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Bible Encyclopedias
Magnetism
1911 Encyclopedia Britannica
or The present article is a digest, mainly from an experimental standpoint, of the leading facts and principles of magnetic science. It is divided into the following sections: 1. General Phenomena.
2. Terminology and Elementary Principles.
3. Magnetic Measurements.
4. Magnetization in Strong Fields.
5. Magnetization in Weak Fields.
6. Changes of Dimensions attending Magnetization.
7. Effects of Mechanical Stress on Magnetization.
8. Effects of Temperature on Magnetism.
9. Magnetic Properties of Alloys and Compounds of Iron. 10. Miscellaneous Effects of Magnetization: Electric Conductivity - Hall Effect - Electro-Thermal Relations - Thermoelectric Quality - Elasticity - Chemical and Voltaic Effects.
I I. Feebly Susceptible Substances.
12. Molecular Theory of Magnetism.
13. Historical and Chronological Notes.
Of these thirteen sections, the first contains a simple description of the more prominent phenomena, without mathematical symbols or numerical data. The second includes definitions of technical terms in common use, together with so much of the elementary theory as is necessary for understanding the experimental work described in subsequent portions of the article; a number of formulae and results are given for purposes of reference, but the mathematical reasoning by which they are obtained is not generally detailed, authorities being cited whenever the demonstrations are not likely to be found in ordinary textbooks. The subjects discussed in the remaining sections are sufficiently indicated by their respective headings. (See also Electro Magnetism, Terrestrial Magnetism, Magneto-Optics and Units.) I. General Phenomena Pieces of a certain highly esteemed iron ore, which consists mainly of the oxide Fe 3 0 4, are sometimes found to possess the power of attracting small fragments of iron or steel. Ore endowed with this curious property was well known to the ancient Greeks and Romans, who, because it occurred plentifully in the district of Magnesia near the Aegean coast, gave it the name of magnes, or the Magnesian stone. In Englishspeaking countries the ore is commonly known as magnetite, and pieces which exhibit attraction as magnets; the cause to which the attractive property is attributed is called magnetism, a name also applied to the important branch of science which has been evolved from the study of phenomena associated with the magnet.
If a magnet is dipped into a mass of iron filings and withdrawn, filings cling to certain parts of the stone in moss-like tufts, other parts remaining bare. There are generally two regions where the tufts are thickest, and the attraction therefore greatest, and between them is a zone in which no attraction is evidenced. The regions of greatest attraction have received the name of poles, and the line joining them is called the axis of the magnet; the space around a magnet in which magnetic effects are exhibited is called the field of magnetic force, or the magnetic field. Up to the end of the 15th century only two magnetic phenomena of importance, besides that of attraction, had been observed. Upon one of these is based the principle of the mariner's compass, which is said to have been known to the Chinese as early as I ioo B.C., though it was not introduced into Europe until more than 2000 years later; a magnet supported so that its axis is free to turn in a horizontal plane will come to rest with its poles pointing approximately north and south. The other phenomenon is mentioned by Greek and Roman writers of the 1st century: a piece of iron, when brought into contact with a magnet, or even held near one, itself becomes " inductively " magnetized, and acquires the power of lifting iron. If the iron is soft and fairly pure, it loses its attractive property when removed from the neighbourhood of the magnet; if it is hard, some of the induced magnetism is permanently retained, and the piece becomes an artificial magnet. Steel is much more retentive of magnetism than any ordinary iron, and some form of steel is now always used for making artificial magnets. Magnetism may be imparted to a bar of hardened steel by stroking it several times from end to end, always in the same direction, with one of the poles of a magnet. Until 1820 all the artificial magnets in practical use derived their virtue, directly or indirectly, from the natural magnets found in the earth: it is now recognized that the source of all magnetism, not excepting that of the magnetic ore itself, is electricity, and it is usual to have direct recourse to electricity for producing magnetization, without the intermediary of the magnetic ore. A wire carrying an electric current is surrounded by a magnetic field, and if the wire is bent into the form of an elongated coil or spiral, a field having certain very useful qualities is generated in the interior. A bar of soft iron introduced into the coil is at once magnetized, the magnetism, however, disappearing almost completely as soon as the current ceases to flow. Such a combination constitutes an electromagnet, a valuable device by means of which a magnet can be instantly made and unmade at will. With suitable arrangements of iron and coil and a sufficiently strong current, the intensity of the temporary magnetization may be very high, and electromagnets capable of lifting weights of several tons are in daily use in engineering works (see Electromagnetism). If the bar inserted into the coil is of hardened steel instead of iron, the magnetism will be less intense, but a larger proportion of it will be retained after the current has been cut off. Steel magnets of great strength and of any convenient form may be prepared either in this manner or by treatment with an electromagnet; hence the natural magnet, or lodestone as it is commonly called, is no longer of any interest except as a scientific curiosity.
Some of the principal phenomena of magnetism may be demonstrated with very little apparatus; much may be done with a small bar-magnet, a pocket compass and a few ounces of iron filings. Steel articles, such as knitting or sewing needles and pieces of flat spring, may be readily magnetized by stroking them with the bar-magnet; after having produced magnetism in any number of other bodies, the magnet will have lost nothing of its own virtue. The compass needle is a little steel magnet balanced upon a pivot; one end of the needle, which always bears a distinguishing mark, points approximately, but not in general exactly, to the north,' the vertical plane through the direction of the needle being termed the magnetic meridian. The bar-magnet, if suspended horizontally in a paper stirrup by a thread of unspun silk, will also come to rest in the magnetic meridian with its marked end pointing northwards. The north-seeking end of a magnet is in English-speaking countries called the north pole and the other end the south pole; in France the names are interchanged. If one pole of the bar-magnet is brought near the compass, it will attract the opposite pole of the compass-needle; and the magnetic action will not be sensibly affected by the interposition between the bar and the compass of any substance whatever except iron or other magnetizable metal. The poles of a piece of magnetized steel may be at once distinguished if the two ends are successively presented to the compass; that end which attracts the south pole of the compass needle (and is therefore north) may be marked for easy identification.
Similar magnetic poles are not merely indifferent to each other, but exhibit actual repulsion. This can be more easily shown if the compass is replaced by a magnetized knitting needle, supported horizontally by a thread. The north pole of the bar-magnet will repel the north pole of the suspended needle, and there will likewise be repulsion between the two south poles. Such experiments as these demonstrate the fundamental law that like poles repel each other; unlike poles attract. It follows that between two neighbouring magnets, the poles of which are regarded as centres of force, there must always be four forces in action. Denoting the two pairs of magnetic poles by N, S and N', S', there is attraction between N and S', and between S and N'; repulsion between N and N', and between S and S'. Hence it is not very easy to determine experimentally the law of magnetic force between poles. The ' In London in 1910 the needle pointed about 16° W. of the geographical north. (See Terrestrial Magnetism.) difficulty was overcome by C. A. Coulomb, who by using very long and thin magnets, so arranged that the action of their distant poles was negligible, succeeded in establishing the law, which has since been confirmed by more accurate methods, that the force of attraction or repulsion exerted between two magnetic poles varies inversely as the square of the distance between them. Since the poles of different magnets differ in strength, it is important to agree upon a definite unit or standard of reference in terms of which the strength of a pole may be numerically specified. According to the recognized convention, the unit pole is that which acts upon an equal pole at unit distance with unit force: a north pole is reckoned as positive (+) and a south pole as negative (-). Other conditions remaining unchanged, the force between two poles is proportional to the product of their strengths; it is repulsive or attractive according as the signs of the poles are like or unlike.
