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, asH,I,B, or a German capital, as ℌ, ℑ, ℬ.3Magnetic Poles and Magnetic Axis.—Aunit magnetic poleis 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 m1and m2are the strengths of two poles,dthe distance between them expressed in centimetres, and f the force in dynes,ƒ = m1m2/ d²(1).The force is one of attraction or repulsion, according as the sign of the product m1m2is negative or positive. The poles at the ends of an infinitely thin uniform magnet, ormagnetic 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 so-called “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 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 thenorthorpositive poleof 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 thesouthornegative 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 theaxis of the magnet.Magnetic Field.—Any space at every point of which there is a finite magnetic force is called afield of magnetic force, or amagnetic field. Thestrengthorintensityof a magnetic field at any point is measured by the force in dynes which a unit pole will experience when placed at that point, thedirectionof the field being the direction in which a positive pole is urged. The field-strength at any point is also called themagnetic forceat 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 letterH, or the German character ℌ. Magnetic force is sometimes, and perhaps more suitably, termedmagnetic intensity; it corresponds to the intensity of gravitygin the theory of heavy bodies (see Maxwell,Electricity and Magnetism, § 12 and § 68, footnote). Aline of forceis a line drawn through a magnetic field in the direction of the force at each point through which it passes. Auniform magnetic fieldis 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 distantrcentimetres from the wire isH = 2i / r,(2)i being the current in C.G.S. units.4The 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 isH = 2πi / 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 l centimetres is (except near the ends)H = 4πin / 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 2πr, and the field inside the coil may be expressed asH = 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 be 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,Mor ℳ, 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. Iflis 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. Theintensity of magnetization, or, more shortly, themagnetizationof a uniformly magnetized body is defined as the magnetic moment per unit of volume, and is denoted by I,I, or ℑ. HenceI = M/v = ml/v = m/a,v being the volume and a the sectional area. If the magnet is not uniform, 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 makesan angle ε with the normal, the normal component of the magnetization, I cos ε, is called thesurface densityof the magnetism, and is generally denoted by σ.Potential and Magnetic Force.—Themagnetic potentialat any point in a magnetic field is the work which would be done against the magnetic forces 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−δV= F, −δV= X, −δV= Y, −δV= Z.δnδxδyδz(6)Surfaces for which the potential is constant are calledequipotential 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 isV = 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′ isV = 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 θ and θ′ with the axis, it is easily shown that the equation to a line of force iscos θ − cos θ′ = constant.(9)Fig.2.Fig.3.At the point where a line of force intersects the perpendicular bisector of the axis r = r′ = r0, say, and cos θ − cos θ′ obviously =l/r0, l being the distance between the poles; l/r0is 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, isV = M cos θ/r²,(10)where θ 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 haveX = −δV=M(3 cos² θ − 1),δxr³Y = −δV=M(3 sin² θ cos θ).δyr³(11)If Fris the force along r and Ftthat along t at right angles to r,Fr= X cos θ + Y sin θ =M2 cos θ,r³(12)Ft= −X sin θ + Y cos θ =Msin θ.r³(13)For the resultant force at P,F = √ (Fr² + Ft²) =M√3 cos² θ + 1.r³(14)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, θ = 0; therefore cos θ = 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, θ = π/2 therefore cos θ = 0, and the point D is at an infinite distance. The magnitude of the force is in this caseFy= M / r³,(16)and its direction is parallel to the axis of the magnet. Although the above useful formulae, (10) 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.Fig.4.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 isMH sin θ,(17)where θ 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 θ and φ 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 φ, isMM′ (sin θ cos φ − 2 sin φ cos θ) / r³.(18)This vanishes if sin θ cos φ = 2 sin φ cos θ,i.e.if tan φ = ½ tan θ, 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′, θ = 0, and the couple becomes2MM′ sin φ/r³,(19)tending to diminish φ; this is called the “end on” position. If the centre of S′N′ is on the perpendicular bisector of SN, θ = ½π, and the couple will beMM′ cos φ/r³,(20)tending to increase φ; 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′ areX = 3MM′ (sin θ sin φ − 2 cos θ cos φ) / r4,(21)Y = 3MM′ (sin θ cos φ + sin φ cos θ) / r4.