The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 2 of 35

to impress him most forcibly was, however, the fact that it was only the beginning and ending of the inducing current which had any effect upon the other circuit. He considered that, since the mere cessation of the inducing current was accompanied by a wave of induced current, that could only be because the induced current circuit was, meantime, in a peculiar condition, to which he gave the name of the electro- tonic state, the annulment of which gave rise to a current in the circuit. The same state he considered to be found in a wire or circuit at rest in a magnetic field. The circuit was in the electrotonic state whilst in the field, but withdrawing the circuit or removing the magnetic field annulled the electro- tonic state and gave rise to a current. To use his own words at a later date (Ser. XXVIIL, § 3172, « Exp. Eesearches "), "Mere motion would not generate a relation which had not a foundation in the existence of some previous state ; " and (Ser. XXIX., § 3269, ibid.} " Again and again the idea of an electrotonic state has been forced upon my mind." The mere motion of an external body, such as a copper wire, in a mag- netic field cannot, he considers, be the sole cause of the current, unless there is a previous peculiar state as regards the wire which, when motion is superadded, produces the current. When, however, subsequent thought and diverse experiment had clarified his ideas and adjusted facts in proper relation, he came to see that that which he had denominated the electrotonic state is really the amount of electromagnetie momentum which the circuit possesses in virtue of its being in a magnetic field. In modern language, it is the equivalent of that which is now called the number of lines of magnetic force passing through the circuit. Every line of magnetic force is a closed loop or continuous line, and if we set out at any point on a line of magnetic force and travel forwards along that line we shall come back to that same point again. If this line of force is originated by a permanent magnet or an electromagnet, then part of our journey will be performed through the iron or steel and part through the air or othe'r diamagnetic surround- ing it. If, then, a closed conducting circuit is so situated that the line of force considered passes through it or is linked with it, the line of force and the closed circuit form, as it were, two links of a chain, and cannot be separated except by pulling

8 HISTOEICAL INTRODUCTION.

one through the other (Fig. 1). When they are so pulled through one another the line of force " cuts " and is cut by the circuit. The number of lines of force, therefore, which at any instant are linked with a given circuit represent poten- tially the greatest amount of " cutting " possible. The exist- ence of lines of magnetic force linked with the circuit is an essential antecedent to the appearance of a current of induction in that circuit when removed from the magnetic field. At a later stage of his investigations Faraday was able to modify his earlier notions of the electrotonic state, and learnt to look on the induced current appearing under these circumstances as due not to a state of things in the circuit, but to a condition of things outside the circuit, or, more precisely, to the relation in which the circuit stands to the magnetic field of force around it.

FIG. 1.

In the 28th and 29th series of his "Experimental Ee- searches," Faraday exhausted all possible means of experi- ment in proving that this conception of the linking or un- linking of loops of force and loops of conducting circuits was an unerring guide to the solution of all problems of electro- magnetic induction. The circuit being given, he was able to show by a course of rigid demonstration that the process of linking with it a loop of magnetic force was always accompanied by the passage of a wave of current round the circuit in one direction, and the unlinking was invariably associated with the flow of an opposite pulsation of electricity. Moreover, and most important of all, he built up a quantitative conception around the term «« a line of magnetic force," so that it came to him to mean not merely a geometrical line or a direction, but a definite physical magnitude, which represented the product of a certain area of space, and a certain mean intensity of mag-

HISTORICAL INTRODUCTION. 9

netic force over that area.* Armed with this idea, he proceeded to show that the quantity of electricity represented by each •current of induction is the numerical equivalent of the " number of lines of force " which are linked or unlinked with the circuit by any operation. He found that this hypothesis never failed to enable him to render a satisfactory and a logical explanation of all his results, and with this clue in hand he could find his way about amidst the entanglements of experimental inquiry, and return always from each fresh excursion after fact with new confirmation of its consistency, and with fresh power to predict the results of other experiments.

