration.

482.] In all actual experiments the current forms a closed circuit of finite dimensions. We shall therefore compare the magnetic action of a finite circuit with that of a magnetic shell of which the circuit is the bounding edge.

It has been shewn by numerous experiments, of which the earliest are those of Ampère, and the most accurate those of Weber, that the magnetic action of a small plane circuit at distances which are great compared with the dimensions of the circuit is the same as that of a magnet whose axis is normal to the plane of the circuit, and whose magnetic moment is equal to the area of the circuit multiplied by the strength of the current.

If the circuit be supposed to be filled up by a surface bounded by the circuit and thus forming a diaphragm, and if a magnetic shell of strength $i$ coinciding with this surface be substituted for the electric current, then the magnetic action of the shell on all distant points will be identical with that of the current.

483.] Hitherto we have supposed the dimensions of the circuit to be small compared with the distance of any part of it from the part of the field examined. We shall now suppose the circuit to be of any form and size whatever, and examine its action at any point $P$ not in the conducting wire itself. The following method, which has important geometrical applications, was introduced by Ampère for this purpose.

Conceive any surface $S$ bounded by the circuit and not passing through the point $P$ . On this surface draw two series of lines crossing each other so as to divide it into elementary portions, the dimensions of which are small compared with their distance from $P$ , and with the radii of curvature of the surface.

Round each of these elements conceive a current of strength $i$ to flow, the direction of circulation being the same in all the elements as it is in the original circuit.

Along every line forming the division between two contiguous elements two equal currents of strength $i$ flow in opposite directions.

The effect of two equal and opposite currents in the same place is absolutely zero, in whatever aspect we consider the currents. Hence their magnetic effect is zero. The only portions of the elementary circuits which are not neutralized in this way are those which coincide with the original circuit. The total effect of the elementary circuits is therefore equivalent to that of the original circuit.

484.] Now since each of the elementary circuits may be considered as a small plane circuit whose distance from $P$ is great compared with its dimensions, we may substitute for it an elementary magnetic shell of strength $i$ whose bounding edge coincides with the elementary circuit. The magnetic effect of the elementary shell on $P$ is equivalent to that of the elementary circuit. The whole of the elementary shells constitute a magnetic shell of strength $i$ , coinciding with the surface $S$ and bounded by the original circuit, and the magnetic action of the whole shell on $P$ is equivalent to that of the circuit.

It is manifest that the action of the circuit is independent of the form of the surface $S$ , which was drawn in a perfectly arbitrary manner so as to fill it up. We see from this that the action of a magnetic shell depends only on the form of its edge and not on the form of the shell itself. This result we obtained before, at Art. 410, but it is instructive to see how it may be deduced from electromagnetic considerations.

The magnetic force due to the circuit at any point is therefore identical in magnitude and direction with that due to a magnetic shell bounded by the circuit and not passing through the point, the strength of the shell being numerically equal to that of the current. The direction of the current in the circuit is related to the direction of magnetization of the shell, so that if a man were to stand with his feet on that side of the shell which we call the positive side, and which tends to point to the north, the current in front of him would be from right to left.

485.] The magnetic potential of the circuit, however, differs from that of the magnetic shell for those points which are in the substance of the magnetic shell.

If

\omega

is the solid angle subtended at the point

P

by the magnetic shell, reckoned positive when the positive or austral side of the shell is next to

P

, then the magnetic potential at any point not in the shell itself is

\omega \phi

, where

\phi

is the strength of the shell. At any point in the substance of the shell itself we may suppose the shell divided into two parts whose strengths are

\phi _{1}

and

\phi _{2}

, where

\phi _{1}+\phi _{2}=\phi

such that the point is on the positive side of

\phi _{1}

and on the negative side of

\phi _{2}

. The potential at this point is

\omega (\phi _{1}+\phi _{2})-4\pi \phi _{2}

.

