Since we are concerned with the quantity of work expended upon a unit pole, and not with the force exerted upon it, the name magneto-motive force' is not very well chosen; in this respect, however, it is on the same footing as the term electromotive force applied to the corresponding electrical quantity.

114. Magnetic resistance. The magnetic resistance p is defined as the reciprocal of the magnetic permeability or 'conductivity.' Accordingly we write :

The greater the magnetic resistance of a substance, the smaller is its permeability; for air, paper, brass, p is large, while for soft iron it is small. The following roughly quantitative experiment is designed to show the marked effect produced upon a magnetic circuit by the interposition of even a thin layer of a substance whose magnetic resistance is high.

S

Experiment 50.-To the poles n and s of a small horse-shoe magnet (fig. 33) is applied a soft iron keeper, semicircular in form and having a loop attached to it, from which a scale pan may be hung by means of a hook. In addition to this undivided keeper, there is another of the same shape and weight, but having only its outer portions l, and l, (fig. 33) made of iron. In the middle, between 1, and l2, a piece of brass m is interposed, and the brass ring h is held in position by two screws S1, S2, so that it helps to hold the several portions of the keepers together. We have thus a keeper m in which a body m of high magnetic resistance is interposed in the course of the lines of force which traverse the strongly paramagnetic portions, and l2. We now proceed to determine the pull required to detach the keeper from the magnet (a) when the ordinary keeper is used; (b) when the divided keeper is used; (c) when a sheet of paper is interposed between the iron keeper and the ends of the magnet (d) using the divided keeper under the same conditions. Numerical example:

FIG. 33

$2

Lifting power of a horse-shoe magnet

(a) 2.220 kilogr. +20 grams (weight of keeper and scale pan); (b) 0.580 kilogr., only about of the preceding.

(c) 1·100 kilogr., about half the value in (a).

(d) 0.190 kilogr., only of the full lifting power.

The lifting power is not proportional to the number of lines of force; but the experiment shows the great decrease in the flux of induction produced by an increase in the resistance of the circuit.

When some portion of a magnetic circuit is of small permeability, the resulting resistance is greater the longer the portion in question, and less the greater its crosssection; for when the cross-section is large, the same flux of induction can be spread over a greater area; the portion difficult to magnetise being then less strongly magnetised than if the whole system of lines of force were concentrated on a very small cross-sectional area. If Ро denotes a factor characteristic of the material, the length of the piece in question, and w its cross-section, we can write for the magnetic resistance

l w= Po

(13)

W

The factor p is the resistance of a block of the substance, whose length is 1 cm. and cross-section 1 cm.2 It is called the specific magnetic resistance,' and is identical with p as defined by (12).

pas

If a magnetic circuit consists of a number of successive parts, whose separate resistances are w1, w2, W3 resistance of the entire circuit is

the

115. Fundamental law of the magnetic circuit. If M is the magneto-motive force around a circuit, made up of tubes of induction, and having the magnetic resistance W, the flux of induction I is given by the ratio of M to W.

The flux of induction I, that is the number of lines of

induction within a given tube of induction, is equal to the magneto-motive force M, divided by the total magnetic resistance W which the lines of induction encounter; that is

M
I=
W

(14)

This law was formulated in 1886 by J. and E. HOPKINSON and G. KAPP. It is analogous to Ohm's law in electricity. But whereas in the latter case the resistance is a true constant, the magnetic resistance p=1/μ must, in accordance with § 111, be regarded as a function of the field intensity.

E.-Energy in the magnetic field

116. Work of magnetisation.-In magnetising a piece of steel, a certain amount of energy must be expended, such as we measure in the simplest case by the product of a force and a displacement. If the method employed is that of rubbing the bar to be magnetised with a permanent magnet, we shall have to consider the mutual action of two systems of lines of force: one arising from the permanent magnet, the other from the newly acquired poles of the piece of steel which we are magnetising. Supposing that during the operation the centre of gravity of the moving bar remains always at the same height, and neglecting the frictional forces which have to be overcome, there is still a certain amount of work to be done against the tensions and pressures which accompany the lines of force. If the north pole is being used to produce the magnetisation, it has to be moved away from that end of the steel where a south pole has been produced, and towards the northseeking end; to accomplish which, work must be expended.

The process of magnetising a bar of steel consists in imparting to it a certain quantity of energy in a special form.

The work which must be expended in the process is called the work of magnetisation; no change is produced in the mass of the bar, as determined by the balance ($20).

In accordance with the principle of the conservation of energy, no work is lost during the transformation. So long as it is not further changed into other forms, the transformed energy remains permanently in the magnetised bar.

If a body be magnetised alternately in opposite senses, after some time it becomes perceptibly heated, a considerable portion of the work of magnetisation being transformed into heat energy.

117. Localisation of the energy of the field.-A portion of the work expended in magnetisation will be consumed in changing the disposition of the molecules, uniting them to form those chains of molecules which distinguish a magnetised from an unmagnetised body. Another portion of the work of magnetisation is stored in the field surrounding the magnet. The space around a magnet-pole is in a certain sense impenetrable to the field of another pole of like name.

The impenetrability implies the existence of a volumedistribution of energy. We must conceive of the magnetic energy as residing in the separate volume-elements of the magnetic field.

The pressures and tensions to which this form of energy is related may be made directly evident (QUINCKE). The existence of the field changes the optical behaviour of the medium, which is evidence that this same special form of energy resides in the field itself.

118. Mechanical example of the localisation of energy.To one end of an elastic wire is attached a load p, considerable in comparison with the weight of the wire. The wire, being fixed at the other end, will be extended by a length 1, proportional to the weight p; so that we may write pel. The factor is constant within wide limits; it depends on the length, cross-section and material of the wire, that is, on the special properties of the medium in which energy is to be stored. To extend the wire by a further amount dl requires the expenditure of work dA = force × displacement = p.dl =ɛl.dl, which depends on l as well as

on dl. The quantity of work dA required to produce an additional extension dl is accordingly greater, the greater the force p already applied. Corresponding to any given extension of the wire, the amount of work which has been expended and is stored up in the wire, is proportional to the square of l. For on integrating the expression dA, we obtain

A = el2;

the wire having no energy of extension when 1=0. Hence I determines the amount of elastic energy residing in the wire.

The value for dA may be resolved into factors in two

Each expression for the energy is the product of two factors; the form A=el. 1/2 having the peculiarity that the two factors are not independent of one another.

119. Measure of the energy of the magnetic field. That the accumulation of magnetic energy in a field must be accompanied by effects somewhat similar to those in our illustrative example (§ 118) is made clear to us when we consider the lines of force, the deformations which they suffer when a pole (for example a unit pole) is introduced into the field, and the pressures and tensions which we know to exist. Since a pole in a magnetic field experiences a force which is greater the more closely the lines of force are packed, that is, the greater the field-intensity , it follows that the introduction of new lines of force, and with them new energy, into the field, is more difficult the higher the value which the field-intensity has already reached.

Let be the field-intensity and B the magnetic induc