So strong became at last his conviction that these lines of force could hardly have such powers if they were mere geometrical conceptions, like lines of latitude and longitude, that he gives expression to it by speaking of them as physical lines of force. He intends to imply that he thinks " a line of force " must be taken to be a definite action going on in a certain region of space, and that, whatever may be its real nature, we must accord to it a definite physical character in some sort or sense, as much as we do an electric current of unit strength flowing along a prescribed circuit. Faraday was not a professed mathematician, and it was perhaps fortunate that his inability to employ the mechanical aid of symbolic reasoning forced him to make clear to himself each step by experimental demonstration. He was thereby compelled to keep to the main track of discovery, and prevented from deviating into the more abstract lines of thought. The special abilities of Kelvin and Helmholtz, and subse- quently those of Clerk Maxwell, were, however, directed to the complete elucidation of these conceptions of Faraday, and the great treatise of Maxwell, as he himself has stated, was under- taken mainly with the hope of making these ideas the basis of a mathematical method. The one cardinal principle which may be said to be at the base of the mode of viewing electrical and magnetic phenomena introduced by these investigators is the denial of action at finite distances, and accounting for the phenomena by the assumption of the existence of a medium

• Faraday's notion of " a line of force " was at first merely a geometrical conception, representing a certain line of action, but his ultimate applica- tions of the term showed that he had come to think of it as a surface integral.

10 HISTORICAL INTRODUCTION.

which is the active agent in the transmission of energy from one place to another, and which is itself capable of storing up energy in a potential and kinetic form.

The mathematical methods and hypotheses of the French school of physicists, represented chiefly by Ampere, Arago, Poisson, and Coulomb, consisted in the assumption that material particles in special states, called electric and magnetic, could act on one another at finite distances without any intervening mechanism according to certain laws of force varying with the distance. Faraday may be said to have raised the standard of revolt against this notion, and indeed he was able to quote in his support the great authority of Newton in rejecting the idea that matter could act on matter across intervening distance without aid from any mechanism. He never considers bodies as existing with nothing between them but their distance, and acting on one another according to some function of that dis- tance. He conceives all space as a field of force, the lines of force being in general curved, and those due to any body ex- tending from it on all sides, their direction being modified by the presence of other bodies. A magnet, an electrified conductor, or a wire conveying an electric current, are thus the focus and source of a system of radiations of force lines or loops which are to be thought of as part and parcel of it. This force- system is capable of deformation or change by the presence of other bodies, but it moves with the magnet, electrified body, or current-carrying wire. These force radiations penetrate sur- rounding bodies, and the apparent actions between bodies at a distance are in reality actions due to immediate action of the field of force of one body upon the other at the place where it is. Then rises for solution the important problem : What are these lines of force ? Faraday answered the question by saying that they consist in some sort of operation or action going on in a medium along certain lines or axes, and Maxwell added to this the suggestion that the electromagnetic medium must be identical with the medium postulated to account for the phenomena of light.

The question which yet remains unanswered is : What is the nature of the action or operation along certain lines in this medium which causes a line of force to exist ? The future of electric and magnetic investigation will, perhaps, conduct us

HISTORICAL INTRODUCTION. 11

step by step to the solution of this supremely important problem.

§ 3. Henry's Investigations. — At the same time that Fara- day was pursuing in England a career of triumphant discovery in the field of electromagnetic science a young philosopher of hardly less intellectual power, but more limited opportunities for research, was following hard on the same path of investi- gation in America. The name of Joseph Henry is one which we must link with that of our own great countryman as a co- worker, nay, even an anticipator in some things, in the region of fundamental discovery in electromagnetism.

To Henry clearly belongs the credit of having improved Sturgeon's electromagnet by substituting for the single layer of copper wire wound on the iron horse-shoe a spool or bobbin of insulated copper wire. By this means he made what he then called intensity magnets, or electromagnets, suitable for excitation by an intensity battery or battery of many cells. Henry in this manner constructed in 1829 or 1830 a very large electromagnet, capable of supporting a weight of GOOlb. or 700lb. Before having any knowledge of Faraday's experi- ments, and guided apparently by the notion that as electric currents can produce magnetism, so magnetism should be able to generate electric currents, Henry experimented as follows : — A piece of wire about 30ft. long and covered with an elastic varnish was closely coiled round the middle of the soft iron armature of this large electromagnet. The wire was wound upon itself so as only to occupy about lin. in length of the armature which was 7in. in all its length. The ends of this wire were connected by long copper wires with a distant galvanometer. The armature with its coil was laid upon the poles of the electromagnet, and the galvanic plates connected with the helix of the electromagnet immersed in the trough of acid. At the moment of immersion the needle of the galvanometer was seen to be deflected about 30deg., but it immediately returned to its normal position. On withdrawing the battery plates from the acid it was noticed that the gal- vanometer needle made a sudden deflection in the opposite direction of about 20deg. A similar effect was produced by pulling off or putting on the armature whilst the magnet

12 HISTORICAL INTRODUCTION.

remained excited. Henry, in Ms account of this experiment, savs ;_« From the foregoing facts it appears that a current of electricity is produced for an instant in a helix of copper wire surrounding a piece of soft iron whenever magnetism is induced in the iron, and a current in the opposite direction when the magnetism ceases ; also, that an instantaneous current in one or other direction accompanies every change in the magnetic intensity of the iron."