On the negative side of the shell the potential becomes $\phi (\omega -4\pi )$ In this case therefore the potential is continuous, and at every point has a single determinate value. In the case of the electric circuit, on the other hand, the magnetic potential at every point not in the conducting wire itself is equal to $i\omega$ , where $i$ is the strength of the current, and $\omega$ is the solid angle subtended by the circuit at the point, and is reckoned positive when the current, as seen from $P$ , circulates in the direction opposite to that of the hands of a watch.

The quantity $i\omega$ to is a function having an infinite series of values whose common difference is $4\pi i$ . The differential coefficients of $i\omega$ with respect to the coordinates have, however, single and determinate values for every point of space.

486.] If a long thin flexible solenoidal magnet were placed in the neighbourhood of an electric circuit, the north and south ends of the solenoid would tend to move in opposite directions round the wire, and if they were free to obey the magnetic force the magnet would finally become wound round the wire in a close coil. If it were possible to obtain a magnet having only one pole, or poles of unequal strength, such a magnet would be moved round and round the wire continually in one direction, but since the poles of every magnet are equal and opposite, this result can never occur. Faraday, however, has shewn how to produce the continuous rotation of one pole of a magnet round an electric current by making it possible for one pole to go round and round the current while the other pole does not. That this process may be repeated in definitely, the body of the magnet must be transferred from one side of the current to the other once in each revolution. To do this without interrupting the flow of electricity, the current is split into two branches, so that when one branch is opened to let the magnet pass the current continues to flow through the other. Faraday used for this purpose a circular trough of mercury, as shewn in Fig. 23, Art. 491. The current enters the trough through the wire $AB$ , it is divided at $B$ , and after flowing through the arcs $BQP$ and $BRP$ it unites at $P$ , and leaves the trough through the wire $PO$ , the cup of mercury $O$ , and a vertical wire beneath $O$ , down which the current flows.

The magnet (not shewn in the figure) is mounted so as to be capable of revolving about a vertical axis through

O

, and the wire

OP

revolves with it. The body of the magnet passes through the aperture of the trough, one pole, say the north pole, being beneath the plane of the trough, and the other above it. As the magnet and the wire

OP

revolve about the vertical axis, the current is gradually transferred from the branch of the trough which lies in front of the magnet to that which lies behind it, so that in every complete revolution the magnet passes from one side of the current to the other. The north pole of the magnet revolves about the descending current in the direction N.E.S.W. and if

\omega

,

\omega ^{\prime }

are the solid angles (irrespective of sign) subtended by the circular trough at the two poles, the work done by the electromagnetic force in a complete revolution is

mi(4\pi -\omega -\omega ^{\prime })

,

where

m

is the strength of either pole, and

i

the strength of the current.

487.] Let us now endeavour to form a notion of the state of the magnetic field near a linear electric circuit.

Let the value of $\omega$ , the solid angle subtended by the circuit, be found for every point of space, and let the surfaces for which $\omega$ is constant be described. These surfaces will be the equipotential surfaces. Each of these surfaces will be bounded by the circuit, and any two surfaces, $\omega _{1}$ and $\omega _{2}$ , will meet in the circuit at an angle ${\frac {1}{2}}(\omega _{1}-\omega _{2})$ .

Figure XVIII, at the end of this volume, represents a section of the equipotential surfaces due to a circular current. The small circle represents a section of the conducting wire, and the horizontal line at the bottom of the figure is the perpendicular to the plane of the circular current through its centre. The equipotential surfaces, 24 of which are drawn corresponding to a series of values of $\omega$ differing by ${\frac {\pi }{6}}$ , are surfaces of revolution, having this line for their common axis. They are evidently oblate figures, being flattened in the direction of the axis. They meet each other in the line of the circuit at angles of 15°.

The force acting on a magnetic pole placed at any point of an equipotential surface is perpendicular to this surface, and varies inversely as the distance between consecutive surfaces. The closed curves surrounding the section of the wire in Fig. XVIII are the lines of force. They are copied from Sir W. Thomson's Paper on 'Vortex Motion^[2].' See also Art. 702.