If a wire of soft iron is substituted for the suspended magnetic needle, either pole of the bar-magnet will attract either end of the wire indifferently. The wire will in fact become temporarily magnetized by induction, that end of it which is nearest to the pole of the magnet acquiring opposite polarity, and behaving as if it were the pole of a permanent magnet. Even a permanent magnet is susceptible of induction, its polarity becoming thereby strengthened, weakened, or possibly reversed. If one pole of a strong magnet is presented to the like pole of a weaker one, there will be repulsion so long as the two are separated by a certain minimum distance. At shorter distances the magnetism induced in the weaker magnet will be stronger than its permanent magnetism, and there will be attraction; two magnets with their like poles in actual contact will always cling together unless the like poles are of exactly equal strength. Induction is an effect of the field of force associated with a magnet. Magnetic force has not merely the property of acting upon magnetic poles, it has the additional property of producing a phenomenon known as magnetic induction, or magnetic flux, a physical condition which is of the nature of a flow continuously circulating through the magnet and the space outside it. Inside the magnet the course of the flow is from the south pole to the north pole; thence it diverges through the surrounding space, and again converging, re-enters the magnet at the south pole. When the magnetic induction flows through a piece of iron or other magnetizable substance placed near the magnet, a south pole is developed where the flux enters and a north pole where it leaves the substance. Outside the magnet the direction of the magnetic induction is generally the same as that of the magnetic force. A map indicating the direction of the force in different parts of the field due to a magnet may be constructed in a very simple manner. A sheet of cardboard is placed above the magnet, and some iron filings are sifted thinly and evenly over the surface: if the cardboard is gently tapped, the filings will arrange themselves in a series of curves, as shown in fig. I. This experiment suggested to Faraday the conception of ' r,? ir`"' lines of force," of which the curves formed by the filings afford a rough indication; Faraday's lines are howeve confined to the plane of the cardboard, but occur in the whole of the space around the magnet. A line of force may be defined as an imaginary line so drawn that its direction at every point of its course coincides with the direction of the magnetic force at that point. Through any point in the field one such line can be drawn, but not more than one, for the force obviously cannot have more than one direction; the lines therefore never intersect. A line of force is regarded as proceeding from the north pole towards the south pole of the magnet, its direction being that in which an isolated north pole would be urged along FIG. I.
it. A south pole would be urged oppositely to the conventional " direction " of the line; hence it follows that a very small magnetic needle, if placed in the field, would tend to set itself along or tangentially to the line of force passing through its centre, as may be approximately verified if the compass be placed among the filings on the cardboard. In the internal field of a long coil of wire carrying an electric current, the lines of force are, except near the ends, parallel to the axis of the coil, and it is chiefly for this reason that the field due to a coil is particularly well adapted for inductively magnetizing iron and steel. The older operation of magnetizing a steel bar by drawing a magnetic pole along it merely consists in exposing successive portions of the bar to the action of the strong field near the pole.
Faraday's lines not only show the direction of the magnetic force, but also serve to indicate its magnitude or strength in different parts of the field. Where the lines are crowded together, as in the neighbourhood of the poles, the force is greater (or the field is stronger) than where they are more widely separated; hence the strength of a field at any point can be accurately specified by reference to the concentration of the lines. The lines presented to the eye by the scattered filings are too vague and ill-defined to give a satisfactory indication of the field-strength (see Faraday, Experimental Researches, § 3 2 37) though they show its direction clearly enough. It is however easy to demonstrate by means of the compass that the force is much greater in some parts of the field than in others. Lay the compass upon the cardboard, and observe the rate at which its needle vibrates after being displaced from its position of equilibrium; this will vary greatly in different regions. When the compass is far from the magnet, the vibrations will be comparatively slow; when it is near a pole, they will be exceedingly rapid, the frequency of the vibrations varying as the square root of the magnetic force at the spot. In a refined form this method is often employed for measuring the intensity of a magnetic field at a given place, just as the intensity of gravity at different parts of the earth is deduced from observations of the rate at which a pendulum of known length vibrates.
It is to the non-uniformity of the field surrounding a magnet that the apparent attraction between a magnet and a magnetizable body such as iron is ultimately due. This was pointed out by W. Thomson (afterwards Lord Kelvin) in 1847, as the result of a mathematical investigation undertaken to explain Faraday's experimental observations. If the inductively magnetized body lies in a part of the field which happens to be uniform there will be no resulting force tending to move the body, and it will not be " attracted." If however there is a small variation of the force in the space occupied by the body, it can be shown that the body will be urged, not necessarily towards a magnetic pole, but towards places of stronger magnetic force. It will not in general move along a line of force, as would an isolated pole, but will follow the direction in which the magnetic force increases most rapidly, and in so doing it may cross the lines of force obliquely or even at right angles.
If a magnetized needle were supported so that it could move freely'about its centre of gravity it would not generally settle with its axis in a horizontal position, but would come to rest with its north-seeking pole either higher or lower than its centre. For the practical observation of this phenomenon it is usual to employ a needle which can turn freely in the plane of the magnetic meridian upon a horizontal axis passing through the centre of gravity of the needle. The angle which the magnetic axis makes with the plane of the horizon is called the inclination or Along an irregular line encircling the earth in the neighbourhood of the geographical equator the needle takes up a horizontal position, and the dip is zero. At places north of this line, which is called the magnetic equator, the north end of the needle points downwards, the inclination generally becoming greater with increased distance from the equator. Within a certain small area in the Arctic Circle (about 97° W. long., 70 N. lat.) the north pole of the needle points vertically downwards, the dip being 90 0. South of the magnetic equator the south end of the needle is always inclined downwards, and there is a spot within the Antarctic Circle (148° E. long., 74° S. lat.) where the needle again stands vertically, but with its north end directed upwards. All these observations may be accounted for by the fact first recognized by W. Gilbert in 1600, that the earth itself is a great magnet, having its poles at the two places where the dipping needle is vertical. To be consistent with the terminology adopted in Britain, it is necessary to regard the pole which is geographically north as being the south pole of the terrestrial magnet, and that which is geographically south as the north pole; in practice however the names assigned to the terrestrial magnetic poles correspond with their geographical situations. Within a limited space, such as that contained in a room, the field due to the earth's magnetism is sensibly uniform, the lines of force being parallel straight lines inclined to the horizon at the angle of dip, which at Greenwich in 1910 was about 67°. It is by the horizontal component of the earth's total force that the compass-needle is directed.