This very lucid statement of experiments, made probably in August, 1831, shows that Henry was at least an independent discoverer of the induction of electric currents. In April, 1832, an account reached him of Faraday's discovery in the previous year, and Henry then repeated his former experiments, and was able by means of larger helices of wire wound on the armature of his electromagnet to greatly increase the magni- tude of the induced current. Henry, therefore, not only dis- covered independently the facts of electromagnetic induction, but correctly interpreted them as well. He early laid a firm grasp upon the essential principles involved, and he came almost within reach of anticipating that discovery which is, and will remain, the crowning glory of his illustrious rival. Between 1831 and 1840, or later, Henry continued to add fresh knowledge to the original facts, and in a later chapter a description will be given of his important investigations on the self and mutual induction of conducting circuits.

CHAPTER II.

ELECTKO-MAGNETIC INDUCTION.

§ 1. Magnetic Force and Magnetic Fields. — Certain sub- stances, such as iron, nickel, cobalt, steel, and some of their compounds, particularly a native oxide of iron, possess peculiar physical properties, and either exist in, or can be put into, a condition in which they are said to be magnetised. When in this condition they exhibit physical qualities which are called magnetic properties, the most obvious of which is the power of producing attraction and repulsion upon other magnetic substances. Some bodies, notably hardened steel, can acquire marked permanent magnetic qualities. The neighbourhood round these bodies when in this state, and within which they exercise these actions, is called a magnetic field. If a small magnetised steel needle is suspended freely at its centre of gravity and held in a magnetic field it is found that it takes up- a certain direction under the influences of forces acting upon it. If disturbed from this position it returns to it again.

It is found that there is a line in the needle round which it can be revolved without changing the set of that line when the needle is left free to obey the forces acting upon it. The direction of this line in the needle is called its magnetic axis. Oersted discovered that a magnetic field exists in the neigh- bourhood of a conductor conveying an electric current, and that it imposes a certain directive influence upon a magnetic needle held near to it. If a small steel magnetised needle is placed in any region containing either conductors carrying electric currents or substances in a permanent magnetic state- it is found that at every point of the field the magnetic axis of this small exploring needle takes up a definite position if it is

14 ELECTRO-MAGNETIC INDUCTION.

freely suspended so as to be removed from the influence of gravity. The direction so assumed by its magnetic axis is called the direction of tlie magnetic force at that point. The magnetic force has at every point in the magnetic field of these active agents a certain direction. On examining the behaviour towards one another of two magnetised steel needles we find that their magnetic properties are exhibited chiefly at the two extremities, and these are called the magnetic poles. The two poles of a magnetic needle are not identical in quality. If a uniformly magnetised steel needle is broken in the middle, the ends where it is broken immediately become new magnetic poles, whereas before rupture that portion of the needle exhibited no apparently active magnetic properties. If the two poles which make their appearance at the broken ends are tested it will be found that they attract one another. If these poles are placed one centimetre apart and the force with which they attract one another measured in absolute units, the square root of the number which expresses this attraction is called the numerical value of the strength of these poles. Hence, a unit magnetic pole is a pole which at a unit of distance attracts another unit pole of opposite kind with a unit of force. The earth as a whole is a magnetic body, and if a small magnetic needle is freely suspended at its centre of gravity, its magnetic axis assumes a certain position at each point on the earth's surface which is called the direction of the terrestrial magnetic force at that point. The pole of the needle which points in our latitude in any direction north of the true east and west line is called the north pole or north-seeking pole of the needle. If we take a very long thin magnetised needle, called for shortness a magnetic filament, we can employ one pole of it, say the north pole, for exploration in a field, whilst the other pole is so far removed as not to be affected. If such a pole, called for shortness a free north pole, is placed in any magnetic field it is acted upon by the magnetic force and urged to move in the direction of this force. If this free north pole is a pole of unit strength, then the force dynami- cally measured in absolute units which acts upon it is called the numerical measure of the magnetic force at that point.