Action of an Electric Circuit on any Magnetic System.

488.] We are now able to deduce the action of an electric circuit on any magnetic system in its neighbourhood from the theory of magnetic shells. For if we construct a magnetic shell, whose strength is numerically equal to the strength of the current, and whose edge coincides in position with the circuit, while the shell itself does not pass through any part of the magnetic system, the action of the shell on the magnetic system will be identical with that of the electric circuit.

Reaction of the Magnetic System on the Electric Circuit.

489.] From this, applying the principle that action and reaction are equal and opposite, we conclude that the mechanical action of the magnetic system on the electric circuit is identical with its action on a magnetic shell having the circuit for its edge.

The potential energy of a magnetic shell of strength

\phi

placed in a field of magnetic force of which the potential is

V

, is, by Art. 410,

M=\phi \iint \left(l{\frac {dV}{dx}}+m{\frac {dV}{dy}}+n{\frac {dV}{dz}}\right)\,ds

,

where

l

,

m

,

n

are the direction-cosines of the normal drawn from the positive side of the element

dS

of the shell, and the integration is extended over the surface of the shell. Now the surface-integral

N=\iint (la+mb+nc)\,ds

,

where

a

,

b

,

c

are the components of the magnetic induction, represents the quantity of magnetic induction through the shell, or, in the language of Faraday, the number of lines of magnetic induction, reckoned algebraically, which pass through the shell from the negative to the positive side, lines which pass through the shell in the opposite direction being reckoned negative.

Remembering that the shell does not belong to the magnetic system to which the potential $V$ is due, and that the magnetic force is therefore equal to the magnetic induction, we have

$a=-{\frac {dV}{dx}},\quad b=-{\frac {dV}{dy}},\quad c=-{\frac {dV}{dz}},$

and we may write the value of $M$ ,

$M=-\phi N$ .

If $\delta x_{1}$ represents any displacement of the shell, and $X_{1}$ the force acting on the shell so as to aid the displacement, then by the principle of conservation of energy,

$X_{1}\delta x_{1}+\delta M=0$ ,

or $X=\phi {\frac {\delta N}{\delta x}}$ .

We have now determined the nature of the force which corresponds to any given displacement of the shell. It aids or resists that displacement accordingly as the displacement increases or diminishes $N$ , the number of lines of induction which pass through the shell.

The same is true of the equivalent electric circuit. Any displacement of the circuit will be aided or resisted accordingly as it increases or diminishes the number of lines of induction which pass through the circuit in the positive direction.

We must remember that the positive direction of a line of magnetic induction is the direction in which the pole of a magnet which points north tends to move along the line, and that a line of induction passes through the circuit in the positive direction, when the direction of the line of induction is related to the direction of the current of vitreous electricity in the circuit as the longitudinal to the rotational motion of a right-handed screw. See Art. 23.

490.] It is manifest that the force corresponding to any displacement of the circuit as a whole may be deduced at once from the theory of the magnetic shell. But this is not all. If a portion of the circuit is flexible, so that it may be displaced independently of the rest, we may make the edge of the shell capable of the same kind of displacement by cutting up the surface of the shell into a sufficient number of portions connected by flexible joints. Hence we conclude that if by the displacement of any portion of the circuit in a given direction the number of lines of induction which pass through the circuit can be increased, this displacement will be aided by the electromagnetic force acting on the circuit.

Every portion of the circuit therefore is acted on by a force urging it across the lines of magnetic induction so as to include a greater number of these lines within the embrace of the circuit, and the work done by the force during this displacement is numerically equal to the number of the additional lines of induction multiplied by the strength of the current.

Let the element $ds$ of a circuit, in which a current of strength $i$ is flowing, be moved parallel to itself through a space $\delta x$ , it will sweep out an area in the form of a parallelogram whose sides are parallel and equal to $ds$ and $\delta x$ respectively.

