This condition of things is represented in the figure, where the two ends of the bar are shown. It is reasonable to suppose that, except on the surface, the currents of neighbouring molecules would neutralise each other with respect to external effect; since, as shown, the neighbouring pairs of contiguous currents would be equal and opposite. There would, therefore, be left only the outside ring of molecular currents; and these may be supposed to be equivalent to one continuous circular current. Thus, if we consider the bar to be cut into flat discs as in § 1 above, each disc would be equivalent to a plane circular current; and the whole bar would be equivalent to a solenoid, as regards the external field.

§ 5. Solenoid, and Hollow Cylindrical Magnet, Contrasted.— In fig. i. we represent in section a solenoid, and a few of its lines

N

FIG. i.

of force are also given, the arrows representing their + direction. It will be seen that inside the solenoid the lines run in a contrary direction to that which they have outside.

In fig. ii. is given in section a hollow cylindrical magnet, in like manner. Here, the lines along the inside have the same direction as outside. The outside field has the same general character. But while all the lines of the solenoid run continuously through and round the hollow tube, in the case of the hollow magnet, on the other hand, all the lines run into, and end in, the solid steel that forms the side of the hollow cylinder. In fact, we must remember that the complete solenoid is equivalent to a magnet, as regards external field, but any longitudinal slip of it taken alone is not equivalent to a magnet; while each such

longitudinal slip of the hollow magnet is itself a complete magnet, the hollow cylinder being not a simple whole, but being a compound arrangement formed of a system of magnets arranged as are the staves in a barrel.

§ 6. Matter Placed in a Uniform Magnetic Field of Force.Let us consider a uniform field of magnetic force, and a cylinder of any material, whose two ends are plane faces standing perpendicular to its axis, so placed as to lie with its axis along the lines of force of the field.

In the case of soft iron, or of steel previously unmagnetised, we find that the magnetisation is such that we have evident magnetism at the two end-surfaces only; if we neglect, as relatively unimportant, the irregularities that occur at the edges where the molecules are free towards the outside and in contact with other molecules towards the inside. Over these end-surfaces the density p of magnetisation (see Chapter XVIII. § 7) will be approximately uniform. There is no experimental reason for supposing but that cylinders of any material are, if sensibly magnetised at all, magnetised in a similar manner to the above.

The value of this density p depends (i.) upon the field-strength I, and (ii.) upon the nature of the material of which the cylinder is composed. Now experiment indicates that, so long as the bar is far from saturation (see Chapter I.), then p is directly proportional to the field-strength I. Thus we may write

p = k I

where k is a quantity depending upon the nature of the material. When the field-strength I unity, then k is numerically equal to p.

=

This quantity is called the coefficient of magnetisation of that material, and is measured by the value which has when I unity.

Without at présent discussing whether the following assumptions are physically possible (we shall see later that they are), let us assume that k may be a + quantity, zero, or a quantity; and let us consider what would be the observed condition of the cylinder in the three cases respectively.

(I.) Let k = a + quantity. This would make positive. That is, remembering what is the direction of the lines of force, we should observe a north-seeking polarity at the end lying furthest down the lines of force, and a south-seeking polarity at the other end. Or this case is the usual one of soft iron or other magnetic matter placed in a field of force.

(II.) Let k = zero.-Here we observe no polarity, since p = o. That is, the material is one whose presence in the field makes no difference to it.

(III.) Let k = a quantity. In this case p will have a contrary sign to that which it had in case (I.). Or the material would be one in which induction takes place in a contrary direction to that observed in iron; thus a north-seeking pole of a magnet would, at any rate apparently, induce in a bar of such material a north-seeking pole at the end nearest to the former.

In order that we may now use convenient names for different classes of bodies, we shall to some extent forestall what will be discussed more fully later on in this Chapter.

We may, therefore, state that there is in the first place a class of bodies for which k is +; or for which induction takes place down the lines of force, as in the case of iron. Such bodies, of which iron (including steel) is by far the most important, are called magnetic, or more properly paramagnetic. In the case of very pure soft iron, k has a large value; thus the presence of a bar of such iron in a magnetic field may increase the field-strength near the poles even fifty-fold.

is

There is, secondly, another large class of bodies for which k - ; or for which induction takes place up the lines of force, or in the contrary direction to the above. For such bodies k is very small. For example, if a bar of bismuth be placed in a magnetic field, this field will appear to be slightly weakened near the poles of the bar, in virtue of the opposed induced polarity of the bar ;

but, from the smallness of k, the field as a whole is but very slightly affected. Such bodies are called diamagnetic. further discussion see §§ 14 and 15.)

(For

§ 7. Movements of Small Bodies in a Non-Uniform Magnetic Field. Let us now consider a field that is not uniform, and a small body placed in the field. By small body we here mean one so small with respect to the whole field that it can be considered to be all of it in a stronger or weaker part of the field at the same time; and yet not so small but that one side of it is, in our nonuniform field, in a part of the field of somewhat different strength to that in which the other side finds itself. We will consider what will be its behaviour.

(I.) Small magnetic bodies.—It can be shown that a small magnetic body, such as an iron pellet, for example, is urged from weaker to stronger parts of the field.

Thus, if a small iron pellet be presented to a pole of a magnet (the usual case of a non-uniform field) there will be induced an opposed polarity on the side next to the pole, and a similar polarity on the side more remote. The former will be attracted, and the latter will be repelled, by the pole; these two polarities are equal in magnitude but opposite in sign. Now the former polarity is in a stronger field than is the latter, and hence attrac· tion will predominate, and the pellet will move towards the pole.

(II.) Small diamagnetic bodies.—For similar reasons a small diamagnetic body is urged from a stronger into a weaker part of the field.

§ 8. The Setting' of a Long Body in a Uniform Magnetic Field. Next let us consider the case of a long cylinder placed in a uniform magnetic field, at an angle with the lines of force of the field. We will represent our cylinder as composed of a series of small spheres placed near to one another. This is a convenient representation, and though not an accurate one, will not invalidate the very general results at which we shall arrive.

(I.) A magnetic cylinder.-Fig. i. represents a cylinder of iron. Each of the little spheres A BCD would, if it stood alone, be magnetised in the direction of the lines of force of the field. This is represented by the lettering n s in each. There is, however, inductive action between the spheres, each n or s inducing an s or n

respectively in the nearest portion of the neighbouring sphere. The total result will be that each little sphere is magnetised, not along the lines of the field, but in a direction represented by n' s' of fig. ii., lying between the direction of the field and the direction of the cylinder.

Thus each little sphere is acted upon by a couple tending to drag n's' into the direction of the lines of the field (see Chapter II. $ 12). Hence there will be a couple acting upon the whole

cylinder, and there will be stable equilibrium only when this lies along the lines of force. It is to be noticed that in this position the external and internal actions concur to give the maximum magnetisation.

When the cylinder is perpendicular to the field there is also equilibrium, but unstable. In this position the internal induction acts against the external, and the magnetisation is at a minimum. We may therefore state that.

A magnetic cylinder tends to set along the lines of force of a uniform field; that is, to assume the position in which its magnetisation is at a maximum.

(II.) A diamagnetic cylinder.-In this case the external and internal induction will both be the

reverse of what it was in case (I.). Thus we must interchange the letters n and s in the above given figures; and must further remember that each n or s in one sphere induces an ʼn or s respectively in the nearest portion of the neighbouring sphere. The total result will be that each little sphere will be magnetised

FIG. iii.