The direction in which a free north pole tends to move is called the positive direction of the magnetic force at that

ELEGTEO-MAGNETIG INDUCTION. 15

point. The magnetic force at any point in the magnetic field of magnetic bodies, whether magnetised substances or conductors conveying electric currents, is thus a quantity which has direction as well as magnitude, and we have defined above how both of these can be measured. By means of a free north magnetic pole of unit strength we may thus explore and define a magnetic field at every point.

A magnetic field in which the magnetic force is the same in magnitude and direction at every point is called a uniform magnetic field. The magnitude of the magnetic force at any point is a measure of the strength of the magnetic field at that point.

There are several simple and yet important cases in which it is possible to calculate the strength of the magnetic field or the magnetic force at certain assigned points in the neighbourhood of conductors conveying electric currents. The pre-determination of the field strength at points near to magnets and conductors conveying electric currents is, generally speaking, except in these simple cases, a very difficult matter.

The Magnetic Force near to a very long Straight Wire conveying an Electric Current.

If a current flows in a thin circular wire we may call a very short length of this conductor, denoted by d s, an element of the circuit or of the current. Ampere showed by a classical series of experiments that the magnetic force due to an element of a current at any point near it was numerically equal to the product of the strength of the current, the length of the element, and the sine of the angle between the direction of the element of the circuit and the line joining the centre of that element with the point, and inversely as the square of this distance. Thus, if ds (Fig. 2) represents the element P of a circuit in which is flowing a current of strength I in absolute electromagnetic measure, and if a is the angle which any line 0 P makes with the direction of the element, and r is the length of the line OP, then the magnetic force at the point 0 due to that element of the current is numerically equal to I d s sin a

16 ELECTRO-MAGNETIC INDUCTION.

and this force is in a direction at right angles to the plane- containing the element of the circuit and the line joining it to the given point. Starting with this fundamental law, we can deduce expressions for the strength of the magnetic field due to currents flowing in conductors of certain forms at certain assigned points. Consider, for instance, a very long, practically infinite straight wire in which a current is flowing, the return.

wire being at a very great distance. Take any point P in the neighbourhood of this conductor (Fig. 3). It is required to find the magnetic force at the point P. Draw P M perpen- dicular to the wire from P. Let N N' be any element of the conductor. Then the magnetic force at P due to the element ds = N N' of the conductor is in a direction at right angles to the plane of the paper, and if the length N P is called r and.

FIG. 3.

the angle P N M is called a, the magnetic force at P due to tha element N N' is equal numerically to

d s sin a

(1)

where I is the current flowing in the element. Let P M be denoted by p. In order to find the magnetic force due to the whole wire at P we have to integrate the above expression throughout the whole length of the wire. To do this we

ELECTRO-MAGNETIC INDUCTION. 17

transform it as follows :— Let the angle M P N = 0, then N P N' is the increment of this angle, call it d6. From the geometry of the figure it is easily seen that sin a = cos 0, and that when

ds is very small J^ - I ,or ^ = d±. rdv p r2 p

Hence substituting these values for sin a and ds/r? in equation (1), we have as the expression for the value of the magnetic force at P, duo to the element N N' of the current, the formula

rf F - 1 cos ^ cld

P

The magnetic force due to the whole infinitely long straight current is obtained by integrating this expression between the

limits 6 = 0 and 6 = 1L and then doubling this value. Hence the magnetic force of the whole wire at P is equal to

— f^cos8dO=—. . (2)

P J0 P

In other words, the magnetic force at any point due to the current I flowing in an infinitely long straight conductor is in magnitude inversely proportional to the perpendicular dis- tance of the point from the wire ; and, as regards direction, it is everywhere perpendicular to the plane containing the wire, and the perpendicular let fall on it from the given point. This conclusion was experimentally verified by Biot and Savart by vibrating a small magnetic needle at different distances from a long straight current, and counting the square of the numbei of oscillations in a given time made by the said small needle in these different positions. If the current in the wire is measured in amperes, then, since ten amperes equal one unit current in absolute electromagnetic measure, and if A is the current so measured in amperes, the magnetic force at any point p centimetres from the wire is equal to

A^=l A

10 p 5 p '

Thus the magnetic force due to a current of one ampere flowing in a long straight wire at a point one centimetre from the wire is equal to one-fifth of a unit of magnetic force. This is nearly equal to the value of the earth's horizontal magnetic

