This expression shows that to determine the ratio of H to M it is only necessary to measure the distance r and the angle of deflection a. Having by the first experiment determined the product HM and it only remains to multiply the two together

by the second the ratio

H

M'

to find H2 and so determine H.

UNIT TUBES OF FORCE.

502. Number of Lines of Force.-Up to this point lines of force have been regarded as simply expressing the direction of the force in the magnetic field. We must now follow Faraday in a very remarkable development of the idea.

In a stream of water flowing steadily lines may be imagined drawn which at every point are in the direction of flow, and which may be called stream lines. An infinite number of such lines may be drawn. The whole stream may then be conceived to be divided up into tubes of flow by means of surfaces which everywhere coincide with stream lines. These tubes of flow may be taken of such a size that each will transmit the same quantity of water per second, say 1 cubic foot. Then, where the stream is most rapid, the cross sections of the tubes of flow will be smallest and they will widen out as the velocity diminishes. The whole number of such tubes in the stream will be equal to the number of cubic feet of water transmitted per second. These tubes of flow may be called unit tubes, and the number of them crossing perpendicularly a surface one square foot in area is equal to the number of cubic feet of water crossing that area per second. Thus the number of unit tubes passing perpendicularly through a unit surface at any point in the stream is equal to the velocity at that point.

Now in the same way the magnetic field may be conceived as divided up into unit tubes by means of surfaces parallel to the lines of force. And it may be proved that where such a unit tube is smaller the field is more intense, and where it widens out the strength of field is less, just as the velocity varies in case of the stream of water. So that it is possible to take these tubes of such a size that the number passing perpendicularly through a square centimeter of surface at any point may be equal to the strength of the magnetic field at that point.

We may imagine that each unit tube is represented by a line of force drawn through its center or axis, and when the phrase number of lines of force is used it refers to such lines.

Using the term in this way, it is clear that in the case shown in figure 272 more lines of force pass through a card in position C than in position A, as the force is greater at C than at A, and consequently there are more lines of force to the square centimeter. Clearly, also, fewer lines of force pass through the card in position B than in A and most of all in position D. If the card were placed parallel to the lines of force none at all would pass through it.

The number of lines of force through A is found by taking the average strength of the field at the surface A and multiplying this by the area of A in square centimeters, since the number of lines

N

A

FIG. 272.

S

per square centimeter is equal to the strength of the field at that point. If the surface is oblique to the lines of force as at B, the number of lines of force passing through it will be found by multiplying the number in the perpendicular position A by the cosine of the angle a, or, what comes to the same thing, multiply the average strength of the field at the surface B by the projection of that surface on a plane at right angles to the lines of force.

Problems.

1. How many lines of force pass through a square meter of floor area where the total strength of the earth's magnetic field is 0.6 and the lines of force are inclined 60° from the horizontal?

2. How many lines of force in this case would pass through an area of one square meter on an east and west wall, and how many in case the wall ran north and south?

503. Lines of Force Inside a Magnet.-The lines of force of a magnet are not to be supposed as only outside of it. If we imagine a minute magnetic needle placed in a crack extending

across the magnet it will be acted on most powerfully by the poles N'S' on each side of the crack, but it will also be affected by the attraction of the end poles N and S of the magnet. In consequence of the superior influence of the poles N'S', it will set its north pole toward the left or S'. If we now imagine the cleft shifted along toward the north end of the magnet, the force inside the cleft will become less because it will be nearer to N, which pole tends to make the needle point in the opposite direction. But still the needle will point from right to left. When the cleft comes infinitely near to the end N, the magnetism of N

and of S', which form two opposite and equally magnetized layers, will neutralize each other so that the effect is the same as though the needle were just outside the magnet at N. We see in this way that the force in the cleft is absolutely continuous with that outside of the magnet: there is no abrupt change in passing through the surface. The force in such a cleft is called the magnetic induction and the lines of force outside of a magnet form continuous closed curves with the lines of induction inside of the magnet. The lines of force outside are also called lines of induction, as there is no distinction between the two except inside of a magnetic medium. What is called the positive direction of these lines is from the north to the south pole outside of the magnet. Of course as many lines of force as emerge from the north pole enter at the south pole, and all the lines of force or induction in the magnet pass through its middle section. Looked at in this way, the poles are seen to be simply those regions where the lines leave or enter the magnet, and the

most intensely magnetized portion of the magnet is the center where the lines of induction are closest together. If a little block could be cut from the center of the magnet without disturbing its magnetism, it would be found a more powerful magnet than a similar block cut from any other part where the lines of induction are not so close.

Warning. In using the words entering and emerging with reference to lines of force nothing like flow or motion must be supposed; when what we arbitrarily call the positive direction of the line of force is toward a surface, it is spoken of as entering it; and when that direction is away from a surface the line of force may be said to leave the surface or emerge from it.

504. Influence of the Shape of a Magnet on Its Power and Retentiveness. A short thick bar of steel is more difficult to magnetize strongly than a long thin one and loses its magnetism more easily. A thick magnet may be thought of as made up of a bundle of thin ones of the same length. But it is clear that in such a bundle each little magnet would tend to set up lines of force down through its neighbor in such direction as to oppose or weaken the others' magnetism.

Thus there is a demagnetizing tendency which is greatest in a short thick magnet. Horseshoe magnets are long and have their poles close together and consequently there is very little demagnetizing tendency. There is, however, a tendency for the lines of force in this case to pass across on the inside of the poles instead of out at the ends. A soft-iron block placed across the poles, and called an armature or keeper, provides an easy path for the lines of force from one end around to the other and thus tends to keep the poles near the ends.

FIG. 274.

505. Ring Magnet.-A uniform ring of iron or steel may be magnetized by means of an electric current so that the lines of force are circles entirely within the substance of the ring. In such a case the magnet has no poles as there are no places where the lines of force enter or leave the ring. Such a magnet has no external field of force and would not act on a magnetic needle placed near it, and yet it is magnetized, as will be evident if it is broken, for in that case each half will show two poles.

MAGNETIC INDUCTION.

506. Induction Studied by Iron Filings. If the lines of force of a horseshoe magnet are examined by means of iron filings on a plate of glass, as described in §486, and if a bar of soft iron is then placed a short distance in front of the poles of the magnet and the field again examined in the same way, a notable change will be observed. The lines of force are bent toward the two ends of the soft-iron bar as though they could be established in the iron more easily than in the surrounding medium. And the softer the iron and the more easily it is magnetized, the greater the number of lines of force that will pass through it rather than the more resisting medium around it. Thus the presence of the

N

S

FIG. 275.

iron makes the field of force weaker beyond it, and the nearer the iron bar is to the poles of the magnet the more lines of force will be drawn into it and the fewer there will be in other parts of the field.

507. Permeability. The ease with which lines of force may be established in any medium as compared with a vacuum has been called by Lord Kelvin the permeability of the medium. Thus iron has a permeability several hundred times greater than air. Most other substances have a permeability which is sensibly the same as air or vacuum, and therefore the magnetic field is practically the same in wood, glass, or water as in air.

A hydraulic analogy may aid in forming a clear conception of this subject. Imagine a stream of water continually flowing out of the north of the horseshoe magnet (Fig. 275) and entering its south pole. Suppose the medium surrounding the magnet was of a uniform porous nature that opposed considerable resistance to the flow from N to S. The lines along which the flow would take place would be like the lines of force in the field before the soft iron was introduced. Now imagine a cavity to be made in the porous medium having just the size and position of the soft-iron bar. The lines of flow would now tend toward this cavity through which the liquid would flow freely and a correspondingly smaller flow would take place in other regions.