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CHAPTER I

INTRODUCTION

§ 1. The Electromagnetic Field. The electric current, or, more generally, electricity in motion, is the only known source of any kind of magnetism, and more particularly also of terrestrial magnetism,' as may, with great probability, be assumed. Magnetic iron ore and other bodies occurring in nature in the magnetic condition manifestly owe their magnetism to that of the earth, or, in some cases, no doubt, to the direct action of electrical discharges.'1

We start, therefore, by taking the fact as known, that a conductor along which a current is passing produces in its vicinity a peculiar condition which is called an electromagnetic, or, more briefly, a magnetic field. The air which in the ordinary conditions of experiment occupies this space plays only a very subordinate part, to which we shall afterwards refer (§ 7). In the phenomena to be subsequently described we shall assume that they take place in a vacuum.

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The condition in question manifests itself among other things by the fact that, on the one hand, forces are exerted in the netic field on other conductors carrying currents; and, on the other, that momentary currents are induced in conductors when, and only in so far as, the condition in question is altered, either as regards position or value, and particularly when it either suddenly appears or completely vanishes. Movable conductors conveying a current are therefore put in motion. On the other hand, momentary currents are induced in movable con

Compare W. von Siemens, Wied. Ann. vol. 24, p. 94, 1885.

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ductors whenever the magnetic condition with reference to them is altered by their motion. This must suffice for the general characterisation of the phenomena in question, the experimental details of which must be assumed to be known.

Both these classes of phenomena, the electrodynamic and the inductive, are equally well suited theoretically for a complete determination of the magnetic condition. A whole series of methods for practically attaining this object has been developed, which we shall consider more closely further on (Chapter X.).

§ 2. The Magnetic Condition as a Directed Quantity.--For our present object, which is mainly theoretical, the following elementary arrangement will be sufficient. Let a metal wire be bent so as to form a plane loop, which encloses the area S; let what is called the secondary circuit, of which it forms a part, have the resistance R. Let the momentary current induced in the wire cause a quantity of electricity, Q, to be displaced, the absolute value of which can be measured by any suitable arrangement. By means of such a movable coil, which is an 'exploring coil,' we can investigate the magnetic field, and, as it were, make a topographical survey of it.

We have in the first place to investigate what takes place if we leave the small exploring coil in one place, and only alter its direction. This is defined by the direction of the perpendicular on one side to the plane of the coil. If we let the perpendicular sweep through all possible directions in space, we find that there are two positions, and these two exactly opposite each other, in which a maximum quantity of electricity is induced, when the current is made or broken, in what is called the primary conductor.

In all other directions of the perpendicular to the coil smaller quantities of electricity are obtained; they are, in fact, in each case proportional to the cosine of the inclination to the direction of maximum induction. It follows from this that for all directions of the perpendicular which lie in the plane at right angles to that special direction the quantity of electricity

1 In practice a ballistic galvanometer is almost always used for such experiments; this itself depends indirectly on actions similar to those here described. Compare Faraday, Exp. Researches, vol. 3, p. 328.

ELEMENTARY CONCEPTIONS OF QUATERNIONS

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induced is zero-that is, no induction takes place. All this tends to indicate that we are here dealing with one of those physical conditions which can only be completely defined by a vector. We must first explain more in detail the important idea expressed by this word, before proceeding to further experiments with the exploring coil.

§ 3. Elementary Conceptions of Quaternions.—Although we shall in the sequel make no use of special quaternion methods, the most elementary extremely useful conceptions and notations of that branch of science will be frequently applied.'

Physical quantities may be divided into two groups, those of the directed and those of the undirected, which are distinguished as vectors and scalars. With regard to scalars nothing need be said; the general properties of physical quantities, of their numerical values, as well as of their units, are supposed to be known. But vectors, from the very fact of their being directed, have, in addition, special properties; we are here chiefly concerned with the law of their geometrical addition.

The sum of two or more vectors is, in general, not equal to the sum of their numerical values. It is obtained in a manner which is generally known, by the way in which a vector quantity of frequent occurrence, force, is geometrically added; that is, by the construction of a parallelogram for two, and of a polygon2 for several vectors. In accordance with this, a vector may, conversely, be resolved into any number of components having given directions, in particular those of the axes of coordinates. The numerical value of a vector component is obtained by multiplying that of the vector itself into the cosine of the angle between the two directions.