18 ELECTRO-MAGNETIC INDUCTION.

force in England. It will be seen that the magnetic force due to powerful currents in long straight cables may be sensible at points very far removed from the cable. This magnetic force is at every point perpendicular to the conductor, and hence the direction of the force of such a straight conductor must be everywhere a tangent to a circle drawn round the wire with its plane perpendicular to the axis of the wire and its centre in that axis. Hence a freely suspended magnetic needle tends to stand perpendicular to a straight conductor when this last is traversed by a current. If a magnetic pole of strength m is placed at any point in the field of such a straight conductor the magnetic force tends to drive the pole in a circle round the

wire with a force equal to m dynamical units or dynes,

P

where p is the distance of the pole from the axis of the wire and I is the absolute value of the current flowing in it. It follows that the lines of magnetic force of such a linear current are circles described round the wire with planes perpendicular to it and centres in the axis of the wire. Oersted was aware of this fact, and he expressly says,* " The electric conflict " (that is, magnetic field) " performs circles round the wire." We may next proceed to determine

The Magnetic Force at the Centre of a Circular Current. If a thin wire is bent into a circle, and a current of slrength I is sent round it, the magnetic force, estimated at the centre of the circle, due to each element of the length of the current, is in a direction at right angles to the plane of the circle. Let ds be an element of length and let r be the radius of the circular wire, then the magnetic force due to ds at the centre

is equal to — 5- . But, since the force due to each element is r*

the same, the magnetic force due to the whole length of the circular wire is equal to

<»)

If the current is measured in amperes and denoted by A, then the magnetic force at the centre of the circular current is

• Annals of Philosophy, Oct., 1820 Vol. XVI., p. 274,

ELECTRO-MAGNETIC INDUCTION. 19

This magnetic force at the centre of a circular current is in a direction perpendicular to the plane of the circle. If the circular current consists of a current of A amperes flowing in a very thin wire wound n times round a circular groove of mean radius r, then the magnetic force at the centre is equal to

^ (5)

5 T

The expression for the magnetic force at a point in the plane of the circle not in the centre is less simple (see Appendix, Note A).

The Magnetic Force due to a Circular Current of n turns at a

point on its axis out of its own plane.

The third case of importance is to find the value of the magnetic force due to a circular current at a point on a line

FIG. 4.

drawn through its centre and perpendicular to its plane. Let the circular current be X Y Z (Fig. 4), and let P be any point on a line 0 P drawn through the centre 0 and perpen- dicular to the plane of X Y Z. The magnetic force at P, due to an element ds of the circuit at X, acts along a line perpendicular to X P, and is in the plane of X 0 P. If r stands for 0 X, and x for 0 P, the magnetic force due to the element ds at P resolved in the direction 0 P is equal to

(

where I is the strength of the current in the element. The above is equal to

Irds

20 ELECTRO-MAGNETIC INDUCTION.

and hence the magnetic force due to one whole turn of the

conductor is equal to

I 2 TT r2

If the circuit makes n turns, the magnetic force at P, due to the current I flowing n times round the circular conductor X Y Z estimated in the direction 0 P, is equal to

27T«I ^_, (6)

This, then, is the expression for the magnetic force, due to a circular current of n turns at a point on its axis but outside of its own plane. The calculation of the magnetic force due to the circular currrent at points other than those on the axis 0 P, is a much more difficult matter. In the above formula I is measured in the electromagnetic units. If the current is measured in amperes and denoted by A, then (6) becomes

^A_£_^ (7)

The above expression may be put into another useful form.

r2 ^mce ?~2 2\I

is the differential with respect to x of x

which last, as can be seen from Fig. 4, is equal to cos X P 0, we may write (7) in the form

*nAA(cos0) (8)

where 0 stands for the angle X P 0.

The Magnetic Force due tn a long closely-coiled Helical Current

at point* on the axis near the centre.

Another useful case in which it is possible to calculate the magnetic force due to a current is in the case of points in the in- terior of a very long closely-coiled helical current called a sole- noid. Such a case is practically realised by coiling insulated wire round a tube. Let the length of the helix be I, and let there be N turns of wire per unit of length. Then if a slice of

ELECTRO-MAGNETIC INDUCTION. 21

this helix is considered of thickness d x, the number of turns of wire in this slice is N d x. Let a current I flow through the wire. Take a point on the axis of the helix somewhere near the centre (see Fig. 5), and take any element of length of the helix at a distance x from this point. Then by (8) the mag- netic force due to this element of length of the helix at the point P is

dx where 6 is the angle 0 P X.