(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 asimple magnetic solenoid, and the product aI is called thestrengthof the solenoid. A magnet which consists entirely of such solenoids, having their ends either upon the surface or closed upon themselves, is called asolenoidal 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 amagnetic shell; the constant product hI is thestrengthof the shell and is generally denoted by Φ or φ. The potential at any point due to a magnetic shell is the product of its strength into the solid angle ω subtended by its edge at the given point, or V = Φω. 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 alamellar 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 isV =4⁄3πa³I cos θ/r²,(23)θ being the inclination of r to the magnetic axis. Since4⁄3πa³I is the moment of the sphere (= volume × 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, oppositeto the direction of magnetization, and equal to4⁄3πI. 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 to2⁄3a from the centre, and the magnet will behave externally like a simple solenoid of length4⁄3a. 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 =4⁄3πac²I = −4⁄3πac²FN.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.5According to F. W. G. Kohlrausch6the distance between the poles of a cylindrical magnet the length of which is from 10 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.7In 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 calledmagnetic 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 1 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 non-magnetic 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 ℬ) is briefly called theinduction, or, especially by electrical engineers, theflux-density. The direction of magnetic induction may be indicated bylines 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 atube of induction. The cross-section 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 intounit 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 1 − 4π, so thatI = (B − H) / 4π,(24)orB = H + 4πI.(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. Theresultant 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 Hi, that of the external field by H0, 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 themagnetizing force.Permeability and Susceptibility.—The ratio B/H is called thepermeabilityof the medium in which the induction is taking place, and is denoted by μ. The ratio I/H is called thesusceptibilityof the magnetized substance, and is denoted by κ. HenceB = μH and I = κH.(26)Alsoμ =B=H + 4πI= 1 + 4πκ,HH(27)andκ =μ − 14π(28)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 1. The permeability of most material substances differs very slightly from unity, being a little greater than 1 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 κ expresses the numerical relation of the magnetization to the magnetizing force. From the equation κ = (μ − 1)/4π, it follows that the magnetic susceptibility of a vacuum (where μ = l) is 0, that of a diamagnetic substance (where μ < l) has a negative value, while the susceptibility of paramagnetic and ferromagnetic substances (for which μ > 1) is positive. No substance has yet been discovered having a negative susceptibility sufficiently great to render the permeability (= 1 + 4πκ) 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 magneto-motive force, to the electromotive force in the circuit. The principal points of difference are that (1) 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 seeElectro-Magnetism.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 ofhysteresis(Gr.ὑστερέω, 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 theresidual magnetization. The ratio of the residual magnetization to its previous maximum value measures theretentiveness, orretentivity, of the metal.8Steel, 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). Thecoercive force, orcoercivity, 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 H0, due to the external field, and of another force, Hi, arising from the induced magnetization of the body. Since Higenerally tends to oppose the external force, thus making H less than H0, it may be called thedemagnetizing 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 elongatedellipsoid of revolutionplaced in a uniform field H0, with its axis of revolution parallel to the lines of force. The magnetization at any point inside the ellipsoid will then beI =κH01 + κN(29)whereN = 4π(1− 1) (1log1 + e− 1),e22e1 − ee being the eccentricity (see Maxwell’sTreatise, § 438). Since I = κH, we haveκH + κNI = κH0,(30)orH = H0− NI,NI being the demagnetizing force Hi. N may be called, after H. du Bois (Magnetic Circuit, p. 33), thedemagnetizing factor, and the ratio of the length of the ellipsoid 2c to its equatorial diameter 2a (= c/a), thedimensional ratio, denoted by the symbolm.Sincee =√(1 −a²)=√(1 −1),c²m²the above expression for N may be writtenN =4π(mlogm+ √(m² − 1)− 1)m² − 12√(m² − 1)m− √(m² − 1)=4π{mlog(m+ √(m² − 1))− 1},m² − 1√(m² − 1)from which the value of N for a given dimensional ratio can be calculated. When the ellipsoid is so much elongated that 1 is negligible in relation tom², the expression approximates to the simpler formN =4π(log 2m− 1)m²(31)In the case of asphere, e = O and N =4⁄3π; therefore from (29)I = κH =κH0=3κH0,1 +4⁄3πκ3 + 4πκ(32)WhenceH =3H0=3H0,3 + 4πκμ + 2(33)andB = μH =3μH0.