tion at some given point, and let us consider a surrounding region (called the region R) so small that nowhere within it does the intensity differ appreciably from 5 or the induction from B. We shall have also Bμh, where μ may or may not be =1. Now suppose that we have a magnet in the form of a divided toroid, the width of the air-gap being I and its area F. Let us further suppose that the degree of magnetisation of the toroid is very small, so that the value of the magnetic induction & across the air-gap is also very small. The strength of pole m of the toroid is given by

4Tm-FdB or m=F

4π

(15)

and we shall suppose the steel to be 'rigidly' magnetised, so that m suffers no change from the action of external fields.

Now introduce the toroid into the field to be examined, the air-gap being made to lie within the region R with its faces perpendicular to the direction of 5 and B, and so that B and B are opposed to one another in direction. Thus though the volume Fl of the air-gap the induction is now B-dB.

Next let the position of the toroid be reversed in the field, in such a manner that each face of the air-gap occupies the position which was just now occupied by the other face. The induction through the air-gap thus becomes B+ dB.

We must now find how much energy has been expended in changing the magnetic induction from B-SB to B + EB, through a volume Fl, the mean value of the field-intensity being . This energy must be equal to the work expended in turning the toroid round against forces due to the surrounding field, an amount of work which is easily calculated. For the nett result of the operation is the displacement of a north pole of strength + m through a length l against the direction of 5, and of a south pole of strength -m through an equal length 1 in the direction of 5. Both these dis

H.

placements demand the expenditure of positive work, the total amount of which for the two poles is

(on substituting for m the value given by (15) ).

The last expression written is the product of three factors the volume of the region affected by the operation, the value of the field-intensity for this region, and the increase 28B of the magnetic induction divided by 4π. Hence if A is the magnetic energy per unit volume at a place where the field-intensity is B and induction B, and if A becomes changed to A+dA when B is changed to H+dB, we shall

The value of A may be deduced from this by integration, supposing μ to be a constant; thus

If we are dealing with air, the factor μ becomes =1, and may be omitted from all the equations.

The expressions for the forms of energy considered here and in § 118 are similar to those for the kinetic energy of a mass m, moving in a determinate direction with velocity v :

The analogy, however, only extends to the form of the expressions. The energy of a stretched body is not spoken of as kinetic, and it is not to be assumed without further inquiry that the energy of a magnetic field is of the ordinary mechanical kind.

[In this paragraph (§ 119) a slight departure has been made from the treatment in the original text.]

143

CHAPTER VII

DYNAMICAL THEORIES OF FIELDS OF FORCE

WHEN a pole is placed in a magnetic field, it experiences a force of determinate magnitude acting in a determinate direction. We have already (§ 31) spoken of the magnetic force as a vector or directed quantity; a magnetic field may be said to be determined by an infinity of such vectors. The question now arises: Can we not discover the nature of those conditions, vectorial in character, which exist at each point of a magnetised medium, and are there not, in particular, some simple types of motion which possess the same character? The latter question is suggested by the development which has taken place in our understanding of other physical phenomena, such as light and heat; the older hypotheses which sought an explanation in the presence of special fluids or corpuscles, having given way to the doctrine of a special kind of motion. The kinetic theories and the mechanical theories connected with them have acquired a great importance; and we will accordingly give an account of the most important which have been formulated in connection with our subject. In doing this, we shall have to depart from a mere description of actual phenomena, and consider points of a more or less conjectural kind.

A.-Symmetrical properties of magnetic fields

In our endeavour to explain magnetic phenomena by means of known mechanical motions, we are guided by a number of fundamental properties, which leave no doubt as to the description of motion to be chosen. We have to

proceed somewhat as we do in mineralogy, when we infer the structure of a crystal from its observed properties. There are also some properties of a magnetic field which imply a certain symmetry in the medium which is the seat of the field; and the conditions obtaining in the separate elements of volume must correspond to these symmetrical relations (CURIE).

120. Axial character of the magnetic lines of force. The simplest kind of vector quantity is the displacement or translation of a small body from a point in space, through a determinate distance in a determinate direction to some other point. It is with vectors of this kind that we are concerned in all phenomena of flow, and we may conceive of the directional properties of a magnetic field as being brought about by the flow of something along the lines of force. We have already made frequent use of terms which are based on this conception, though at the same time we have pointed out that this flow of the lines of force is only a mode of representation. There are certain phenomena which show that we have not to deal with this simplest form of vector, but with vectors of a different kind. For example, there is the rotation of the plane of polarisation produced in a beam of plane-polarised light which is made to pass through a magnetic field along the direction of the lines of force (FARADAY). From this observation, as early as 1856, Sir WILLIAM THOMSON (Lord KELVIN) arrived at the conclusion that the conditions obtaining along a line of force are kinematically comparable with those along an axis of rotation. And in fact rotation about an axis is a phenomenon of a vectorial nature. The direction of the vector is the direction of the axis of rotation, the positive direction of the axis being determined in accordance with some fixed convention. Thus we may agree that a rotation in the given sense, accompanied by a translation along the positive direction of the axis of that rotation, shall constitute a right-handed screw-motion; or we may base our definition on the left-handed screw. Finally, the rotational velocity may be represented by setting off an appropriate number

of units of length along the axis of rotation. Magnetic forces possess the same kind of symmetry as an angular velocity. If we suppose the region with which we are concerned to be 'perverted' by reflection in a plane mirror, rotational movements will be affected differently from simple translations, whose directions we indicate by means of arrows. It will be shown that the same holds good for magnetic fields, from a consideration of the special forms described in Section II. F. KOLÁČEK, to whom we owe the very appropriate term ' axial character of the magnetic lines of force,' has quoted Hall's phenomenon as proving that there is something which rotates about these lines as axes.

6

121. Rotational vectors.-If we conceive of magnetic phenomena as due to some kind of motion which is rotational in character, there must be an invariable relation between the direction of rotation and the direction in which the lines of force are reckoned positively. We have already (§ 28) given a rule for determining the positive sense along a line of force, and we must now agree upon a similar rule for the corresponding rotational sense.

The rotation which we conceive to take place about a line of force would appear to be in the direction of clockhands to an eye looking along the line in the positive sense, that is, from the north pole towards the south pole.

The propriety of this convention will appear in Section II.

It will be well to construct a small model (fig. 34), so as to be always able to recall easily the rule just given.

The cylindrical rod A A carries a spike S at its upper end, and at its middle is a wooden disc H to whose circumference two tin arrows P1, P2 are fastened. On looking along AA towards the spike S (direction of lines of force) we see the lower side of the disc, whose rotation, suggested by P, and P2, is in the direction of the hands of a clock.

FIG. 34

L