We shall later on become acquainted with very simple methods of superposing two or more magnetic fields. The magnetic condition is determined by the total number of lines of force per unit of cross-sectional area, the important magnitude thus measured being called the magnetic induction.' To show its significance as clearly as possible, we shall make use of a simple example. Let there be a toroid, uniformly magnetised with intensity I, and let this toroid be cut through at one place, so as to leave two free ends separated by an air-gap. We shall suppose the gap to be so narrow as not to disturb the uniformity of magnetisation. Then in accordance with § 99, there will be 473 lines of force passing across each square centimetre of the gap, and these continue their course to the same number through the substance of the toroid. Upon this system of lines of force we have to superpose whatever system may be due to external causes. In the latter let there be lines of force per cm.2, so that is the strength of the external field in absolute measure. Then within the air gap there will be 473+ lines of force per cm.2, and the same lines continue their course uninterrupted through the substance of the toroid. The magnetic condition is characterised by the magnetic induction Since all substances placed in a magnetic field become magnetised to some intensity I, we see that B is always to be distinguished from 5; when we have confined our attention to the strength of field H, we have tacitly assumed J=0; this being justified by the small magnetisability of air, with which medium we were usually concerned. 107. Dimensions of the magnetic induction.-Since the numerical factor 4π is without dimensions, it follows that B is the sum of two quantities, each of which has the dimensions cm. gr. sec.1 ($100 and $70). In other words, magnetic induction, intensity of magnetisation, and strength of field are all of the same dimensions. When we write B=4πJ+H, we assume that the lines of force arising from I are coincident in direction with those of the magnetic field due to external causes. If instead the two systems of lines ran in opposite directions, 5 would have to be subtracted. The induction B is a directed quantity like the field-intensity ; that is, it has a determinate direction as well as a determinate magnitude. 108. Tubes of induction and flux of induction.-A line of induction is one which at each point satisfies the condition that its tangent is in the same direction as the magnetic induction. As in the case of lines of force, the number of lines of induction which cut (normally) through unit area, measures the strength of the induction. A space enclosed by a tubular surface, made up entirely of lines of induction, is called a tube of induction. There is also a quantity analogous to the flux of force, namely, the flux of induction I, which for each tube is the product of the induction B and the cross-section w of the tube; that is, I= Bw. Since, in accordance with § 106, it is always with the induction that we are concerned, it is really only tubes of induction that can be drawn, and all the diagrams of Chapter V. are essentially diagrams of the flux of induction. It is only when we are dealing with a medium for which B- that we can identify the flux of induction with the flux of force; but since this condition is very nearly fulfilled in the case of such a feebly magnetisable substance as air, we were able to speak of our diagrams as relating to the flux of force. Throughout every tube of induction, the flux of induction I is always absolutely constant. = 109. Measure of magnetic permeability.-The magnetic induction B furnishes a convenient means of expressing numerically the value of the magnetic permeability. Since B denotes the total number of lines of force which cut through each unit of cross-sectional area within a body placed in a magnetic field, the ratio "=B/H (8) furnishes a measure of the extent to which the body has collected together the lines of force of the field H, and caused them to pass more thickly through it. This effect of the body we attribute to its greater permeability. It is accordingly to be measured by the ratio of the actual number of lines of force B per unit area, to the number 5 which would have been present in the absence of the body from the field. Since Band H are of the same dimensions, the permeability is without dimensions, in other words it is a pure number. If we put μ=1 for air (the medium in which we generally work), then μ for all paramagnetic substances μ>1, for all diamagnetic substances μ‹ 1. For good soft wrought iron μ may reach as high a value as 2,000 or 3,000, while for bismuth μ=0.99982. In a medium whose permeability is unity, the flux of induction becomes identical with the flux of force, and the tubes of induction with the tubes of force. 110. Susceptibility. In the last paragraph we considered the ratio of the induction to the field-intensity; but for some purposes it is more convenient to measure the ratio of the intensity of magnetisation I to the field-intensity H. This latter ratio is called the magnetic susceptibility κ, so that K=I/H. (9) In accordance with the assumption which we have made, K=0 for air, while for paramagnetic substances x is positive (>0), In the case of soft iron, & lies between 200 and 300. Since I and are of the same dimensions, x is like μ a pure number. 111. Relation between permeability and susceptibility. Since the permeability μ=B/H, the susceptibility = 3/5, and B-473+, we have = The magnetic character of a substance is determined by either of the quantities μ or κ, and the last written equations enable us, when one is given, to determine the other. The two quantities μ and are not in general constant for any given substance, but are functions of the field intensity , so that the permeability and susceptibility of a substance are different according as it is placed in a weak field or a strong one. D.-The fundamental property of the magnetic induction 112. Magnetic circuits. When we defined the magnetic induction by the relation B=4πI+H, we were dealing with a definite position in the field, where the magnetic conditions due respectively to the internal lines of force and the external field have to be superposed. In many cases, however, especially in those which are of the greatest practical importance, it is more convenient to consider as a whole the system of tubes of induction, which bend round so as to form closed circuits. This is possible, since we know of a quantity which remains constant along a tube of induction, and whose value is characteristic of the tube. This quantity is the flux of induction I=Bw. The lines of induction of a magnet pass through the interior as well as through the surrounding field, any bundle of the lines which we may consider separately passing from the surrounding space into the magnet and out again into the surrounding space without any change in their number. In a medium of permeability 1 we found that the flux of force along each tube was constant so long as no sources and sinks were encountered. The constancy of the flux of induction along each tube of induction is absolute, and holds good under all conditions. It is the law of the conservation of the flux of induction with which we are concerned when the field extends through media for which the permeability μ has different values, so that at the surfaces which separate such media from one another there are sources and sinks of lines of force. When tubes of induction pass from a medium of small permeability (such as air) into one of higher permeability (such as iron or steel) they become more closely crowded together, so that the induction B attains a higher value; when, leaving this medium, they enter one which is feebly paramagnetic or is diamagnetic, they diverge again correspondingly. Each tube of induction, taken in its entirety, is an annular portion of space, possessing the property that through every cross-section of it the flux of induction I has the same value. A collection of tubes of induction lying side by side, and occupying a finite portion of space, constitutes what is called a magnetic circuit.' A horse-shoe magnet with its keeper attached, closed or divided toroids, all furnish examples of magnetic circuits. In the lastmentioned case the lines of force which traverse the air-gap are continued through the substance of the toroid, where the high value of the induction is largely due to the high magnetisation. As other examples of magnetic circuits may be mentioned the system of magnets in a dynamo machine and the iron cores of transformers. 113. Magneto-motive force.—This quantity may be taken as a measure of the total magnetising influence along a closed tube of induction. If at a place where the fieldintensity is units, we move the unit pole through a distance s in the direction of the magnetic force, the work done by magnetic actions upon the pole will be s. If the unit pole is moved through a further distance along the tube of induction, a further amount of work will be done, and this must be added to the former amount. If the pole has accomplished a complete circuit of the tube, returning once more to its starting point, the total work done is a quantity characteristic of the tube of induction. It has been called the magneto-motive force,' M, and is a quantity of work referred to unit pole, that is it has the dimensions (work strength of pole) or M has the dimensions: [L] [M] [T]'[L]÷[L] [M]' [T]'=[L]' [M]' [T]-1 |