PART V.-MAGNETIC INDUCTION

CHAPTER XI

MAGNETIC INDUCTION

501. Intensity of Magnetisation. We have already referred, in § 417, to the fact that a piece of iron when placed in a magnetic field becomes magnetised by induction, and we now have to investigate this phenomenon of induced magnetism in more detail.

We have defined the magnetic force at a given point in air as the force in dynes which would act on a unit pole placed at the point, and we have seen how the direction of the force which would act on a north pole when placed anywhere in the air surrounding a magnet may be mapped out by means of magnetic lines of force. We also found that in the air surrounding the magnet these lines of force ran from the north pole of the magnet to the south pole. In the case of electro-static lines of force, since there is no force exerted within a closed conductor, we did not have to consider the forms of the lines of force within a conducting body, so that a line of force originated at a positively electrified conductor and ended at the surface of a negatively electrified body. In the case of the magnetic lines of force due to an electric current (§ 472) we, however, found that they consisted of closed curves, that is, each line of force is continuous, and has neither beginning nor end. We are thus led to consider whether the lines of magnetic force due to magnets are like lines of electro-static force, originating at a north pole and ending up at a south pole, or whether they are like the magnetic lines of force of a current and are continuous curves. Suppose that a long thin steel rod is magnetised uniformly, then there will be a pole at each end and magnetic lines of force will run, in the surrounding air, from the north pole to the south pole. Now suppose that the magnet is bent into the form of a circle, the two poles being brought into contact with one another. Before the poles were brought into contact there were a number of lines of force passing through the air, but when the poles are in contact practically no lines of force pass through the air. The magnet, however, is still magnetised, for if the poles are separated, or if it is cut at any other place and the ends are separated, lines of force will at once pass through the air. We are thus led to conclude that the magnetic

lines of force exist even when they do not pass through the air, and that when the poles are in contact the lines run round the steel ring thus formed. When the poles are separated the lines of force still form continuous lines, the direction in the air being from the north pole to the south, while for the remainder of their course they run in the substance of the steel, the direction here being from the south pole to the north.

Just as in the case of electro-static lines of force we were able by means of the consideration of tubes of force to represent the strength of the electro-static field at any point, so by the consideration of tubes of magnetic force we can represent the strength of the magnetic field at every point. A unit magnetic tube of force is bounded by lines of magnetic force, and is such that the product of the magnetic force at any point of the tube into the cross-section of the tube at that point is equal to unity.

Attention must be paid to the difference between the methods in which the electro-static and the magnetic unit tubes are defined. Since ́ the electro-static tube starts and ends at given points, we are able to define the unit tube by the quantity of electrification at the ends. Thus defined, we have seen that the product of the electric force at a point in a tube into the cross-section of the tube at that point is equal to 4π. In the magnetic case, on the other hand, the tubes being endless, we have to adopt another method of defining the unit tube, namely, that the product of the magnetic force into the cross-section should be constant. The constant usually adopted being, however, not 4′′, but unity. Greater uniformity, no doubt, would be secured by altering the definition of the electro-static unit tube so as to conform to the definition of the magnetic tube in the manner given at the end of § 455. Usage having decreed that the definitions we have adopted should be used, it seems hopeless to attempt to make any change. The difference in the definitions will account for the frequent occurrence of the quantity 47 in formulæ dealing with one kind of tube, but not in the corresponding formula dealing with the other kind of tube.

If a long thin magnet is taken of which the pole strength is m and a sphere of radius r, where is small, be described with the pole as centre, the force at any point on this sphere is m/r2, since the other pole is at a great distance, so that it exerts no appreciable force. If a is the crosssection of a tube of force at the surface of the sphere, then by the definition of the unit tube we have

Hence, since the area of the surface of a sphere of radius r is 412, the number of tubes of force which cross the surface of the sphere is

or

4m; that is, there are 4

a

tubes of force which leave the north pole of

a magnet, of which the pole strength is m. Since all these tubes of force return to the north pole through the substance of the iron, there will be 4 tubes of force passing through the magnet, these tubes of force being all due to the magnetism of the magnet itself.

If we accept the hypothesis of molecular magnets (§ 420), the strength of the pole " either measures the strength of the molecular magnets, or, if we assume that they are of constant strength, the proportion of them which are turned with their axes in one direction. Hence for a given magnet the degree to which the steel is magnetised will depend on the number of tubes of force which pass through the substance of the magnet, that is, on the closeness of the packing of the tubes. This closeness depends on the strength of the poles of the magnet and also on the cross-section of the magnet. If then s is the cross-section of the magnet, and is the strength of either of the poles, the degree to which the steel is magnetised is measured by 4m/s. Now if / is the length of the magnet, and its magnetic moment is M, we have m=M. Hence the degree of magnetisation of the steel is measured by 4πMils, or since, if I' is the volume of M the magnet, V=Is by 4π Thus the degree of magnetisation of the V steel is proportional to the quotient of the magnetic moment by the volume, and this quantity is called the intensity of magnetisation of the steel. That the value of this quantity does not depend on the form of the magnet can be seen by the following considerations. If a magnet were taken of length 72 and cross-section 25, so that the volume was the same as before, and were magnetised so that its moment was 1, the strength of each pole would be M'27, that is, 2m, and the number of tubes of force passing through the magnet would be 4′′ × 2, and the number of tubes of force passing through unit area of the cross-section would be Sam/25 or 4mm/s, which is the same number as obtained before. Since the moment and the volume are the same as before, the intensity of magnetisation is the same as before, and we have just seen that the closeness of the tubes of force is the same.

