6

Meaning of the term Magnetic Polarization.'

382.] In speaking of the state of the particles of a magnet as magnetic polarization, we imply that each of the smallest parts into which a magnet may be divided has certain properties related to a definite direction through the particle, called its Axis of Magnetization, and that the properties related to one end of this axis are opposite to the properties related to the other end.

The properties which we attribute to the particle are of the same kind as those which we observe in the complete magnet, and in assuming that the particles possess these properties, we only assert what we can prove by breaking the magnet up into small pieces, for each of these is found to be a magnet.

Properties of a Magnetized Particle.

383.] Let the element dx dydz be a particle of a magnet, and let us assume that its magnetic properties are those of a magnet the strength of whose positive pole is m, and whose length is ds. Then if P is any point in space distant from the positive pole and from the negative pole, the magnetic potential at P will be m due to the positive pole, and due to the negative pole, or

m

m

--

додо

(1)

If ds, the distance between the poles, is very small, we may put

r-rds cos e,

(2)

where is the angle between the vector drawn from the magnet to P and the axis of the magnet, or

384.] The product of the length of a uniformly and longitudinally magnetized bar magnet into the strength of its positive pole is called its Magnetic Moment.

Intensity of Magnetization.

The intensity of magnetization of a magnetic particle is the ratio of its magnetic moment to its volume. We shall denote it by I. The magnetization at any point of a magnet may be defined by its intensity and its direction. Its direction may be defined by its direction-cosines A, p, v.

385.]

COMPONENTS OF MAGNETIZATION.

9

Components of Magnetization.

The magnetization at a point of a magnet (being a vector or directed quantity) may be expressed in terms of its three components referred to the axes of coordinates. Calling these A, B, C, A = Iλ, B = Iμ, C = Iv,

and the numerical value of I is given by the equation

I2 = A2 + B2 + C2.

(4)

(5)

385.] If the portion of the magnet which we consider is the differential element of volume dady dz, and if I denotes the intensity of magnetization of this element, its magnetic moment is Idxdy dz. Substituting this for m ds in equation (3), and remembering that

r cos e = λ(-x) +μ (n− y) + v(5—2),

(6)

where έ, n, $ are the coordinates of the extremity of the vector r drawn from the point (x, y, z), we find for the potential at the point (,, ) due to the magnetized element at (x, y, z),

8V =

{A (§−x)+B (n− y) + C (5—2)} — dx dy dz.

+C

(7)

To obtain the potential at the point (§: n, $) due to a magnet of finite dimensions, we must find the integral of this expression for every element of volume included within the space occupied by the magnet, or

V

=

= [[] { 4 (E− x) + B (n − y) + C (5−2)} — dæ dy dz. (8)

Integrating by parts, this becomes

where the double integration in the first three terms refers to the surface of the magnet, and the triple integration in the fourth to the space within it.

If l, m, n denote the direction-cosines of the normal drawn outwards from the element of surface dS, we may write, as in Art. 21, the sum of the first three terms,

SSU A + m B +

! + m B + nC) / ds,

where the integration is to be extended over the whole surface of the magnet.

If we now introduce two new symbols σ and p, defined by the

the expression for the potential may be written

v = f f = ds + [ffe dødy dz.

V

386.] This expression is identical with that for the electric potential due to a body on the surface of which there is an electrification whose surface-density is σ, while throughout its substance there is a bodily electrification whose volume-density is p. Hence, if we assume σ and p to be the surface- and volume-densities of the distribution of an imaginary substance, which we have called 'magnetic matter,' the potential due to this imaginary distribution will be identical with that due to the actual magnetization of every element of the magnet.

The surface-density is the resolved part of the intensity of magnetization I in the direction of the normal to the surface drawn outwards, and the volume-density p is the convergence' (see Art. 25) of the magnetization at a given point in the magnet.

This method of representing the action of a magnet as due to a distribution of magnetic matter' is very convenient, but we must always remember that it is only an artificial method of representing the action of a system of polarized particles.

On the Action of one Magnetic Molecule on another.

387.] If, as in the chapter on Spherical Harmonics, Art. 129,

where l, m, n are the direction-cosines of the axis, then the potential due to a magnetic molecule at the origin, whose axis is parallel to h1, and whose magnetic moment is m1, is

where A, is the cosine of the angle between h, and r.

(2)

Again, if a second magnetic molecule whose moment is m2, and whose axis is parallel to h2, is placed at the extremity of the radius vector r, the potential energy due to the action of the one magnet on the other is

m1 m2 =

2.3

(3)

(4)

12

where is the cosine of the angle which the axes make with each other, and A1, A2 are the cosines of the angles which they make with r.

Let us next determine the moment of the couple with which the first magnet tends to turn the second round its centre.

Let us suppose the second magnet turned through an angle do in a plane perpendicular to a third axis h, then the work done

against the magnetic forces will be

d W
аф

do, and the moment of the

The actual moment acting on the second magnet may therefore be considered as the resultant of two couples, of which the first acts in a plane parallel to the axes of both magnets, and tends to increase the angle between them with a force whose moment is

while the second couple acts in the plane passing through

(6)

and

the axis of the second magnet, and tends to diminish the angle between these directions with a force

where (rh1), (r h2), (h1 h2) denote the angles between the lines r, h1, h2.

To determine the force acting on the second magnet in a direction parallel to a line 3, we have to calculate

=

(8)

(9)

3 λg (μ12—5λ12) +3 μ13 λ2+3 μ23

If we suppose the actual force compounded of three forces, R, H1 and Д, in the directions of r, h1 and h2 respectively, then the force in the direction of h is

Since the direction of h is arbitrary, we must have

The force R is a repulsion, tending to increase r; H1 and H1⁄2 act on the second magnet in the directions of the axes of the first and second magnet respectively.

This analysis of the forces acting between two small magnets was first given in terms of the Quaternion Analysis by Professor Tait in the Quarterly Math. Journ. for Jan. 1860. See also his work on Quaternions, Art. 414.

Particular Positions.

388.] (1) If λ, and λ, are each equal to 1, that is, if the axes of the magnets are in one straight line and in the same direction, 1, and the force between the magnets is a repulsion M12 =

The negative sign indicates that the force is an attraction.

(13)

(2) If A, and A, are zero, and 12 unity, the axes of the magnets are parallel to each other and perpendicular to r, and the force is a repulsion

3 m 1 m 2
go4

(14)

to the first magnet. This is equivalent to a single force

3 m1 m2 p4

acting parallel to the direction of the axis of the second magnet, and cutting r at a point two-thirds of its length from my.

Thus in the figure (1) two magnets are made to float on water, ma