If the magnetic induction is denoted by ${\mathfrak {B}}$ , and if its direction makes an angle $\epsilon$ with the normal to the parallelogram, the value of the increment of $N$ corresponding to the displacement is found by multiplying the area of the parallelogram by ${\mathfrak {B}}\cos \epsilon$ . The result of this operation is represented geometrically by the volume of a parallelepiped whose edges represent in magnitude and direction $\delta x$ , $ds$ , and ${\mathfrak {B}}$ , and it is to be reckoned positive if when we point in these three directions in the order here given the pointer moves round the diagonal of the parallelepiped in the direction of the hands of a watch. The volume of this parallelepiped is equal to $X\,\delta x$ .

If $\theta$ is the angle between $ds$ and ${\mathfrak {B}}$ , the area of the parallelogram is $ds\cdot {\mathfrak {B}}\sin \theta$ , and if $\eta$ is the angle which the displacement $dx$ makes with the normal to this parallelogram, the volume of the parallelepiped is

$ds\cdot {\mathfrak {B}}\sin \theta \cdot \delta x\cos \eta =\delta N$ .

Now $X\,\delta x=i\,\delta N=i\,ds\cdot {\mathfrak {B}}\sin \theta \,\delta x\cos \eta$ , and $X=i\,ds\cdot {\mathfrak {B}}\sin \theta \cos \eta$ is the force which urges $ds$ , resolved in the direction $\delta x$ .

The direction of this force is therefore perpendicular to the parallelogram, and is equal to $i\cdot ds\cdot {\mathfrak {B}}\sin \theta$ .

This is the area of a parallelogram whose sides represent in magnitude and direction $i\,ds$ and ${\mathfrak {B}}$ . The force acting on $ds$ is therefore represented in magnitude by the area of this parallelogram, and in direction by a normal to its plane drawn in the direction of the longitudinal motion of a right-handed screw, the handle of which is turned from the direction of the current $i\,ds$ to that of the magnetic induction ${\mathfrak {B}}$ .
Fig. 22. We may express in the language of Quaternions, both the direction and the magnitude of this force by saying that it is the vector part of the result of multiplying the vector $i\,ds$ , the element of the current, by the vector ${\mathfrak {B}}$ , the magnetic induction.

491.] We have thus completely determined the force which acts on any portion of an electric circuit placed in a magnetic field. If the circuit is moved in any way so that, after assuming various forms and positions, it returns to its original place, the strength of the current remaining constant during the motion, the whole amount of work done by the electromagnetic forces will be zero. Since this is true of any cycle of motions of the circuit, it follows that it is impossible to maintain by electromagnetic forces a motion of continuous rotation in any part of a linear circuit of constant strength against the resistance of friction, &c.

It is possible, however, to produce continuous rotation provided that at some part of the course of the electric current it passes from one conductor to another which slides or glides over it.

Fig. 23. When in a circuit there is sliding contact of a conductor over the surface of a smooth solid or a fluid, the circuit can no longer be considered as a single linear circuit of constant strength, but must be regarded as a system of two or of some greater number of circuits of variable strength, the current being so distributed among them that those for which $N$ is increasing have currents in the positive direction, while those for which $N$ is diminishing have currents in the negative direction.

Thus, in the apparatus represented in Fig. 23, $OP$ is a moveable conductor, one end of which rests in a cup of mercury $O$ , while the other dips into a circular trough of mercury concentric with $O$ .

The current $i$ enters along $AB$ , and divides in the circular trough into two parts, one of which, $x$ , flows along the arc $BQP$ , while the other, $y$ , flows along $BRP$ . These currents, uniting at $P$ , flow along the moveable conductor $PO$ and the electrode $OZ$ to the zinc end of the battery. The strength of the current along $OP$ and $OZ$ is $x+y$ or $i$ .

Here we have two circuits, $ABQPOZ$ , the strength of the current in which is $x$ , flowing in the positive direction, and $ABRPOZ$ , the strength of the current in which is $y$ , flowing in the negative direction.

Let ${\mathfrak {B}}$ be the magnetic induction, and let it be in an upward direction, normal to the plane of the circle.