On account of the essential differences between the mathematical operations to be performed with scalars and vectors, it is desirable to be able to see from the symbol for a quantity to which of the two groups it belongs. Hence it is usual to denote, by German capitals, those quantities the vector character of which

For further details reference must be made to the important works of Grassman, of Hamilton, and of Tait. See also O. Heaviside, Electromagnetic Theory, London, 1893.

2 That is in the general case of a broken polygon of straight lines in space.

is to be clearly shown.' We shall adopt this plan, and refer to Chapter III. for further geometrical considerations as to

vectors.

§ 4. Magnetic Intensity.-After this unavoidable digression. we return to the magnetic vector. We now attempt to determine its numerical value, by placing the exploring coil in the direction of maximum induction, and investigating on what variable quantities the quantity Q of induced electricity depends. We shall then find that it is proportional to the area S of the winding, and inversely proportional to the resistance R. These factors, which have obviously no connection with the magnetic condition, we eliminate by forming the expression QRS; we have then to consider this as the absolute measure for the magnetic condition, and accordingly put

It is here to be observed that if Q, R, and S are expressed in any consistent system of measurement, the expression QRS measures the magnetic condition also in this system. In such considerations as we are here concerned with, it is, in fact, the practice to adopt exclusively the electromagnetic C.G.S. system.2 In accordance with this S in the above equation is expressed in square centimetres, Q in decacoulombs, R in millimicrohms.

The quantity thus defined absolutely, we shall call the intensity of the magnetic field; the symbol chosen denotes its vector character. Its direction is that of the perpendicular on one side to the coil in the position of maximum induction, and with the condition that the sense of the current, induced by the cessation of the field, stands in the same geometrical relation to the direction of the field, as the sense in which the hands of a clock move, to the direction from the dial towards the works.

1 This practice was introduced by Maxwell in his Treatise on Electricity and Magnetism. Instead of this, we find in many English authors block letters in the middle of the text, which, however, are scarcely an ornament; the choice of a notation is, of course, a somewhat unimportant matter of taste. 2 It is not necessary to enter here more minutely upon the theory of absolute systems of measurement, as there are several excellent special works on the subject.

MAGNETIC FIELD OF STRAIGHT CONDUCTORS

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If we imagine lines of intensity' in space-that is, curves whose tangent at each point gives the direction of the intensity --we have by these a means of making evident the distribution of the direction of the magnetic vector in space. We can in many cases, as we shall afterwards see, draw conclusions as to the numerical value of a given vector from the course of such groups of lines.

A method frequently used for graphically representing lines of intensity in two dimensions consists in using the finest dust from iron filings; when this is scattered on a sheet of stout paper, which is then gently tapped, the dust arranges itself in the direction of these lines, and can afterwards be fixed.

§ 5. Magnetic Field of Straight Conductors. The more detailed geometrical investigation of the field which is produced by linear conductors of various shapes in the space near them we need not go into. It will be sufficient to mention briefly a few special cases which we most frequently meet with.

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Here again, as has been before stated, all the equations are to be interpreted in the electromagnetic C.G.S. system. Currents, for instance, are then to be measured in deca-ampères, and linear dimensions in centimetres. On this depends the simplicity of the equation, and the avoidance of arbitrary constants.

A. Straight Element of Current.-This, which is exclusively a mathematical abstraction, cannot physically be realised; it presents, however, considerable interest, for by integrating the corresponding elementary equation over closed conductors we obtain results which are susceptible of exact experimental confirmation. A straight infinitely short element of conductor of length dL, conveying a current, produces at a point at a distance

In ordinary language we frequently speak of magnetic force and lines of force, the latter expression being also frequently used for those curves which we shall more logically introduce as lines of induction (§ 61). Maxwell himself has, however, given the preference to the word ‘intensity,' as undoubtedly follows from the second edition of his Treatise-so far as the author himself revised it (see particularly 1, § 12). E. Cohn, ‘Systematik der Elektricität,' Wied. Ann. vol. 40, p. 628, 1890, as well as Hertz, Untersuchungen, p. 30, Leipzig, 1892, agree with this.

2 Compare Mascart and Joubert, Electricité et Magnétisme, vol. 1, §§ 442-506, Paris, 1882.