The magnets hitherto considered have been assumed to have each two poles, the one north and the other south. It is possible that there may be more than two. If, for example, a knitting needle is stroked with the south pole of a magnet, the strokes being directed from the middle of the needle towards the two extremities alternately, the needle will acquire a north pole at each end and a south pole in the middle. By suitably modifying the manipulation a further number of consequent poles, as they are called, may be developed. It is also possible that a magnet may have no poles at all. Let a magnetic pole be drawn several times around a uniform steel ring, so that every part of the ring may be successively subjected to the magnetic force. If the operation has been skilfully performed the ring will have no poles and will not attract iron filings. Yet it will be magnetized; for if it is cut through and the cut ends are drawn apart, each end will be found to exhibit polarity. Again, a steel wire through which an electric current has been passed will be magnetized, but so long as it is free from stress it will give no evidence of magnetization; if, however, the wire is twisted, poles will be developed at the two ends, for reasons which will be explained later. A wire or rod in this condition is said to be circularly magnetized; it may be regarded as consisting of an indefinite number of elementary ring-magnets, having their axes coincident with the axis of the wire and their planes at right angles to it. But no magnet can have a single pole; if there is one, there must also be at least a second, of the opposite sign and of exactly equal strength. Let a magnetized knitting needle, having north and south poles at the two ends respectively, be broken in the middle; each half will be found to possess a north and a south pole, the appropriate supplementary poles appearing at the broken ends. One of the fragments may again be broken, and again two bipolar magnets will be produced; and the operation may be repeated, at least in imagination, till we arrive at molecular magnitudes and can go no farther. This experiment proves that the condition of magnetization is not confined to those parts where polar phenomena are exhibited, but exists throughout the whole body of the magnet; it also suggests the idea of molecular magnetism, upon which the accepted theory of magnetization is based. According to this theory the molecules of any magnetizable substance are little permanent magnets the axes of which are, under ordinary conditions, disposed in all possible directions indifferently. The process of magnetization consists in turning round the molecules by the application of magnetic force, so that their north poles may all point more or less approximately in the direction of the force; thus the body as a whole becomes a magnet which is merely the resultant of an immense number of molecular magnets.
In every magnet the strength of the south pole is exactly equal to that of the north pole, the action of the same magnetic force upon the two poles being equal and oppositely directed. This may be shown by means of the uniform field of force due to the earth's magnetism. A magnet attached to a cork and [[[Terminology And Principles]] floated upon water will set itself with its axis in the magnetic meridian, but it will be drawn neither northward nor southward; the forces acting upon the two poles have therefore no horizontal resultant. And again if a piece of steel is weighed in a delicate balance before and after magnetization, no change whatever in its weight can be detected; there is consequently no upward or downward resultant force due to magnetization; the contrary parallel forces acting upon the poles of the magnet are equal, constituting a couple, which may tend to turn the body, but not to propel it.
Iron and its alloys, including the various kinds of steel, though exhibiting magnetic phenomena in a pre-eminent degree, are not the only substances capable of magnetization. Nickel and cobalt are also strongly magnetic, and in 1903 the interesting discovery was made by F. Heusler that an alloy consisting of copper, aluminium and manganese (Heusler's alloy), possesses magnetic qualities comparable with those of iron. Practically the metals iron, nickel and cobalt, and some of their alloys and compounds constitute a class by themselves and are called ferromagnetic substances. But it was discovered by Faraday in 1845 that all substances, including even gases, are either attracted or repelled by a sufficiently powerful magnetic pole. Those substances which are attracted, or rather which tend, like iron, to move from weaker to stronger parts of the magnetic field, are termed paramagnetic; those which are repelled, or tend to move from stronger to weaker parts of the field, are termed diamagnetic. Between the ferromagnetics and the paramagnetics there is an enormous gap. The maximum magnetic susceptibility of iron is half a million times greater than that of liquid oxygen, one of the strongest paramagnetic substances known. Bismuth, the strongest of the diamagnetics, has a negative susceptibility which is numerically 20 times less than that of liquid oxygen.
Many of the physical properties of a metal are affected by magnetization. The dimensions of a piece of iron, for example, its elasticity, its thermo-electric power and its electric conductivity are all changed under the influence of magnetism. On the other hand, the magnetic properties of a substance are affected by such causes as mechanical stress and changes of temperature. An account of some of these effects will be found in another section.' 2. Terminology And Elementary Principles In what follows the C.G.S. electromagnetic system of units will be generally adopted, and, unless otherwise stated, magnetic substances will be assumed to be isotropic, or to have the same physical properties in all directions.
Vectors
Physical quantities such as magnetic force, magnetic induction and magnetization, which have direction as well as magnitude, are termed vectors; they are compounded and resolved in the same manner as mechanical force, which is itself a vector. When the direction of any vector quantity denoted by a symbol is to be attended to, it is usual to employ for the symbol either a block letter, as H, I, B, or a German capital, as j, ,3?
Magnetic Poles and Magnetic Axis
A unit magnetic pole is that which acts on an equal pole at a distance of one centimetre with a force of one dyne. A pole which points north is reckoned positive, one which points south negative. The action between any two magnetic poles is mutual. If m, and m 2 are the strengths of two poles, d the distance between them expressed in centimetres, and f the force in dynes, f=ml m2/d2 (I) The force is one of attraction or repulsion, according as the sign of the product m l m 2 is negative or positive. The poles at the ends of an infinitely thin uniform magnet, or magnetic filament, would act as definite centres of force. An actual magnet may generally be regarded as a bundle of magnetic filaments, and those portions of the surface of the magnet where the filaments terminate, and socalled " free magnetism " appears, may be conveniently called poles or polar regions. A more precise definition is the following: When the magnet is placed in a uniform field, the parallel forces acting on the positive poles of the constituent filaments, whether the filaments ' For the relations between magnetism and light see Magnetooptics.
2 Clerk Maxwell employed German capitals to denote vector quantities. J. A. Fleming first recommended the use of blockletters as being more convenient both to printers and readers.
terminate outside the magnet or inside, have a resultant, equal to the sum of the forces and parallel to their direction, acting at a certain point N. The point N, which is the centre of the parallel forces, is called the north or positive pole of the magnet. Similarly, the forces acting in the opposite direction on the negative poles of the filaments have a resultant at another point S, which is called the south or negative pole. The opposite and parallel forces acting on the poles are always equal, a fact which is sometimes expressed by the statement that the total magnetism of a magnet is zero. The line joining the two poles is called the axis of the magnet.
Magnetic Field
Any space at every point of which there is a finite magnetic force is called a field of magnetic force, or a magnetic field. The strength or intensity of a magnetic field at any point is measured by the force in dynes which a unit pole will experience when placed at that point, the direction of the field being the direction in which a positive pole is urged. The field-strength at any point is also called the magnetic force at that point; it is denoted by H, or, when it is desired to draw attention to the fact that it is a vector quantity, by the block letter H, or the German character ,C. Magnetic force is sometimes, and perhaps more suitably, termed magnetic intensity; it corresponds to the intensity of gravity g in the theory of heavy bodies (see Maxwell, Electricity and Magnetism, § 12 and § 68, footnote). A line of force is a line drawn through a magnetic field in the direction of the force at each point through which it passes. A uniform magnetic field is one in which H has everywhere the same value and the same direction, the lines of force being, therefore, straight and parallel. A magnetic field is generally due either to a conductor carrying an electric current or to the poles of a magnet. The magnetic field due to a long straight wire in which a current of electricity is flowing is at every point at right angles to the plane passing through it and through the wire; its strength at any point distant r centimetres from the wire is H = 21/r, (2) i being the current in C.G.S. units. 3 The lines of force are evidently circles concentric with the wire and at right angles to it; their direction is related to that of the current in the same manner as the rotation of a corkscrew is related to its thrust. The field at the centre of a circular conductor of radius r through which current is passing is H = 27ri/r, (3) the direction of the force being along the axis and related to the direction of the current as the thrust of a corkscrew to its rotation. The field strength in the interior of a long uniformly wound coil containing n turns of wire and having a length of 1 centimetres is (except near the ends) H = 41rin/l. (4) In the middle portion of the coil the strength of the field is very nearly uniform, but towards the end it diminishes, and at the ends is reduced to one-half. The direction of the force is parallel to the axis of the coil, and related to the direction of the current as the thrust of a corkscrew to its rotation. If the coil has the form of a ring of mean radius r, the length will be 21rr, and the field inside the coil may be expressed as H = 2ni/r. (5) The uniformity of the field is not in this case disturbed by the influence of ends, but its strength at any point varies inversely as the distance from the axis of the ring. When therefore sensible uniformity is desired, the radius of the ring should he large in relation to that of the convolutions, or the ring should have the form of a short cylinder with thin walls. The strongest magnetic fields employed for experimental purposes are obtained by the use of electromagnets. For many experiments the field due to the earth's magnetism is sufficient; this is practically quite uniform throughout considerable spaces, but its total intensity is less than half a unit.