μ + 2(34)Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of μ 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 acylindrical rodplaced longitudinally in a uniform field H0is not uniform, being greatest at the ends and least in the middle part. Denoting its mean value byHi, and that of the demagnetizing factor byN, we haveH = H0−Hi= H0−NI.(35)Du Bois has shown that when the dimensional ratio m (= length/diameter) exceeds 100,Nm² = constant = 45, and hence for long thin rodsN= 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 ofm²Nfor cylindrical rods, and of N andm²N for ellipsoids of revolution, are given in the following useful table (loc. cit.p. 41):—Demagnetizing Factors.m.Cylinder.Ellipsoid.N.m²N.N.m²N.012.5664012.566400.5——6.5864—1——4.1888—5——0.7015—100.216021.60.254925.5150.120627.10.135030.5200.077531.00.084834.0250.053333.40.057936.2300.039335.40.043238.8400.023838.70.026642.5500.016240.50.018145.3600.011842.40.013247.5700.008943.70.010149.5800.006944.40.008051.2900.005544.80.006552.51000.004545.00.005454.01500.002045.00.002658.32000.001145.00.001664.03000.0005045.00.0007567.54000.0002845.00.0004572.05000.0001845.00.0003075.010000.0000545.00.0000880.0In 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 Hi= 0 and H = H0.The residual magnetization Irretained by a bar of ferromagnetic metal after it has been removed from the influence of an external field produces a demagnetizing forceNIr, 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 V1and V2, the work done upon the magnet, and therefore its potential energy, will beW = m (V2− V1) = m dV,which may be writtenW = m dsdV= MdV= −MH0= −vIH0,dsdswhere M is the moment of the magnet, v the volume, I the magnetization, and H0the magnetic force along ds. The small magnet may be a sphere rigidly magnetized in the direction of H0; 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 havedW = −vI dH0= −vκH0dH0;1 +4⁄3πκwhenceW = −vκH²0.21 +4⁄3πκ(37)The mechanical force acting on the sphere in the direction of displacement x isF = −dW= vκdH²0.dx1 +4⁄3πκdx(38)If H0is constant, the force will be zero; if H0is variable, the sphere will tend to move in the direction in which H0varies most rapidly. The coefficient κ / (1 +4⁄3πκ) 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;9hence the forces acting on any small body will be in the same directions as in the case of a sphere.10Directing 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 shown11that 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 κ will, when surrounded by a medium having the susceptibility κ′, behave towards a magnet as if its susceptibility were κa= (κ − κ′) / (1 + 4πκ′). Since 1 + 4πκ′ can never be negative, the apparent susceptibility κawill be positive or negative according as κ is greater or less than κ′. 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.12Circular 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 isH = 2ir / a²(39)it therefore varies directly as the distance from the axis, where it is zero.13If 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,14the 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 sphere15of which the outer radius is R and the inner radius r is placed in a uniform field H0, the field inside will also be uniform and in the same direction as H0, and its value will be approximatelyHi=H0.1 +2⁄9(μ − 2) (1 − r³/R³)(40)For a cylinder placed with its axis at right angles to the lines of force,Hi=H0.1 +1⁄4(μ − 2) (1 − r²/R²)(41)These expressions show that the thicker the screen and the greater its permeability μ, the more effectual will be the shielding action. Since μ 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 threeprincipal axesat 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 haveI1= κ1H, I2= κ2H, I3= κ3H,the three susceptibilities κ being in general unequal, though in some cases two of them may have the same value. For crystalline bodies the value of κ (+ 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 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.16
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, asH,I,B, or a German capital, as ℌ, ℑ, ℬ.3
Magnetic Poles and Magnetic Axis.—Aunit magnetic poleis 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 m1and m2are the strengths of two poles,dthe distance between them expressed in centimetres, and f the force in dynes,
ƒ = m1m2/ d²
(1).
The force is one of attraction or repulsion, according as the sign of the product m1m2is negative or positive. The poles at the ends of an infinitely thin uniform magnet, ormagnetic 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 so-called “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 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 thenorthorpositive poleof 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 thesouthornegative 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 theaxis of the magnet.