While

OP

moves through an angle

\theta

in the direction opposite to that of the hands of a watch, the area of the first circuit increases by

{\frac {1}{2}}OP^{2}.\theta

, and that of the second diminishes by the same quantity. Since the strength of the current in the first circuit is

x

, the work done by it is

{\frac {1}{2}}x.OP^{2}.\theta .{\mathfrak {B}}

, and since the strength of the second is

-y

, the work done by it is

{\frac {1}{2}}y.OP^{2}.\theta {\mathfrak {B}}

. The whole work done is therefore

{\frac {1}{2}}(x+y)OP^{2}.\theta {\mathfrak {B}}

⁠ or ⁠

{\frac {1}{2}}i.OP^{2}.\theta {\mathfrak {B}}

,

depending only on the strength of the current in

PO

. Hence, if

i

is maintained constant, the arm

OP

will be carried round and round the circle with a uniform force whose moment is

{\frac {1}{2}}i.OP^{2}{\mathfrak {B}}

. If, as in northern latitudes,

{\mathfrak {B}}

acts downwards, and if the current is inwards, the rotation will be in the negative direction, that is, in the direction

PQBR

.

492.] We are now able to pass from the mutual action of magnets and currents to the action of one current on another. For we know that the magnetic properties of an electric circuit $C_{1}$ , with respect to any magnetic system $M_{2}$ , are identical with those of a magnetic shell $S_{1}$ , whose edge coincides with the circuit, and whose strength is numerically equal to that of the electric current. Let the magnetic system $M_{2}$ be a magnetic shell $S_{2}$ , then the mutual action between $S_{1}$ and $S_{2}$ is identical with that between $S_{1}$ and a circuit $C_{2}$ , coinciding with the edge of $S_{2}$ and equal in numerical strength, and this latter action is identical with that between $C_{1}$ and $C_{2}$ .

Hence the mutual action between two circuits, $C_{1}$ and $C_{2}$ , is identical with that between the corresponding magnetic shells $S_{1}$ and $S_{2}$ .

We have already investigated, in Art. 423, the mutual action of two magnetic shells whose edges are the closed curves $s_{1}$ and $s_{2}$ .

If we make

M=\int _{0}^{s_{2}}\int _{0}^{s_{1}}{\frac {\cos \epsilon }{r}}\,ds_{1}\,ds_{2}

,

where

\epsilon

is the angle between the directions of the elements

ds_{1}

and

ds_{2}

, and

r

is the distance between them, the integration being extended once round

s_{2}

and once round

s_{1}

, and if we call

M

the potential of the two closed curves

s_{1}

and

s_{2}

, then the potential energy due to the mutual action of two magnetic shells whose strengths are

i_{1}

and

i_{2}

bounded by the two circuits is

-i_{1}i_{2}M

,

and the force

X

, which aids any displacement

\delta x

, is

i_{1}i_{2}{\frac {\delta M}{\delta x}}

.

The whole theory of the force acting on any portion of an electric circuit due to the action of another electric circuit may be deduced from this result.

493.] The method which we have followed in this chapter is that of Faraday. Instead of beginning, as we shall do, following Ampère, in the next chapter, with the direct action of a portion of one circuit on a portion of another, we shew, first, that a circuit produces the same effect on a magnet as a magnetic shell, or, in other words, we determine the nature of the magnetic field due to the circuit. We shew, secondly, that a circuit when placed in any magnetic field experiences the same force as a magnetic shell. We thus determine the force acting on the circuit placed in any magnetic field. Lastly, by supposing the magnetic field to be due to a second electric circuit we determine the action of one circuit on the whole or any portion of the other.

494.] Let us apply this method to the case of a straight current of infinite length acting on a portion of a parallel straight conductor.

Let us suppose that a current $i$ in the first conductor is flowing vertically downwards. In this case the end of a magnet which points north will point to the right-hand of a man looking at it from the axis of the current.