Magnetic Moment and Magnetization
The moment, M, M or V, of a uniformly and longitudinally magnetized bar-magnet is the product of its length into the strength of one of its poles; it is the moment of the couple acting on the magnet when placed in a field of unit intensity with its axis perpendicular to the direction of the field. If 1 is the length of the magnet, M = ml. The action of a magnet at a distance which is great compared with the length of the magnet depends solely upon its moment; so also does the action which the magnet experiences when placed in a uniform field. The moment of a small magnet may be resolved like a force. The intensity of magnetization, or, more shortly, the magnetization of a uniformly magnetized body is defined as the magnetic moment per unit of volume, and is denoted by I, I, or „a. Hence I = M/v = ml/v = m/a, v being the volume and a the sectional area. If the magnet is not uni - form, the magnetization at any point is the ratio of the moment of an element of volume at that point to the volume itself, or I = m.ds/dv. where ds is the length of the element. The direction of the magnetization is that of the magnetic axis of the element;'in isotropic substances it coincides with the direction of the magnetic force at the point. If the direction of the magnetization at the surface of a magnet makes 3 The C.G.S. unit of current =10 amperes.
an angle e with the normal, the normal component of the magnetization, I cos e, is called the surface density of the magnetism, and is generally denoted by a.
Potential and Magnetic Force
The magnetic potential at any point in a magnetic field is the work which would be done against the magnetic fdrees in bringing a unit pole to that point from the boundary of the field. The line through the given point along which the potential decreases most rapidly is the direction of the resultant magnetic force, and the rate of decrease of the potential in any direction is equal to the component of the force in that direction. If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y ,-s Surfaces for which the potential is constant are called equipotential surfaces. The resultant magnetic force at every point of such a surface is in the direction of the normal (n) to the surface; every line of force therefore cuts the equipotential surfaces at right angles. The potential due to a single pole of strength m at the distance r from the pole is V = m/ r , (7) the equipotential surfaces being spheres of which the pole is the centre and the lines of force radii. The potential due to a thin magnet at a point whose distance from the two poles respectively is r and r' is V =m(l/r=l/r') (8) When V is constant, this equation represents an equipotential surface.
The equipotential surfaces are two series of ovoids surrounding the two poles respectively, and separated by a plane at zero potential passing perpendicularly through the middle of the axis. If r and r' make angles 0 and 0 with the axis, it is easily shown that the equation to a line of force is cos 0 - cos B'= constant. (9) At the point where a line of force intersects the perpendicular bisector of the axis r=r'=r o , say, and cos 0 - cos 0 obviously =l/r o, l being FIG. 2. FIG. 3.
the distance between the poles; l/r o is therefore the value of the constant in (9) for the line in question. Fig. 2 shows the lines of force and the plane sections of the equipotential surfaces for a thin magnet with poles concentrated at its ends. The potential due to a small magnet of moment M, at a point whose distance from the centre of the magnet is r, is V=M cos 0/r 2 , (io) where 0 is the angle between r and the axis of the magnet. Denoting the force at P (see fig. 3) by F, and its components parallel to the co-ordinate axes by X and Y, we have X= - ax = M(3 cos' 0 - I), Y= - y = M (3 sin 0 cos 0. If F T is the force along r and F t that along t at right angles to r, F r =X cos 0+ Y sin 0=M 2 cos 0, F t = - X sin 0+ Y cos 0 = - r 3 sin 0. For the resultant force at P, F=-VF r 2. -+F t 2 =M 13 cos t 0+1. The direction of F is given by the following construction: Trisect OP at C, so that OC =OP/3; draw CD at right angles to OP, to cut the axis produced in D; then DP will be the direction of the force at P. For a point in the axis OX, 0 =0; therefore cos 0 = 1, and the point D coincides with C; the magnitude of the force is, from (14), Fx=2M / r3, (15) its direction being along the axis OX. For a point in the line OY bisecting the magnet perpendicularly, 0 =42 therefore cos 0 =0, and the point D is at an infinite distance. The magnitude of the force is in this case F5 = M/r 3, (16) and its direction is parallel to the axis of the magnet. Although the above useful formulae, (io) to (15), are true only for an infinitely small magnet, they may be practically applied whenever the distance r is considerable compared with the length of the magnet.
Couples and Forces between Magnets
If a small magnet of moment M is placed in the sensibly uniform field H due to a distant magnet, the couple tending to turn the small magnet upon an axis at right angles to the magnet and to the force is MH sin 0, (17) where 0 is the angle between the axis of the magnet and the direction of the force. In fig. 4 S'N' is a small magnet of moment M', and SN a distant fixed magnet of moment M; the axes of SN and S'N' make angles of 0 and 4 respectively with the line through their middle points. It can be deduced from (17), (12) and (13) that the couple on S'N' due to SN, and tending to increase 4), is MM' (sin 0 cos 4-2 sin 4) cos 0)/r'. (18) This vanishes if sin 0 cos 4)=2 sin 4 cos 0, i.e. if tan 4)=1 tan 0, S'N' being then along a line of force, a result which explains the construction given above for finding the direction of the force F in (14). If the axis of SN produced passes through the centre of S'N', 0 =0, and the couple becomes 2MM'sin4)/r 3, (19) tending to diminish cp; this is called the " end on " position. If the centre of S'N' is on the perpendicular bisector of SN, 0 = Zir, and the couple will be MM 'cos cp/r 3 , (20) tending to increase 4); this is the " broadside on " position. These two positions are sometimes called the first and second (or A and B) principal positions of Gauss. The components X, Y, parallel and perpendicular to r, of the force between the two magnets SN and S'N' are X =3MM'(sin 0 sin 4)-2 cos 0 cos 4)/r 4 , (21) Y=3MM'(sin 0 cos 4-{-sin 4 cos 0)/r 4 . (22) It will be seen that, whereas the couple varies inversely as the cube of the distance, the force varies inversely as the fourth power.
Distributions of Magnetism
A magnet may be regarded as consisting of an infinite number of elementary magnets, each having a pair of poles and a definite magnetic moment. If a series of such elements, all equally and longitudinally magnetized, were placed end to end with their unlike poles in contact, the external action of the filament thus formed would be reduced to that of the two extreme poles. The same would be the case if the magnetization of the filament varied inversely as the area of its cross-section a in different parts. Such a filament is called a simple magnetic solenoid, and the product aI is called the strength of the solenoid. A magnet which consists entirely of such solenoids, having their ends either upon the surface or closed upon themselves, is called a solenoidal magnet, and the magnetism is said to be distributed solenoidally; there is no free magnetism in its interior. If the constituent solenoids are parallel and of equal strength, the magnet is also uniformly magnetized. A thin sheet of magnetic matter magnetized normally to its surface in such a manner that the magnetization at any place is inversely proportional to the thickness h of the sheet at that place is called a magnetic shell; the constant product hI is the strength of the shell and is generally denoted by 4, or 4. The potential at any point due to a magnetic shell is the product of its strength into the solid angle w subtended by its edge at the given point, or V = Fu. For a given strength, therefore, the potential depends solely upon the boundary of the shell, and the potential outside a closed shell is everywhere zero. A magnet which can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, is called a lamellar magnet, and the magnetism is said to be distributed lamellarly. A magnet consisting of a series of plane shells of equal strength arranged at right angles to the direction of magnetization will be uniformly magnetized.