Magnetic Field.—Any space at every point of which there is a finite magnetic force is called afield of magnetic force, or amagnetic field. Thestrengthorintensityof a magnetic field at any point is measured by the force in dynes which a unit pole will experience when placed at that point, thedirectionof the field being the direction in which a positive pole is urged. The field-strength at any point is also called themagnetic forceat 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 letterH, or the German character ℌ. Magnetic force is sometimes, and perhaps more suitably, termedmagnetic intensity; it corresponds to the intensity of gravitygin the theory of heavy bodies (see Maxwell,Electricity and Magnetism, § 12 and § 68, footnote). Aline of forceis a line drawn through a magnetic field in the direction of the force at each point through which it passes. Auniform magnetic fieldis 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 distantrcentimetres from the wire is
H = 2i / r,
(2)
i being the current in C.G.S. units.4The 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 = 2πi / 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 l centimetres is (except near the ends)
H = 4πin / 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 2πr, 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 be 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,Mor ℳ, 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. Iflis 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. Theintensity of magnetization, or, more shortly, themagnetizationof a uniformly magnetized body is defined as the magnetic moment per unit of volume, and is denoted by I,I, or ℑ. Hence
I = M/v = ml/v = m/a,
v being the volume and a the sectional area. If the magnet is not uniform, 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 makesan angle ε with the normal, the normal component of the magnetization, I cos ε, is called thesurface densityof the magnetism, and is generally denoted by σ.
Potential and Magnetic Force.—Themagnetic potentialat any point in a magnetic field is the work which would be done against the magnetic forces 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
(6)
Surfaces for which the potential is constant are calledequipotential 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 θ and θ′ with the axis, it is easily shown that the equation to a line of force is
cos θ − cos θ′ = constant.
(9)
At the point where a line of force intersects the perpendicular bisector of the axis r = r′ = r0, say, and cos θ − cos θ′ obviously =l/r0, l being the distance between the poles; l/r0is 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 θ/r²,
(10)
where θ 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
(11)
If Fris the force along r and Ftthat along t at right angles to r,
(12)
(13)
For the resultant force at P,
(14)
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, θ = 0; therefore cos θ = 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, θ = π/2 therefore cos θ = 0, and the point D is at an infinite distance. The magnitude of the force is in this case
Fy= M / r³,
(16)
and its direction is parallel to the axis of the magnet. Although the above useful formulae, (10) 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 θ,
(17)
where θ 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 θ and φ 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 φ, is
MM′ (sin θ cos φ − 2 sin φ cos θ) / r³.
(18)
This vanishes if sin θ cos φ = 2 sin φ cos θ,i.e.if tan φ = ½ tan θ, 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′, θ = 0, and the couple becomes
2MM′ sin φ/r³,
(19)
tending to diminish φ; this is called the “end on” position. If the centre of S′N′ is on the perpendicular bisector of SN, θ = ½π, and the couple will be
MM′ cos φ/r³,
(20)
tending to increase φ; 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 θ sin φ − 2 cos θ cos φ) / r4,
(21)
Y = 3MM′ (sin θ cos φ + sin φ cos θ) / r4.
(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 asimple magnetic solenoid, and the product aI is called thestrengthof the solenoid. A magnet which consists entirely of such solenoids, having their ends either upon the surface or closed upon themselves, is called asolenoidal 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 amagnetic shell; the constant product hI is thestrengthof the shell and is generally denoted by Φ or φ. The potential at any point due to a magnetic shell is the product of its strength into the solid angle ω subtended by its edge at the given point, or V = Φω. 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 alamellar 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 =4⁄3πa³I cos θ/r²,
(23)
θ being the inclination of r to the magnetic axis. Since4⁄3πa³I is the moment of the sphere (= volume × 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, oppositeto the direction of magnetization, and equal to4⁄3πI. 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 to2⁄3a from the centre, and the magnet will behave externally like a simple solenoid of length4⁄3a. 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 =4⁄3πac²I = −4⁄3πac²FN.
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.5According to F. W. G. Kohlrausch6the distance between the poles of a cylindrical magnet the length of which is from 10 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.7In 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 calledmagnetic 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 1 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 non-magnetic 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 ℬ) is briefly called theinduction, or, especially by electrical engineers, theflux-density. The direction of magnetic induction may be indicated bylines 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 atube of induction. The cross-section 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 intounit 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 1 − 4π, so that
I = (B − H) / 4π,
(24)
or
B = H + 4πI.
(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. Theresultant 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 Hi, that of the external field by H0, 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 themagnetizing force.
Permeability and Susceptibility.—The ratio B/H is called thepermeabilityof the medium in which the induction is taking place, and is denoted by μ. The ratio I/H is called thesusceptibilityof the magnetized substance, and is denoted by κ. Hence
B = μH and I = κH.
