IDEAL SIMPLE MAGNET. 661 If a piece of magnetized steel be suspended at its centre of gravity, so as to be free to turn all ways about it, the effect of the earth's magnetism upon it consists in a tendency for a particular line through this centre of gravity to take a determinate direction, which is the direction of terrestrial magnetic force. When the line is placed in any other position, the couple tending to bring it back is proportional to the sine of the angle between the two positions, and is the V same for all directions of deviation. The line which possesses this property is the magnetic axis of the body, and the name is sometimes given to all lines parallel to it. If the piece of steel be uniformly magnetized, this axis is the direction of magnetization; or the direction of magnetization is the common direction of all those lines which tend to place themselves along lines of force in a field1 where the lines of force are parallel. 674. Ideal Simple Magnet: Thin Bar, uniformly and longitudinally Magnetized. The mutual actions of magnets admit of very accurate expression when the magnets are very thin in comparison with their length, uniform in section, and uniformly magnetized in the direction of their length. Such bars, which may be called simple magnets, behave as if their forces resided solely in their ends, which may therefore in the strictest sense be called their poles. The two poles of any one such bar are equal in strength; that is to say, one of them attracts a pole of another simple magnet with the same force with which the other repels it at the same distance. In the language of the two-fluid theory, the two fluids destroy one another except at the two ends, and the quantities which reside at the ends are equal but of opposite sign. The same number which denotes the quantity of fluid at either pole, denotes the strength of the pole, or, as it is often called, the strength of the magnet. Its definition is best expressed by saying that the force between a pole of one simple magnet and a pole of another, is the product of their strengths divided by the square of the distance between them.2 1 A field of force is any region of space traversed by lines of force; or, in other words, any region pervaded by force of attraction or repulsion. A magnetic field is any region pervaded by magnetic force. All space in the neighbourhood of the earth is a magnetic field, and within moderate distances the lines of force in it may be regarded as parallel, unless artificial magnets or pieces of iron are present to produce disturbance. 2 We here, and throughout the remainder of this chapter, ignore the existence of induction, which, however, is not altogether absent even in the hardest steel. The effect of induction is always to favour attraction. The attractions will therefore be somewhat stronger, and the repulsions somewhat weaker, than our theory supposes. The force which a pole of a simple magnet experiences in a magnetic field, is the product of the strength of the pole and the intensity of the field. This rule applies to the force which a pole experiences from the earth's magnetism, the intensity of the field being in this case the intensity of terrestrial magnetic force; and, from the uniformity of the field, the forces on the two poles are in this case equal, constituting a couple, whose arm is the line joining the poles multiplied by the sine of the angle which this line makes with the lines of force. The product of the line joining the two poles by the strength of either pole is called the moment of the magnet, and it is evident, from what has just been said, that the continued product of the moment of the magnet, the intensity of terrestrial magnetic force, and the sine of the angle between the length of the magnet and the lines of force, is equal to the moment of the couple which the earth's magnetism exerts upon the magnet. 675. Compound Magnet of Uniform Magnetization.-Any magnet which is not a simple magnet in the sense defined in § 674 may be called a compound magnet. It is convenient to define the moment of a compound magnet by the condition stated in the concluding words of that section, so that the moments of different magnets, whether simple or compound, may be compared by comparing the couples exerted on them by terrestrial magnetism when their axes are equally inclined to the lines of force. If a number of simple magnets of equal strength be joined end to end, with their similar poles pointing the same way, there will be mutual destruction of the two imaginary fluids at every junction, and the system will constitute one simple magnet of the same strength as any one of its components; but its moment will evidently be the sum of their moments. If any number of simple magnets be united, either end to end or side to side, provided only that they are parallel, and have their similar poles turned the same way, the resultant couple exerted upon the whole system by terrestrial magnetism will (§ 27) be the sum of the separate couples exerted on each simple magnet, and the moment of the system will be the sum of the moments of its parts. But any piece of uniformly magnetized material may be regarded as being thus built up, and hence, if different portions be cut from the same uniformly magnetized mass, their moments will be simply proportional to their volumes. The quotient of moment by volume, for uniformly magnetized mass, is called intensity of magnetization. any 676. Actual Magnets.