It can be shown that uniform magnetization is possible only when the form of the body is ellipsoidal. (Maxwell, Electricity and Magnetism, II., § 437). The cases of greatest practical importance are those of a sphere (which is an ellipsoid with three equal axes) and an ovoid or prolate ellipsoid of revolution. The potential due to a uniformly magnetized sphere of radius a for an external point at a distance r from the centre is V =:I ra 3 I cos 0/r 22 , (23) 0 being the inclination of r to the magnetic axis. Since 7ra'I is the moment of the sphere (=volume X magnetization), it appears from (10) that the magnetized sphere produces the same external effect as a very small magnet of equal moment placed at its centre and magnetized in the same direction; the resultant force therefore is the same as in (14). The force in the interior is uniform, opposite (6) (II) [[[Terminology And Principles]] to the direction of magnetization, and equal to 3rI. When it is desired to have a uniform magnet with definitely situated poles, it it usual to employ one having the form of an ovoid, or elongated ellipsoid of revolution, instead of a rectangular or cylindrical bar. If the magnetization is parallel to the major axis, and the lengths of the major and minor axes are 2a and 2C, the poles are situated at a distance equal to 3a from the centre, and the magnet will behave externally like a simple solenoid of length 3a. The internal force F is opposite to the direction of the magnetization, and equal to NI, where N is a coefficient depending only on the ratio of the axes. The moment = airac 2 I = - 3nac2FN.
The distribution of magnetism and the position of the poles in magnets of other shapes, such as cylindrical or rectangular bars, cannot be specified by any general statement, though approximate determinations may be obtained experimentally in individual cases.' According to F. W. G. Kohlrausch 2 the distance between the poles of a cylindrical magnet the length of which is from io to 30 times the diameter, is sensibly equal to five-sixths of the length of the bar. This statement, however, is only approximately correct, the distance between the poles depending upon the intensity of the magnetization. 3 In general, the greater the ratio of length to section, the more nearly will the poles approach the end of the bar, and the more nearly uniform will be the magnetization. For most practical purpose a knowledge of the exact position of the poles is of no importance; the magnetic moment, and therefore the mean magnetization, can always be determined with accuracy.
Magnetic Induction or Magnetic Flux. - When magnetic force acts on any medium, whether magnetic, diamagnetic or neutral, it produces within it a phenomenon of the nature of a flux or flow called magnetic induction (Maxwell, loc. cit., § 428). Magnetic induction, like other fluxes such as electrical, thermal or fluid currents, is defined with reference to an area; it satisfies the same conditions of continuity as the electric current does, and in isotropic media it depends on the magnetic force just as the electric current depends on the electromotive force. The magnitude of the flux produced by a given magnetic force differs in different media. In a uniform magnetic field of unit intensity formed in empty space the induction or magnetic flux across an area of I square centimetre normal to the direction of the field is arbitrarily taken as the unit of induction. Hence if the induction per square centimetre at any point is denoted by B, then in empty space B is numerically equal to H; moreover in isotropic media both have the same direction, and for these reasons it is often said that in empty space (and practically in air and other nonmagnetic substances) B and H are identical. Inside a magnetized body, B is the force that would be exerted on a unit pole if placed in a narrow crevasse cut in the body, the walls of the crevasse being perpendicular to the direction of the magnetization (Maxwell, § § 399, 604); and its numerical value, being partly due to the free magnetism on the walls, is generally very different from that of H. In the case of a straight uniformly magnetized bar the direction of the magnetic force due to the poles of the magnet is from the north to the south pole outside the magnet, and from the south to the north inside. The magnetic flux per square centimetre at any point (B, B, or 0) is briefly called the induction, or, especially by electrical engineers, the flux-density. The direction of magnetic induction may be indicated by lines of induction; a line of induction is always a closed curve, though it may possibly extend to and return from infinity. Lines of induction drawn through every point in the contour of a small surface form a re-entrant tube bounded by lines of induction; such a tube is called a tube of induction. The crosssection of a tube of induction may vary in different parts, but the total induction across any section is everywhere the same. A special meaning has been assigned to the term " lines of induction." Suppose the whole space in which induction exists to be divided up into unit tubes, such that the surface integral of the induction over any cross-section of a tube is equal to unity, and along the axis of each tube let a line of induction be drawn. These axial lines constitute the system of lines of induction which are so often referred to in the specification of a field. Where the induction is high the lines will be crowded together; where it is weak they will be widely separated, the number per square centimetre crossing a normal surface at any point being always equal to the numerical value of B. The induction may therefore be specified as B lines per square centimetre. The direction of the induction is also of course indicated by the direction of the lines, which thus serve to map out space in a convenient manner. Lines of induction are frequently but inaccurately spoken of as lines of force.
When induction or magnetic flux takes place in a ferromagnetic metal, the metal becomes magnetized, but the magnetization at any point is proportional not to B, but to B - H. The factor of proportionality will be I-41r, so that ' The principal theoretical investigations are summarized in Mascart and Joubert's Electricity and Magnetism, i. 391-398 and ii. 646-657. The case of a long iron bar has been experimentally studied with great care by C. G. Lamb, Proc. Phys. Soc., 18 99, 16, 509.
Wied. Ann., 1884, 22, 411.
See C. G. Lamb, loc. cit. p. 518.
I = (B - H)/47r, (24) or B = H +41rI. (25) Unless the path of the induction is entirely inside the metal, free magnetic poles are developed at those parts of the metal where induction enters and leaves, the polarity being south at the entry and north at the exit of the flux. These free poles produce a magnetic field which is superposed upon that arising from other sources. The resultant magnetic field, therefore, is compounded of two fields, the one being due to the poles, and the other to the external causes which would be operative in the absence of the magnetized metal. The intensity (at any point) of the field due to the magnetization may be denoted by H i, that of the external field by Ho, and that of the resultant field by H. In certain cases, as, for instance, in an iron ring wrapped uniformly round with a coil of wire through which a current is passing, the induction is entirely within the metal; there are, consequently, no free poles, and the ring, though magnetized, constitutes a poleless magnet. Magnetization is usually regarded as the direct effect of the resultant magnetic force, which is therefore often termed the magnetizing force. Permeability and Susceptibility. - The ratio B/H is called the permeability of the medium in which the induction is taking place, and is denoted byµ. The ratio I/H is called the susceptibility of the magnetized substance, and is denoted by «. Hence B =µH and I =KH. (26) Also A = B = H H4?rI _ I +41K, (27) and (28) 471 Since in empty space B has been assumed to be numerically equal to H, it follows that the permeability of a vacuum is equal to i. The permeability of most material substances differs very slightly from unity, being a little greater than I in paramagnetic and a little less in diamagnetic substances. In the case of the ferromagnetic metals and some of their alloys and compounds, the permeability has generally a much higher value. Moreover, it is not constant, being an apparently arbitrary function of H or of B; in the same specimen its value may, under different conditions, vary from less than 2 to upwards of 5000. The magnetic susceptibility expresses the numerical relation of the magnetization to the magnetizing force. From the equation K=(µ - I)/47r, it follows that the magnetic susceptibility of a vacuum (where µ = I) is o, that of a diamagnetic substance (where ,u < I) has a negative value, while the susceptibility of paramagnetic and ferromagnetic substances (for which µ> I) is positive. No substance has yet been discovered having a negative susceptibility sufficiently great to render the permeability (= I +471K) negative.