The lines of magnetic induction are therefore horizontal circles, having their centres in the axis of the current, and their positive direction is north, east, south, west.

Let another descending vertical current be placed due west of the first. The lines of magnetic induction due to the first current are here directed towards the north. The direction of the force acting on the second current is to be determined by turning the handle of a right-handed screw from the nadir, the direction of the current, to the north, the direction of the magnetic induction. The screw will then move towards the east, that is, the force acting on the second current is directed towards the first current, or, in general, since the phenomenon depends only on the relative position of the currents, two parallel currents in the same direction attract each other.

In the same way we may shew that two parallel currents in opposite directions repel one another.

495.] The intensity of the magnetic induction at a distance $r$ from a straight current of strength $i$ is, as we have shewn in Art. 479,

$2{\frac {i}{r}}.$

Hence, a portion of a second conductor parallel to the first, and carrying a current $i'$ in the same direction, will be attracted towards the first with a force

$F=2ii'{\frac {a}{r}},$

where $a$ is the length of the portion considered, and $r$ is its distance from the first conductor.

Since the ratio of $a$ to $r$ is a numerical quantity independent of the absolute value of either of these lines, the product of two currents measured in the electromagnetic system must be of the dimensions of a force, hence the dimensions of the unit current are

$[i]=[F^{\frac {1}{2}}]=[M^{\frac {1}{2}}L^{\frac {1}{2}}T^{-1}].$

496.] Another method of determining the direction of the force which acts on a current is to consider the relation of the magnetic action of the current to that of other currents and magnets.

If on one side of the wire which carries the current the magnetic action due to the current is in the same or nearly the same direction as that due to other currents, then, on the other side of the wire, these forces will be in opposite or nearly opposite directions, and the force acting on the wire will be from the side on which the forces strengthen each other to the side on which they oppose each other.

Thus, if a descending current is placed in a field of magnetic force directed towards the north, its magnetic action will be to the north on the west side, and to the south on the east side. Hence the forces strengthen each other on the west side and oppose each other on the east side, and the current will therefore be acted on by a force from west to east. See Fig. 22, p. 138.

In Fig. XVII at the end of this volume the small circle represents a section of the wire carrying a descending current, and placed in a uniform field of magnetic force acting towards the left-hand of the figure. The magnetic force is greater below the wire than above it. It will therefore be urged from the bottom; towards the top of the figure.

497.] If two currents are in the same plane but not parallel, we may apply this principle. Let one of the conductors be an infinite straight wire in the plane of the paper, supposed horizontal. On the right side of the current the magnetic force acts downward, and on the left side it acts upwards. The same is true of the magnetic force due to any short portion of a second current in the same plane. If the second current is on the right side of the first, the magnetic forces will strengthen each other on its right side and oppose each other on its left side. Hence the second current will be acted on by a force urging it from its right side to its left side. The magnitude of this force depends only on the position of the second current and not on its direction. If the second current is on the left side of the first it will be urged from left to right.

Hence, if the second current is in the same direction as the first it is attracted, if in the opposite direction it is repelled, if it flows at right angles to the first and away from it, it is urged in the direction of the first current, and if it flows toward the first current, it is urged in the direction opposite to that in which the first current flows.

In considering the mutual action of two currents it is not necessary to bear in mind the relations between electricity and magnetism which we have endeavoured to illustrate by means of a right-handed screw. Even if we have forgotten these relations we shall arrive at correct results, provided we adhere consistently to one of the two possible forms of the relation.

498.] Let us now bring together the magnetic phenomena of the electric circuit so far as we have investigated them.

We may conceive the electric circuit to consist of a voltaic battery, and a wire connecting its extremities, or of a thermoelectric arrangement, or of a charged Leyden jar with a wire connecting its positive and negative coatings, or of any other arrangement for producing an electric current along a definite path.

The current produces magnetic phenomena in its neighbourhood.

