633.]

MAGNETIC ENERGY.

247

Hence, the electrostatic energy of the whole field will be the same if we suppose that it resides in every part of the field where electrical force and electrical displacement occur, instead of being confined to the places where free electricity is found.

The energy in unit of volume is half the product of the electromotive force and the electric displacement, multiplied by the cosine of the angle which these vectors include.

In Quaternion language it is - Į SED.

Magnetic Energy.

632.] We may treat the energy due to magnetization in a similar way. If A, B, C are the components of magnetization and a, ß, y the components of magnetic force, the potential energy of the system of magnets is, by Art. 389,

¥ √ √ √ (4 a + B B+ Cy) dx dy dz,

(6)

the integration being extended over the space occupied by magnetized matter. This part of the energy, however, will be included in the kinetic energy in the form in which we shall presently obtain it.

633.] We may transform this expression when there are no electric currents by the following method.

as is always the case in magnetic phenomena where there are no

currents,

SSS (aa +

(a a+ B)+cy)dxdydz = 0,

the integral being extended throughout all space, or

(9)

SSS {(a + 4 = 4) a + (B + 4 = B) B+(y+47C)y} dxdydz=0. (10)

Hence, the energy due to a magnetic system

1

− + [[] (da+BB+Cy) dedydz = != [[] (a2 + 32 + y2) dzdydz,

Electrokinetic Energy.

634.] We have already, in Art. 578, expressed the kinetic energy of a system of currents in the form

where p is the electromagnetic momentum of a circuit, and i is the strength of the current flowing round it, and the summation extends to all the circuits.

But we have proved, in Art. 590, that p may be expressed as a line-integral of the form

p

dx ds

+ Gdy

ds

where F, G, H are the components of the electromagnetic momentum, A, at the point (xyz), and the integration is to be extended round the closed circuit s.

dx
ds

We therefore find

( + G

If u, v, w are the components of the density of the current at any point of the conducting circuit, and if S is the transverse section of the circuit, then we may write

where the integration is to be extended to every part of space where there are electric currents.

635.] Let us now substitute for u, v, w their values as given by the equations of electric currents (E), Art. 607, in terms of the components a, B, y of the magnetic force. We then have

da
dß
G (da H

dx dy

where the integration is extended over a portion of space including all the currents.

If we integrate this by parts, and remember that, at a great distance from the system, a, B, and y are of the order of magnitude r3, we find that when the integration is extended throughout all space, the expression is reduced to

3

d dz
dF

dx

dy

dz

dx

dF

dy

637.]

ELECTROKINETIC ENERGY.

249

By the equations (A), Art. 591, of magnetic induction, we may substitute for the quantities in small brackets the components of magnetic induction a, b, c, so that the kinetic energy may be written

1

T = = f (a a+bB+cy) dɛdy dz,

(19)

where the integration is to be extended throughout every part of space in which the magnetic force and magnetic induction have values differing from zero.

The quantity within brackets in this expression is the product of the magnetic induction into the resolved part of the magnetic force in its own direction.

In the language of quaternions this may be written more simply,

-S.BH.

where B is the magnetic induction, whose components are a, and is the magnetic force, whose components are a, ß, y.

b, c,

636.] The electrokinetic energy of the system may therefore be expressed either as an integral to be taken where there are electric currents, or as an integral to be taken over every part of the field in which magnetic force exists. The first integral, however, is the natural expression of the theory which supposes the currents to act upon each other directly at a distance, while the second is appropriate to the theory which endeavours to explain the action between the currents by means of some intermediate action in the space between them. As in this treatise we have adopted the latter method of investigation, we naturally adopt the second expression as giving the most significant form to the kinetic energy.

According to our hypothesis, we assume the kinetic energy to exist wherever there is magnetic force, that is, in general, in every part of the field. The amount of this energy per unit of volume

is

1

8 T

SBH, and this energy exists in the form of some kind

of motion of the matter in every portion of space.

When we come to consider Faraday's discovery of the effect of magnetism on polarized light, we shall point out reasons for believing that wherever there are lines of magnetic force, there is a rotatory motion of matter round those lines. See Art. 821.

Magnetic and Electrokinetic Energy compared.

637. We found in Art. 423 that the mutual potential energy

of two magnetic shells, of strengths and p', and bounded by the closed curves s and s' respectively, is

where is the angle between the directions of ds and ds', and r is the distance between them.

We also found in Art. 521 that the mutual energy of two circuits s and s', in which currents i and i' flow, is

If i, ' are equal to 4, ' respectively, the mechanical action between the magnetic shells is equal to that between the corresponding electric circuits, and in the same direction. In the case of the magnetic shells, the force tends to diminish their mutual potential energy, in the case of the circuits it tends to increase their mutual energy, because this energy is kinetic.

It is impossible, by any arrangement of magnetized matter, to produce a system corresponding in all respects to an electric circuit, for the potential of the magnetic system is single valued at every point of space, whereas that of the electric system is many-valued.

But it is always possible, by a proper arrangement of infinitely small electric circuits, to produce a system corresponding in all respects to any magnetic system, provided the line of integration which we follow in calculating the potential is prevented from passing through any of these small circuits. This will be more fully explained in Art. 833.

The action of magnets at a distance is perfectly identical with that of electric currents. We therefore endeavour to trace both to the same cause, and since we cannot explain electric currents by means of magnets, we must adopt the other alternative, and explain magnets by means of molecular electric currents.

633.] In our investigation of magnetic phenomena, in Part III of this treatise, we made no attempt to account for magnetic action at a distance, but treated this action as a fundamental fact of experience. We therefore assumed that the energy of a magnetic system is potential energy, and that this energy is diminished when the parts of the system yield to the magnetic forces which act on them.

If, however, we regard magnets as deriving their properties from electric currents circulating within their molecules, their energy

639.]

AMPÈRE'S THEORY OF MAGNETS.

251

is kinetic, and the force between them is such that it tends to move them in a direction such that if the strengths of the currents were maintained constant the kinetic energy would increase.

This mode of explaining magnetism requires us also to abandon the method followed in Part III, in which we regarded the magnet as a continuous and homogeneous body, the minutest part of which has magnetic properties of the same kind as the whole.

We must now regard a magnet as containing a finite, though very great, number of electric circuits, so that it has essentially a molecular, as distinguished from a continuous structure.

If we suppose our mathematical machinery to be so coarse that our line of integration cannot thread a molecular circuit, and that an immense number of magnetic molecules are contained in our element of volume, we shall still arrive at results similar to those of Part III, but if we suppose our machinery of a finer order, and capable of investigating all that goes on in the interior of the molecules, we must give up the old theory of magnetism, and adopt that of Ampère, which admits of no magnets except those which consist of electric currents.

We must also regard both magnetic and electromagnetic energy as kinetic energy, and we must attribute to it the proper sign, as given in Art. 635.

In what follows, though we may occasionally, as in Art. 639, &c., attempt to carry out the old theory of magnetism, we shall find that we obtain a perfectly consistent system only when we abandon that theory and adopt Ampère's theory of molecular currents, as in Art. 644.

The energy of the field therefore consists of two parts only, the electrostatic or potential energy

W =

¥ [ff (Pƒ + Qg + Rh) dx dy dz,

and the electromagnetic or kinetic energy

ON THE FORCES WHICH ACT ON AN ELEMENT OF A BODY PLACED

IN THE ELECTROMAGNETIC FIELD.

Forces acting on a Magnetic Element.

639.] The potential energy of the element de dy dz of a body magnetized with an intensity whose components are A, B, C, and