Magnetic Circuit
The circulation of magnetic induction or flux through magnetic and non-magnetic substances, such as iron and air, is in many respects analogous to that of an electric current through good and bad conductors. Just as the lines of flow of an electric current all pass in closed curves through the battery or other generator, so do all the lines of induction pass in closed curves through the magnet or magnetizing coil. The total magnetic induction or flux corresponds to the current of electricity (practically measured in amperes); the induction or flux density B to the density of the current (number of amperes to the square centimetre of section); the magnetic permeability to the specific electric conductivity; and the line integral of the magnetic force, sometimes called the magnetomotive force, to the electro-motive force in the circuit. The principal points of difference are that (I) the magnetic permeability, unlike the electric conductivity, which is independent of the strength of the current, is not in general constant; (2) there is no perfect insulator for magnetic induction, which will pass more or less freely through all known substances. Nevertheless, many important problems relating to the distribution of magnetic induction may be solved by methods similar to those employed for the solution of analogous problems in electricity. For the elementary theory of the magnetic circuit see ELECTxoMAGNETISM.
Hysteresis, Coercive Force, Retentiveness
It is found that when a piece of ferromagnetic metal, such as, iron, is subjected to a magnetic field of changing intensity, the changes which take place in the induced magnetization of the iron exhibit a tendency to lag behind those which occur in the intensity of the field - a phenomenon to which J. A. Ewing (Phil. Trans. clxxvi. 524) has given the name of hysteresis (Gr. barepfw, to lag behind). Thus it happens that there is no definite relation between the magnetization of a piece of metal which has been previously magnetized and the strength of the field in which it is placed. Much depends upon its antecedent magnetic condition, and indeed upon its whole magnetic history. A well-known example of hysteresis is presented by the case of permanent magnets. If a bar of hard steel is placed in a strong magnetic field, a certain intensity of magnetization is induced in the bar; but when the strength of the field is afterwards reduced to zero, the magnetization does not entirely disappear. That portion which is permanently retained, and which may amount to considerably more than one-half, is called the residual magnetization. The ratio of the residual magnetization to its previous maximum value measures the retentiveness, or retentivity, of the metal.' Steel, which is well suited for the construction of permanent magnets, is said to possess great " coercive force." To this term, which had long been used in a loose and indefinite manner, J. Hopkinson supplied a precise meaning (Phil. Trans. clxxvi. 460). The coercive force, or coercivity, of a material is that reversed magnetic force which, while it is acting, just suffices to reduce the residual induction to nothing after the material has been temporarily submitted to any great magnetizing force. A metal which has great retentiveness may at the same time have small coercive force, and it is the latter quality which is of chief importance in permanent magnets.
Demagnetizing Force.-It has already been mentioned that when a ferromagnetic body is placed in a magnetic field, the resultant magnetic force H, at a point within the body, is compounded of the force H o, due to the external field, and of another force, Hi, arising from the induced magnetization of the body. Since H, generally tends to oppose the external force, thus making H less than H o, it may be called the demagnetizing force. Except in the few special cases when a uniform external field produces uniform magnetization, the value of the demagnetizing force cannot be calculated, and an exact determination of the actual magnetic force within the body is therefore impossible. An important instance in which the calculation can be made is that of an elongated ellipsoid of revolution placed in a uniform field H o, with its axis of revolution parallel to the lines of force. The magnetization at any point inside the ellipsoid will then be I = HN (29) where N=47r (e2t) (-2-eloI- e - t), e being the eccentricity (see Maxwell's Treatise, § 438). Since I =KH, we have KH I-KNI =KHo, (30) or H =Ho-NI, NI being the demagnetizing force H i. N may be called, after H. du Bois (Magnetic Circuit, p. 33), the demagnetizing factor, and the ratio of the length of the ellipsoid 2c to its equatorial diameter 2a (=c/a), the dimensional ratio, denoted by the symbol nt.
a2 Since e= I -72= V the above expression for N may be written 4 7r nt + 1/m 2 - I N = 11t1-I (2011-- I in - Jnt2 - I I/ 4 7r C = 111 2 - I 1/111 t - log (in + Im 21) - I, from which the value of N for a given dimensional ratio can be calculated. When the ellipsoid is so much elongated that I is negligible in relation to m'-, the expression approximates to the simpler form N=412 (log 201-I).
In the case of a sphere, e= O and N=37r; therefore from (29) I=KH IMir. 3 +t 47fK o, Whence H Ho Ho (33) _ 3 __ 3 3r 47rK /1+2 and B =µH- + 2 Ho.
Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three times that of the external force, and nearly independent of the permeability. The demagnetizing force inside a cylindrical rod placed longitudinally in a uniform field Ho is not uniform, being greatest at the ends and least in the middle part. Denoting its mean value by Hti, and that of the demagnetizing factor by N, we have H=Ho-Hti=Ho-NI. (35) Du Bois has shown that _when the dimensional ratio in (= length/ diameter) exceeds t00, Nm 2 =constant=45, and hence for long thin rods N = 45/ m2. (36) From an analysis of a number of experiments made with rods of different dimensions H. du Bois has deduced the corresponding mean demagnetizing factors. These, together with values of nt 2 N for cylindrical rods, and of N and m 2 N for ellipsoids of revolution, are given in the following useful table (loc. cit. p. 41): Demagnetizing Factors. In the middle part of a rod which has a length of 400 or 500 diameters the effect of the ends is insensible; but for many experiments the condition of endlessness may be best secured by giving the metal the shape of a ring of uniform section, the magnetic field being produced by an electric current through a coil of wire evenly wound round the ring. In such cases H i = o and H =Ho.
The residual magnetization I,. retained by a bar of ferromagnetic metal after it has been removed from the influence of an external field produces a demagnetizing force NI T, which is greater the smaller the dimensional ratio.. Hence the difficulty of imparting any considerable permanent magnetization to a short thick bar not possessed of great coercive force. The magnetization retained by a long thin rod, even when its coercive force is small, is sometimes little less than that which was produced by the direct action of the field.
Demagnetization by Reversals.-In the course of an experiment it is often desired to eliminate the effects of previous magnetization, and, as far as possible, wipe out the magnetic history of a specimen. In order to attain this result it was formerly the practice to raise the metal to a bright red heat, and allow it to cool while carefully guarded from magnetic influence. This operation, besides being very troublesome, was open to the objection that it was almost sure to produce a material but uncertain change in the physical constitution of the metal, so that, in fact, the results of experiments made before and after the treatment were not comparable. Ewing introduced the method (Phil. Trans. clxxvi. 539) of demagnetizing a specimen by subjecting it to a succession of magnetic forces which alternated in direction and gradually diminished in strength from a high value to zero. By means of a simple arrangement, which will be described farther on, this process can be carried out in a few seconds, and the metal can be brought as often as desired to a definite condition, which, if not quite identical with the virgin state, at least closely approximates to it.