If any closed curve be drawn, and the line-integral of the magnetic force taken completely round it, then, if the closed curve is not linked with the circuit, the line-integral is zero, but if it is linked with the circuit, so that the current $i$ flows through the closed curve, the line-integral is $4\pi i$ , and is positive if the direction of integration round the closed curve would coincide with that of the hands of a watch as seen by a person passing through it in the direction in which the electric current flows. To a person moving along the closed curve in the direction of integration, and passing through the electric circuit, the direction of the current would appear to be that of the hands of a watch. We may express this in another way by saying that the relation between the directions of the two closed curves may be expressed by describing a right-handed screw round the electric circuit and a right-handed screw round the closed curve. If the direction of rotation of the thread of either, as we pass along it, coincides with the positive direction in the other, then the line-integral will be positive, and in the opposite case it will be negative.

Fig. 24.

Relation between the electric current and the lines of magnetic induction indicated by a right-handed screw.

499.] Note.—The line-integral $4\pi i$ depends solely on the quantity of the current, and not on any other thing whatever. It does not depend on the nature of the conductor through which the current is passing, as, for instance, whether it be a metal or an electrolyte, or an imperfect conductor. We have reason for believing that even when there is no proper conduction, but merely a variation of electric displacement, as in the glass of a Leyden jar during charge or discharge, the magnetic effect of the electric movement is precisely the same.

Again, the value of the line-integral $4\pi i$ does not depend on the nature of the medium in which the closed curve is drawn. It is the same whether the closed curve is drawn entirely through air, or passes through a magnet, or soft iron, or any other substance, whether paramagnetic or diamagnetic.

500.] When a circuit is placed in a magnetic field the mutual action between the current and the other constituents of the field depends on the surface-integral of the magnetic induction through any surface bounded by that circuit. If by any given motion of the circuit, or of part of it, this surface-integral can be increased, there will be a mechanical force tending to move the conductor or the portion of the conductor in the given manner.

The kind of motion of the conductor which increases the surface- integral is motion of the conductor perpendicular to the direction of the current and across the lines of induction.

If a parallelogram be drawn, whose sides are parallel and proportional to the strength of the current at any point, and to the magnetic induction at the same point, then the force on unit of length of the conductor is numerically equal to the area of this parallelogram, and is perpendicular to its plane, and acts in the direction in which the motion of turning the handle of a right-handed screw from the direction of the current to the direction of the magnetic induction would cause the screw to move.

Hence we have a new electromagnetic definition of a line of magnetic induction. It is that line to which the force on the conductor is always perpendicular.

It may also be defined as a line along which, if an electric current be transmitted, the conductor carrying it will experience no force.

501.] It must be carefully remembered, that the mechanical force which urges a conductor carrying a current across the lines of magnetic force, acts, not on the electric current, but on the conductor which carries it. If the conductor be a rotating disk or a fluid it will move in obedience to this force, and this motion may or may not be accompanied with a change of position of the electric current which it carries. But if the current itself be free to choose any path through a fixed solid conductor or a network of wires, then, when a constant magnetic force is made to act on the system, the path of the current through the conductors is not permanently altered, but after certain transient phenomena, called induction currents, have subsided, the distribution of the current will be found to be the same as if no magnetic force were in action.

The only force which acts on electric currents is electromotive force, which must be distinguished from the mechanical force which is the subject of this chapter.

Fig. 25.

Relations between the positive directions of motion and of rotation indicated by three right-handed screws.

↑ See another account of Örsted's discovery in a letter from Professor Hansteen in the Life of Faraday by Dr. Bence Jones, vol. ii. p. 395.
↑ Trans. R. S. Edin., vol. xxv. p. 217, (1869).

[1] See another account of Örsted's discovery in a letter from Professor Hansteen in the Life of Faraday by Dr. Bence Jones, vol. ii. p. 395.

[2] Trans. R. S. Edin., vol. xxv. p. 217, (1869).

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A Treatise on Electricity and Magnetism/Part IV/Chapter I

PART IV.

ELECTROMAGNETISM.

CHAPTER I.

ELECTROMAGNETIC FORCE.