Let the slice of the helix be taken at successive distances from the point P, bsginning with x = 0 and ending with the end of the helix. The sum of all the magnetic forces due to each element of the helix to the left of the point P is then equal to the integral of (9) taken between the limits 6 = 90deg. or cos 0 = 0 and 0=0,, where Ol is the angle 0' P X', or the angle subtended by half the mean diameter of the end of the

FIG. 5.

helix at the point P. Similarly, to obtain the whole force at P due to the elements of the helix lying to the right of P we have to integrate (9) from 6 = 0 to 6 = 02, where 09 is the angle sub- tended by half the aperture of the other end of the helix at P. Adding these forces together we have as value of the whole magnetic force of the whole helix at P the expression

F = 27rNI(cos fli + cos 02).

If the helix is so long that the half diameter of the aperture of the ends of the helix, as seen from the point P, is practically zero, then 0: and 02 are both practically zero, and therefore cos 0J + COS 02 = 2, nearly, or

F = 47rNI (10)

The magnetic field at P is then equal to 4?r times the absolute current-turns per unit of length. If the current is measured

o O

in amperes, the force is equal to -?rN A, or to -TT times the

5 5

22 ELECTRO-MAGNETIC INDUCTION.

ampere-turns per unit of length of the coil. Since - TT is nearly

1-25, the approximate practical rule for the magnetic force in the neighbourhood of the centre of a long helix of this kind is that the magnetic force is numerically equal to 1£ times the ampere-turns per unit of length of the helix. The above formula is only strictly true for points on the axis of the helix and for helices very long compared with their diameters. It is very nearly true for all points in the interior of a fairly long helix. Thus, for instance, if the helix is twelve diameters long, the magnetic force in the interior throughout one quarter of its length on either side of the central point does not differ by much more than one per cent, from the value it has at the central point. Hence this fact presents us with an easy and practical method of procuring a magnetic field of known strength. On a long pasteboard or metal tube provided with cheeks wind covered copper wire carefully and evenly in any number of layers. Count the turns and layers of wire, and measure the length between the cheeks ; this gives us the turns per unit of length. Then pass a known current through the wire, and calculate by formula (10) the field at the centre. The coil should be at least twelve diameters long. We may approximately apply (10) to calcu- late the magnetic force in the interior of such long bobbins as are used in winding the field-magnets of dynamos. The magnetic force in the interior of a long bobbin is strongest in the centre of the bobbin, and falls off towards either end, and it is slightly stronger at points nearer the wire than on the central axis even at the centre. The complete calculation of the field at any point in the interior or exterior of a not very long helix is a rather difficult matter, but the above formula (10) will be sufficient for most practical purposes.

A final, practical, and useful case is that of the predetermina- tion of the magnetic force in tJie interior of a circular closed solenoid or endless helical current. Let a wooden ring of circular cross-section be wound over closely with insulated wire so that the turns of the wire are contiguous and one or more layers are put on. This is called a circular solenoid, and we can calculate the magnetic force for points in the

ELECTRO-MAGNETIC INDUCTION. 23

interior when the circular solenoid is traversed by a current. Let R be the mean radius of the solenoid, and a that of the mean circular section. If the wire is wound in one layer on a wooden ring, then a will be the mean between the half diameter of the section of the ring and the half diameter measured over all after the wire is wound on it.

The magnetic force is not the same at all points over the circular cross-section of the solenoid. To find out what-it is at any point we may proceed as follows : — A solenoid of any size, meaning by that a spiral current with turns closely adjacent, is electrically equivalent to a bundle of elementary solenoids or spiral currents of exceedingly small cross-section. Consider such a very small-sectioned solenoid, which may be called a spiral filament. It may be obtained in practice by winding insulated wire of small size on a very fine knitting needle as a core, and then withdrawing the needle. The section of this solenoid being very small, the magnetic force in its interior is everywhere nearly the same over the cross-section, and if the spiral is long the force in the centre in the interior is equal to 47rral, where I is the absolute current flowing in the wire, and n is the number of turns per unit of length. Let this long elementary solenoid be bent round into a circle so as to form a closed or endless solenoid, let x be the mean radius of the circle which it forms and let nv be the number of turns of wire of the spiral in an arc of the solenoid equal to one unit angle in circular measure. Then, the number of turns per unit of length of the spiral being n, we have n x = nv and we may write the expression for the magnetic force in the interior of the solenoid as

(11)