Forces acting on a Small Body in the Magnetic Field.-If a small magnet of length ds and pole-strength m is brought into a magnetic field such that the values of the magnetic potential at the negative and positive poles respectively are V 1 and the work done upon the magnet, and therefore its potential energy, will be W =m(V2-Vi) =mdV, which may be written W =m d s- = M d v= - MHo = - vIHo, ds ds where M is the moment of the magnet, v the volume, I the magnetization, and Ho the magnetic force along ds. The small magnet may be a sphere rigidly magnetized in the direction of Ho; if this is replaced by an isotropic sphere inductively magnetized by the field, then, for a displacement so small that the magnetization of the sphere may be regarded as unchanged, we shall have dW = - vIdHo = v I+-, whence W = - 2 I + H2 ° (37) The mechanical force acting on the sphere in the direction of displacement x is 1 Hopkinson specified the retentiveness by the numerical value of the " residual induction " (=47rI).
dW F - d -v 1+ a 7rK dx dH (38) (34) [[[Magnetic Measurements]] If Ho is constant, the force will be zero; if Ho is variable, the sphere will tend to move in the direction in which Ho varies most rapidly. The coefficient K/(i +171-K) is positive for ferromagnetic and paramagnetic substances, which will therefore tend to move from weaker to stronger parts of the field; for all known diamagnetic substances it is negative, and these will tend to move from stronger to weaker parts. For small bodies other than spheres the coefficient will be different, but its sign will always be negative for diamagnetic substances and positive for others; hence the forces acting on any small body will be in the same directions as in the case of a sphere' Directing Couple acting on an Elongated Body. - In a non-uniform field every volume-element of the body tends to move towards regions of greater or less force according as the substance is paramagnetic or diamagnetic, and the behaviour of the whole mass will be determined chiefly by the tendency of its constituent elements. For this reason a thin bar suspended at its centre of gravity between a pair of magnetic poles will, if paramagnetic, set itself along the line joining the poles, where the field is strongest, and if diamagnetic, transversely to the line. These are the " axial " and " equatorial " positions of Faraday. It can be shown 3 that in a uniform field an elongated piece of any non-crystalline material is in stable equilibrium only when its length is parallel to the lines of force; for diamagnetic substances, however, the directing couple is exceedingly small, and it would hardly be possible to obtain a uniform field of sufficient strength to show the effect experimentally.
Relative Magnetization
A substance of which the real susceptibility is will, when surrounded by a medium having the susceptibility k', behave towards a magnet as if its susceptibility were - -}-4,rK'). Since i +47-K' can never be negative, the apparent susceptibility will be positive or negative according as is greater or less than Thus, for example, a tube containing a weak solution of an iron salt will appear to be diamagnetic if it is immersed in a stronger solution of iron, though in air it is paramagnetic.4 Circular Magnetization. - An electric current i flowing uniformly through a cylindrical wire whose radius is a produces inside the wire a magnetic field of which the lines of force are concentric circles around the axis of the wire. At a point whose distance from the axis of the wire is r the tangential magnetic force is H = 21r /a 2 (39) it therefore varies directly as the distance from the axis, where it is zero.' If the wire consists of a ferromagnetic metal, it will become " circularly magnetized by the field, the lines of magnetization being, like the lines of force, concentric circles. So long as the wire (supposed isotropic) is free from torsional stress, there will be no external evidence of magnetism.
Magnetic Shielding
The action of a hollow magnetized shell on a point inside it is always opposed to that of the external magnetizing force, 6 the resultant interior field being therefore weaker than the field outside. Hence any apparatus, such as a galvanometer, may be partially shielded from extraneous magnetic action by enclosing it in an iron case. If a hollow sphere 7 of which the outer radius is R and the inner radius r is placed in a uniform field Ho, the field inside will also be uniform and in the same direction as Ho, and its value will be approximately 3 i - R 3 For a cylinder placed with its axis at right angles to the lines of force, 2 = Ho (41) 2 +4(-2)(i - R2) These expressions show that the thicker the screen and the greater its permeability o, the more effectual will be the shielding action. Since p can never be infinite, complete shielding is not possible.
Magneto-Crystallic Phenomenon
In anisotropic bodies, such as crystals, the direction of the magnetization does not in general coincide with that of the magnetic force. There are, however, always three principal axes at right angles to one another along which the magnetization and the force have the same direction. If each of these axes successively is placed parallel to the lines of force in a uniform field H, we shall have = 12 = 13=K3H, the three susceptibilities being in general unequal, though in some cases two of them may have the same value. For crystalline bodies the value of or -) is nearly always small and constant, the magnetization being therefore independent of the form of the body and proportional to the force. Hence, whatever the position of the body, if the field be resolved into three components parallel to the 1 For all except ferromagnetic substances the coefficient is sensibly equal to See W. Thomson's Reprint, §§ 615, 634-651.
3 Ibid. §§ 646, 684.
4 Faraday, Exp. Res. xxi.
' J. J. Thomson, Electricity and Magnetism, § 205.
Maxwell, Electricity and Magnetism, § 7 H. du Bois, Electrician, 1898, 40, 317.
principal axes of the crystal, the actual magnetization will be the resultant of the three magnetizations along the axes. The body (or each element of it) will tend to set itself with its axis of greatest susceptibility parallel to the lines of force, while, if the field is not uniform, each volume-element will also tend to move towards places of greater or smaller force (according as the substance is paramagnetic or diamagnetic), the tendency being a maximum when the axis of greatest susceptibility is parallel to the field, and a minimum when it is perpendicular to it. The phenomena may therefore be exceedingly complicated.$ 3. Magnetic Measurements Magnetic Moment. - The moment M of a magnet may be determined in many ways,' the most accurate being that of C. F. Gauss, which gives the value not only of M, but also that of H, the horizontal component of the earth's force. The product MH is first determined by suspending the magnet horizontally, and causing it to vibrate in small arcs. If A is the moment of inertia of the magnet, and t the time of a complete vibration, MH = 41r 2 A/t 2 (torsion being neglected). The ratio M/H is then found by one of the magnetometric methods which in their simplest forms are described below. Equation (44) shows that as a first approximation.
M/H = (d 2 -1 2 ) tan 0/2d, where 1 is half the length of the magnet, which is placed in the " broadside-on " position as regards a small suspended magnetic needle, d the distance between the centre of the magnet and the needle, and 0 the angle through which the needle is deflected by the magnet. We get therefore M 2 = MH X M/H = 27r 2 A(d 2 - l 2) 2 tan 0/t 2 d (42) s H 2 =Mhxh/M = 87r 2 Ad/tt 2 (d 2 -1 2) 2 tan B} (43) When a high degree of accuracy is required, the experiments and calculations are less simple, and various corrections are applied. The moment of a magnet may also be deduced from a measurement of the couple exerted on the magnet by a uniform field H. Thus if the magnet is suspended horizontally by a fine wire, which, when the magnetic axis points north and south, is free from torsion, and if 0 is the angle through which the upper end of the wire must be twisted to make the magnet point east and west, then MH = CB, or M = C6/H, where C is the torsional couple for r 0. A bifilar suspension is sometimes used instead of a single wire. If P is the weight of the magnet, l the length of each of the two threads, 2a the distance between their upper points of attachment, and 2b that between the lower points, then, approximately, MH = P(ab/l) sin 0. It is often sufficient to find the ratio of the moment of one magnet to that of another. If two magnets having moments M, M' are arranged at right angles to each other upon a horizontal support which is free to rotate, their resultant R will set itself in the magnetic meridian. Let 0 be the angle which the standard magnet M makes with the meridian, then M'/R = sin 0, and M/R = cos 0, whence M' = M tan 0.