-The definitions and laws of simple magnets are approximately applicable to actual magnets, when magnetized in the usual manner. If an actual bar-magnet in the form of a rectangular parallelepiped were magnetized with perfect uniformity, and in the direction of its length, it might be regarded as made up of a number of simple magnets laid side by side, and its behaviour would be represented by supposing a complete absence of magnetic fluid from all parts of it except its ends (in the strict mathematical sense). One of these terminal faces would be covered with positive, and the other with negative fluid, and if the magnet were broken across at any part of its length, the quantities of positive and negative fluid on the broken ends would be the same as on the ends of the complete magnet. The observed fact that magnets behave as if the fluids were distributed through a portion of their substance in the neighbourhood of the ends, and not confined to the ends strictly so called, indicates a falling off in magnetization towards the extremities, and is approximately represented by conceiving of a number of short magnets laid end to end, and falling off in strength towards the two extremities of the series.1 The resultant force due to the imaginary magnetic fluids which are distributed through the terminal portions of an actual barmagnet is, in the case of actions at a great distance, sensibly the same as if the two portions of fluid were collected at their respective centres of gravity. These two centres of gravity are the poles of the magnet for all actions between the magnet and other magnets at a great distance, and more especially between the magnet and the earth. The moment of any magnet, however irregular in its magnetization, may be defined by reference to the expression given in § 674 for the couple exerted on the body by terrestrial magnetism. This couple is MI sin a, where I denotes the intensity of terrestrial magnetic force, a the inclination of the magnetic axis of the body to the lines of the earth's magnetic force, and M the moment which we are defining. 1 Thus the last magnet at the positive end being weaker than its neighbour, its negative pole will be weaker than its neighbour's positive pole, so that there will be an excess of positive fluid at this junction. Similar reasoning applies to all the junctions near the ends. There will be an excess of positive fluid at all junctions near the positive end, and an excess of negative at all junctions near the negative end. CHAPTER LII. EXPERIMENTAL DETAILS. 677. The Earth's Force simply Directive. The forces which produce the orientation of a magnet depend upon causes of which very little is known. They are evidently connected in some way with the earth, and are accordingly referred to TERRESTRIAL MAGNETISM. We have already stated (§ 673) that the combined effect of the forces exerted by terrestrial magnetism upon a magnetized needle is equivalent to a couple tending to turn the needle into a particular direction, and (§ 676) that in the case of needles magnetized in the ordinary way, there are two definite points or poles (near the two ends of the needle) which may be regarded as the points of application of the two equal forces which constitute the couple. The fact that terrestrial magnetic force simply tends to turn the needle, and not to give it a movement of translation, in other words, that the resultant force (as distinguished from couple) is zero, is completely proved by the two following experiments:— (1) If a bar of steel is weighed before and after magnetization, no change is found in its weight. This proves that the vertical component is zero. (2) If a bar of steel, not magnetized, is suspended by a long and fine thread, the direction of the thread is of course vertical. If the bar is then magnetized, the direction of the thread still remains vertical. The most rigorous tests fail to show any change of its position. This proves that the horizontal component is zero, a conclusion which may be verified by floating a magnet on water by means of a cork. It will be found that there is no tendency to move across the water in any particular direction. 678. Horizontal, Vertical, and Total Intensities.-If S denote the strength of a magnet, and I the intensity of terrestrial magnetic force, HORIZONTAL AND VERTICAL INTENSITIES. 665 each pole of the magnet experiences a force SI, and if L denote the distance between the poles (often called the length of the magnet), the distance between the lines of action of these two parallel and opposite forces may have any value intermediate between L and zero, according to the position in which the needle is held. It will be zero when the line of poles is that of the dipping-needle; it will be L when the line of poles is perpendicular to the dipping-needle; and will be L sin a when the line of poles is inclined at any angle a to the dipping-needle. The force SI upon either pole of the magnet acts in the direction of the dipping-needle; in other words, in the direction of the lines of force due to terrestrial magnetism. Let & denote the dip, that is the inclination of the lines of force to the horizon, then the force SI can be resolved into SI cos & horizontal, and SI sin & vertical. Hence the horizontal and vertical intensities H and V are connected with the total intensity and dip I and ♪ by the two equations 679. Torsion-balance.-Coulomb, in investigating the laws of the mutual action of magnets, employed a torsion-balance scarcely differing from that which he used in his electrical researches. The suspending thread carried, at its lower end, a stirrup on which a magnetized bar was laid horizontally. The torsion-head was so adjusted that one end of the magnet was opposite the zero of the divisions on the glass case when the supporting thread was without torsion. In order to effect this adjustment, the magnet was first suspended by a thread whose torsional power was inconsiderable, so that the magnet placed itself in the magnetic meridian. The case was then turned till its zero came to this position. The torsionless thread was then replaced by a fine metallic wire, and the magnet was replaced by a copper bar of the same weight. The head was. then turned till this bar came into the magnetic meridian, and lastly the magnet was put in the place of the bar. Fig. 424 shows the arrangement adopted for observing the repulsion or attraction between one pole of the suspended magnet and one pole of another magnet placed vertically. Before the insertion of the latter, the suspended magnet was acted on by no horizontal |