The number of tubes of force which pass through the magnet is 4, and the number of tubes per unit area of cross-section is 47m s, but the number of tubes per unit area of the cross-section is also equal to 4–1. Hence I=m's. The quantity m's, which is the pole strength per unit area, is called the surface density of the magnetisation, and from the above relation it is seen that this quantity is numerically equal to the intensity of magnetisation, I. Thus if a disc of iron, of which the area of the face is S, is magnetised transversely, the intensity of magnetisation being I, then the pole strength of either side will be IS.

502. Magnetic Induction.-We have seen that when a piece of iron is placed in a magnetic field it becomes magnetised owing to induction, and we have now to consider how the intensity of the induced magnetism depends on the conditions under which the induction takes place.

In any portion of a magnetic field which is filled with non-magnetic medium, all the tubes of force are due to causes (currents, magnets, &c.) which are external to the portion of the medium considered. If we assume, and we shall see later to what extent this assumption is justified, that in a magnetic medium the molecules of the medium are already magnetised, and that the act of magnetising any given portion of the medium consists of turning these molecular magnets in a given direction, then when such a medium is unmagnetised the molecular magnets will be turned in all directions. Each molecular magnet will have its tubes of force, just as a large magnet, but in the unmagnetised state these tubes of force will be turned in all directions, so that on the whole there will be as many tubes passing through any element of area, taken in the medium, in one direction as in the opposite direction. Where, however, a magnetic medium is placed in a magnetic field a certain proportion of the molecular magnets will be turned in the direction of the lines of force of the field, so that their tubes of force all point in the same direction. This effect is shown in Fig. 407, which represents the result of sprinkling iron filings over a sheet of glass on which were placed a number of small magnets, the axes of which were arranged irregularly in all directions. In Fig. 408 the corresponding figure is shown for the same magnets, but here they have been all arranged with their axes pointing in one direction. It will be observed that now there are lines of force extending to some distance from the group of magnets.

Thus within a magnetisable medium, which is placed in a magnetic field, we have to do with two sets of tubes of force-(1) those which are due to the magnetising field, and which would exist if the magnetic medium were replaced by a non-magnetic medium; (2) those due to the magnetism of the molecules of the medium itself.

Suppose that a long unmagnetised cylindrical bar of soft iron of cross-section s is placed in a uniform magnetic field of strength H, with its length parallel to the lines of force of the field. Then if the cylinder were of an unmagnetisable material, the number of tubes of force due to the field which would cross the cross-section of the cylinder is sH. Owing, however, to the fact that the cylinder becomes magnetised by induction, there will be in addition a certain number of tubes of force both within the cylinder and in the air outside, due to this induced magnetism.

Owing to the induction, poles will be induced at the ends of the iron and these poles will in general produce a force within the material of the iron which will tend to diminish the strength of the inducing field. This disturbing action of the induced poles causes a considerable complication in the consideration of the problem, and so we shall at first consider the case of a very long cylinder of comparatively small crosssection. In this case, if we confine ourselves to a consideration of the state of the iron near the middle of the cylinder, the influence of the

poles, which are by supposition at a considerable distance, may be neglected.

If m is the strength of the poles induced in the iron, then, as we have seen in the last section, there will be 47m lines of force due to this induced magnetism. Since that end of the cylinder which points in the direction in which the lines of force of the field run becomes a north pole, the induced lines of force will run in the air in the opposite direction to the lines of force of the field, but within the iron they will run in the same direction as the lines of force of the field. The number of tubes which pass through the iron is therefore sH tubes, due to the inducing field, and 47m tubes, due to the induced magnetisation, er sH+4m in all. Hence the number of tubes of force which cross unit area of the cross-section of the iron is H+4m's. But the intensity of magnetisation of the iron is equal to m/s, for the lines of force due to the field alone have nothing to do with the magnetism of the material, and in fact remain the same whatever the nature of the material of which the cylinder is composed. Hence if I is the intensity of the magnetism induced in the iron, the number of lines of force which cross unit area of the cross-section of the cylinder is H+41. This quantity is called the induction, B, in the iron, so that

B=H+4πI.

We have in the above spoken of the tubes of force due to the magnetising field, and to the induced magnetism which is induced in the iron, and we have defined the induction as the number of tubes of force which cross unit area at right angles to the tubes. Now although in the example we have taken we have for simplicity supposed that the cylinder of iron was placed with its length parallel to the lines of force of the magnetising field, and was entirely magnetised by induction, so that inside the iron the lines of force due to the field and those due to the induced magnetism were parallel, this is not always so. Thus if the cylinder had been placed with its length inclined to the lines of force of the magnetising field, and been permanently magnetised, the lines of force within the iron due to the permanent magnetism of the iron would not be parallel to those due to the field.

Since in the iron we have always to do with the resultant of the two sets of tubes, that is, with the induction, it is usual to speak of the tubes of induction within any magnetisable material. Thus a line of induction is a curve drawn so that it everywhere indicates the direction of the induction, that is, of the resultant of the field causing the induced magnetisation and that of the magnetisation, both permanent and induced, of the material. A tube of induction is a portion of space bounded by lines of induction, and such that the induction across every cross-section of the tube is equal to unity.

If there are no magnetisable materials in the field, then there will be