A convenient and rapid method of estimating a magnetic moment has been devised by H. Armagnat. 10 The magnet is laid on a table with its north pole pointing northwards. A compass having a very short needle is placed on the line which bisects the axis of the magnet at right angles, and is moved until a neutral point is found where the force due to the earth's field H is balanced by that due to the magnet. If 2l is the distance between the poles m and - in, d the distance from either pole to a point P on the line AB (fig. 5), we have for the resultant force at P R = - 2cosO X m/d' = - elm/d 3 = - M/d3. When P is the neutral point, H is equal and opposite to R; therefore M = Hd 3, or the moment is numerically equal to the cube of the distance from the neutral point to a pole, multiplied by the M. Faraday, Exp. Res. xxii., xxiii.; W. Thomson, Reprint, § 604; J. C. Maxwell, Treatise, § 435; E. Mascart and J. Joubert, Electricity and Magnetism, §§ 3 8 4, 39 6, 1226; A. Winkelmann, Physik, v. 287. See A. Winkelmann, Physik, v. 69-94; Mascart and Joubert, Electricity and Magnetism, ii. 617.
10 Sci. Abs. A, 1906, 9, 225.
H2 _ Ho (40) FIG. 5.
] or horizontal intensity of the earth's force. The distance between the poles may with sufficient accuracy for a rough determination be assumed to be equal to five-sixths of the length of the magnet.
Measurement of Magnetization and Induction
The magnetic condition assumed by a piece of ferromagnetic metal in different circumstances is determinable by various modes of experiment which may be classed as magnetometric, ballistic, and traction methods. When either the magnetization I or the induction B corresponding to a given magnetizing force H is known, the other may be found by means of the formula B = 41rI + H.
Magnetometric Methods. - Intensity of magnetization is most directly measured by observing the action which a magnetized body, generally a long straight rod, exerts upon a small magnetic needle placed near it. The magnetic needle may be cemented horizontally across the back of a little plane or concave mirror, about or $ in. in diameter, which is suspended by a single fibre of unspun silk; this arrangement, when enclosed in a case with a glazed front to protect it from currents of air, constitutes a simple but efficient magnetometer. Deflections of the suspended needle are indicated by the movement of a narrow beam of light which the mirror reflects from a lamp and focusses upon a graduated cardboard scale placed at a distance of a few feet; the angular deflection of the beam of light is, of course, twice that of the needle. The suspended needle is, in the absence of disturbing causes, directed solely by the horizontal component of the earth's field of magnetic force H E, and therefore sets itself approximately north and south. The magnetized body which is to be tested should be placed in such a position that the force H P due to its poles may, at the spot occupied by the suspended needle, act in a direction at right angles to that due to the earth - that is, east and west. The direction of the resultant field of force will then make, with that of H E, an angle 0, such that Hp/H E tan 0, and the suspended needle will be deflected through the same angle. We have therefore HP=HE tan 0. The angle B is indicated by the position of the spot of light upon the scale, and the horizontal intensity of the earth's field H E is known; thus we can at once determine the value of H P, from which the magnetization I of the body under test may be calculated.
In order to fulfil the requirement that the field which a magnetized rod produces at the magnetometer shall be at right angles to that of the earth, the rod may be conveniently placed in any one of three different positions with regard to the suspended needle.
(I) The rod is set in a horizontal position level with the suspended needle, its axis being in a line which is perpendicular to the magnetic meridian, and which passes through the centre of suspension of the needle. This is called the " end-on " position, and is indicated in fig. 6. AB is the rod and C the middle point of its axis; NS is the magnetometer needle; AM bisects the undeflected needle NS at right angles. Let 21=the length of the rod (or, more accurately, the distance between its poles), v= its volume, m and - m the strength of its poles, and let d= the distance CM. For most ordinary purposes the length of the needle may be assumed to be negligible in comparison with the distance between the needle and the rod. We then have approximately for the field at M due to the rod 4 HP= (dm1)2 (d + 1) = m (d 2 _ M Therefore 2m1= = (d2 - 12)2Ht, - (d2-12)2HEtan 0 2d 2d (44) M (d2-12)2HE (45) And I = v - 2dv tan 0, whence we can find the values of I which correspond to different angles of deflection.
(2) The rod may be placed horizontally east and west in such a position that the direction of the undeflected suspended needle bisects it at right angles. This is known as the " broadside-on " position, and is represented in fig. 7. Let the distance of each pole of the rod AB from the centre of the magnetometer needle = d. Then, since Hp, the force at M due to m and - m, is the resultant of d, and - m2, we have (L Hp m d d2 2ml H = d3, the direction being parallel to AB. d3H P = d3H And I = v tan B. v (3) In the third position the test rod is placed vertically with one of its poles at the level of the magnetometer needle, and in the line 7.
drawn perpendicularly to the undeflected needle from its centre of suspension. The arrangement is shown in fig. 8, where AB is the vertical rod and M indicates the position of the magnetometer needle, which is supposed to be perpendicular to the plane of the paper. Denoting the distance AM by d 1 , BM by d2, and AB by 1, we have for the force at M due to the magnetism of the rod H P =d 12 - horizontal component (dla - d 2 3). 3 tan 8, d l - I _ idlv j d23 d2/ I - 1d12H tan ©. This last method of arrangement is called by Ewing the " one-pole method, because the magnetometer deflection is mainly caused by the upper pole of the rod (Magnetic Induction, p. 40). For experiments with long thin rods or wires it has an advantage over the other arrangements in that the position of the poles need not be known with great accuracy, a small upward or downward displacement having little effect upon the magnetometer deflection. On the other hand, a vertically placed rod is subject to the inconvenience that it is influenced by the earth's magnetic field, which is not the case when the rod is horizontal and at right angles to the magnetic meridian. This extraneous influence may, however, be eliminated by surrounding the rod with a coil of wire carrying a current such as will produce in the interior a magnetic field equal and opposite to the vertical component of the earth's field.
If the cardboard scale upon which the beam of light is reflected by the magnetometer mirror is a flat one, the deflections as indicated by the movement of the spot of light are related to the actual deflections of the needle in the ratio of tan 20 to 0. Since 0 is always small, sufficiently accurate results may generally be obtained if we assume. that tan 20 =2 tan 0. If the distance of the mirror from the scale is equal to n scale divisions, and if a deflection 0 of the needle causes, the reflected spot of light to move over s scale divisions, we shall have s/n = tan 20 exactly, s/2n = tan 0 approximately.
We may therefore generally substitute s/2n for tan 0 in the various expressions which have been given for I.
Of the three methods which have been described, the first two are generally the most suitable for determining the moment or the magnetization of a permanent magnet, and the last for studying the changes which occur in the magnetization of a long rod or wire wl?E:n subjected to various external magnetic forces, or, in other words, for determining the relation of I to H. A plan of the apparatus as arranged by Ewing for the latter purpose is shown diagrammatically in fig. 9. The cardboard scale SS is placed above a wooden screen, having in it a narrow vertical slit which permits a beam of light from the lamp L to reach the mirror of the magnetometer M, whence it is reflected upon the scale. A is the upper end of a glass tube, half a metre or so in length, which is clamped in a vertical position. behind the magnetometer. The tube is wound over its whole lengthwith two separate coils of insulated wire, the one being outside the other. The inner coil is supplied, through the intervening apparatus, with current from the battery of secondary cells B,; this produces the desired magnetic field inside the tube. The outer coil derives current, through an adjustable resistance R, from a xvii. I I a (46) FIG. 8.