(26)
Also
(27)
and
(28)
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 1. The permeability of most material substances differs very slightly from unity, being a little greater than 1 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 κ expresses the numerical relation of the magnetization to the magnetizing force. From the equation κ = (μ − 1)/4π, it follows that the magnetic susceptibility of a vacuum (where μ = l) is 0, that of a diamagnetic substance (where μ < l) has a negative value, while the susceptibility of paramagnetic and ferromagnetic substances (for which μ > 1) is positive. No substance has yet been discovered having a negative susceptibility sufficiently great to render the permeability (= 1 + 4πκ) 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 magneto-motive force, to the electromotive force in the circuit. The principal points of difference are that (1) 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 seeElectro-Magnetism.
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 ofhysteresis(Gr.ὑστερέω, 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 theresidual magnetization. The ratio of the residual magnetization to its previous maximum value measures theretentiveness, orretentivity, of the metal.8Steel, 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). Thecoercive force, orcoercivity, 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 H0, due to the external field, and of another force, Hi, arising from the induced magnetization of the body. Since Higenerally tends to oppose the external force, thus making H less than H0, it may be called thedemagnetizing 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 elongatedellipsoid of revolutionplaced in a uniform field H0, with its axis of revolution parallel to the lines of force. The magnetization at any point inside the ellipsoid will then be
(29)
where
e being the eccentricity (see Maxwell’sTreatise, § 438). Since I = κH, we have
κH + κNI = κH0,
(30)
or
H = H0− NI,
NI being the demagnetizing force Hi. N may be called, after H. du Bois (Magnetic Circuit, p. 33), thedemagnetizing factor, and the ratio of the length of the ellipsoid 2c to its equatorial diameter 2a (= c/a), thedimensional ratio, denoted by the symbolm.
Since
the above expression for N may be written
from which the value of N for a given dimensional ratio can be calculated. When the ellipsoid is so much elongated that 1 is negligible in relation tom², the expression approximates to the simpler form
(31)
In the case of asphere, e = O and N =4⁄3π; therefore from (29)
(32)
Whence
(33)
and
(34)
Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of μ 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 acylindrical rodplaced longitudinally in a uniform field H0is not uniform, being greatest at the ends and least in the middle part. Denoting its mean value byHi, and that of the demagnetizing factor byN, we have
H = H0−Hi= H0−NI.
(35)
Du Bois has shown that when the dimensional ratio m (= length/diameter) exceeds 100,Nm² = 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 ofm²Nfor cylindrical rods, and of N andm²N 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 Hi= 0 and H = H0.
The residual magnetization Irretained by a bar of ferromagnetic metal after it has been removed from the influence of an external field produces a demagnetizing forceNIr, 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 V1and V2, the work done upon the magnet, and therefore its potential energy, will be
W = m (V2− V1) = m dV,
which may be written
where M is the moment of the magnet, v the volume, I the magnetization, and H0the magnetic force along ds. The small magnet may be a sphere rigidly magnetized in the direction of H0; 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
whence
(37)
The mechanical force acting on the sphere in the direction of displacement x is
(38)
If H0is constant, the force will be zero; if H0is variable, the sphere will tend to move in the direction in which H0varies most rapidly. The coefficient κ / (1 +4⁄3πκ) 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;9hence the forces acting on any small body will be in the same directions as in the case of a sphere.10
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 shown11that 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 κ will, when surrounded by a medium having the susceptibility κ′, behave towards a magnet as if its susceptibility were κa= (κ − κ′) / (1 + 4πκ′). Since 1 + 4πκ′ can never be negative, the apparent susceptibility κawill be positive or negative according as κ is greater or less than κ′. 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.12
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 = 2ir / a²
(39)
it therefore varies directly as the distance from the axis, where it is zero.13If 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,14the 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 sphere15of which the outer radius is R and the inner radius r is placed in a uniform field H0, the field inside will also be uniform and in the same direction as H0, and its value will be approximately
(40)
For a cylinder placed with its axis at right angles to the lines of force,
(41)
These expressions show that the thicker the screen and the greater its permeability μ, the more effectual will be the shielding action. Since μ 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 threeprincipal axesat 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
I1= κ1H, I2= κ2H, I3= κ3H,
the three susceptibilities κ being in general unequal, though in some cases two of them may have the same value. For crystalline bodies the value of κ (+ 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 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.16
3.Magnetic Measurements
Magnetic Moment.—The moment M of a magnet may be determined in many ways,17the 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 = 4π²A / t² (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 = (d² − l²) tan θ/2d,
where l 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 θ the angle through which the needle is deflected by the magnet. We get therefore
M² = MH × M/H = 2π²A (d² − l²)² tan θ/t²d