ATTRACTIVE ACTION OF CIRCULAR CONDUCTORS 249

and the force for a ferromagnetic sphere is in the direction of increasing values of . We may sum up those consideraHe. tions in the following:

II. In any given field a small ferromagnetic sphere tends always to pass from places of weaker to places of stronger intensity ; and this quite independently of the direction of this vector.

The mechanical force exerted on the sphere further has evidently the scalar potential (§ 39)

The above principle was propounded by Faraday on the basis of his experimental investigations. Its mathematical enunciation is due to Lord Kelvin.1

§ 161. Attractive Action of Circular Conductors on Sphere. Let us apply this fundamental principle to the simple case of a

plane circular conductor carrying the current 1. Let r be the radius of the circle, a the distance along the axis (fig. 46),

22

% = √x2 + r2, the distance of a point on the axis from the

Faraday, Exp. Res., vol. 3, series 21, especially § 2418; Sir W. Thomson, Reprint Electr. and Magnet. §§ 643-646. This potential of the mechanical force must be carefully distinguished from the magnetic potential r (§§ 45, 48); neither are the magnetic isodynamic surfaces directly connected with the ordinary equipotential surfaces.

circumference. The numerical value of the field intensity in a point of the axis (~) [§ 6 C, equation (4)] is

Let the small iron sphere be restricted to motion along the X-axis, for instance, by being compelled to move without friction along a tube. The component of force then acting upon it amounts, according to equations (23) and (26), to

= V
8 п

π VI24

дг -6 a 2
дидх

In the top half of fig. 46 the function z-3 is graphically represented. According to (26) this is to be multiplied by the constant±2 Ir2, in order to have the intensity in a given place. This, as will be seen, attains a maximum in the plane. of the circular conductor [2π I/r, according to equation (5), § 6]; at a distance x = 2r-that is, equal to the diameter of the circle-it amounts to only eight per cent. of that maximum.

=

In the lower half of fig. 46 the fraction a/z is represented. This is 0 in the plane of the circle itself, and then rapidly increases. As a repeated differentiation shows, it attains a maximum for a r√7=0·38 r-while the steepest part of the curve of intensity is at x = 0·5 r—and then gradually diminishes to very small values. Multiplication of that function by 9π VI24 gives the component of force in absolute measure. This is always directed towards the conductor, that is in the sense of increasing values of the intensity of the field; and this independent of its direction. The left half of fig. 46 is omitted, as it is symmetrical with the right half represented.

§ 162. Attractive Action of Coils on Spheres.-The field in the axis of a long, uniformly-wound coil (§ 6 D) may be regarded as the superposition of the fields due to individual turns. The transition from the portion of the field in the middle of the coil, which is known to be uniform, towards the outside is represented by a curve, which is like that of fig. 46. The field at the opening attains half the value of its value in the middle,

ATTRACTIVE ACTION OF COILS ON SPHERES

251

and then rapidly diminishes. A small iron sphere, with its motion again restricted to the axis of the coil,' will, as with the circular conductor, be drawn into the coil. The attractive action is at its maximum near the opening, and then decreases as we approach the region of appreciably uniform intensity, within which a mechanical action is of course no longer exerted on an iron sphere.

An accurate representation of the forces by equations. would be as difficult as it would be without object, as they always depend on the particular dimensions of the coil. If we dispense with uniform winding, we may influence as we like the distribution of the field, and therewith that of the attractive force along the axis. This can be attained by altering the pitch of the winding, so that the number of turns per unit length is a variable quantity. If, for example, we desire for any purpose to produce a steady attractive action within a prescribed range on the axis of the coil, then we must have

in which C and B are constants. No general directions can be given as to how the coil must be wound so as to produce a given field. In any given case this must be determined by actual trial, in which the principles laid down may serve as some guide.

For a given position of the sphere the attractive force, according to equation (27), is, ceteris paribus, proportional to the square of the current in the coil. Saturation can never be obtained with a sphere under the influence of the field of a coil.

The discussions in this paragraph apply not only to small spheres, but, approximately, also to other pieces of iron, the dimensions of which are nearly equal in all directions, and are

1 Such a restriction is here necessary as well as with the circular conductor, because the iron sphere, in accordance with Faraday's principle, would otherwise move towards places of higher intensity on the surface of the conductor; that is, over the inner surface of the coil.

small in comparison with those of the coil. But when one dimension preponderates, and is comparable with the length of the coil, the theory is rather complicated, and we are at present compelled to return to empirical investigation.

§ 163. Attractive Action of Coils on Iron Cores.-We possess numerous measurements on the attraction of 'short' or 'long' cores—that is, of such as are shorter or longer than the coil. We shall pass over the older researches of Hankel, Dub, von Feilitzsch, von Waltenhofen, &c.,' and confine ourselves to the discussion of the more recent systematic ones of Bruger.2

His experimental arrangement will be sufficiently clear from fig. 47; the weight of the core in each case was compensated

by one sliding weight, and the attraction was measured by another weight for various relative positions of the core and of the coil. Equilibrium is obtained, that is to say, the attraction

ceases, when the middle of the core M, supposed to be symmetrical, coincides with the middle of the coil m. That follows from symmetry, as well as from the principle of least magnetic reluctance (§ 158). We determine, therefore, the relative position by the height Y of the middle of the core over the middle of the coil. If the short or long core is moved downwards, the force of suction only attains an appre

ciable value just before the lower end U of the core enters the upper opening o of the coil. It then rises to a maximum, and falls

1 G. Wiedemann, Lehre von der Elektricität, vol. 3, §§ 651-665, 1883. 2 Bruger, Action of Solenoids on Iron Cores of various Shapes. Inaugural Dissertation. Erlangen, 1886. [See also Mr. Mahon, the Electrician, vol. 35, pp. 293, 604; 1895. It would appear possible at present to give a theory describing these actions with more or less approximation.—H. du B.]

POLARISED MECHANISMS

253

off again until the above position of equilibrium is attained. Bruger restricted himself to investigating long coils, and found, in agreement with older statements, that the maximum suction corresponds to about the position in which the lower end of the core U is just on the point of emerging from the lower opening u of the coil (as represented in fig. 47); and this holds for cores. the length of which is twice that of the coil.

Bruger used, among others, a coil with the following constants: length L ̧ 13 cm.; number of turns n = 266; field

=

Hm in the middle of the coil, per ampere,

5m 04 πη
L

=

I'

= about 30 C.G.S.

He obtained, for example, with three cores of about 39 cm. (= 3 L) length the curves given in fig. 47. Y defines the position of the core, X gives the corresponding suction in gramme-weight. In the diagrams of the curve:

A A, full curve: cylindrical core;
BB, dot and dash curve: double cone;
CC, dotted curve: moniliform core;

Hm about 180 C.G.S.

m = about 180 C.G.S. = about 250 C.G.S.

For the details we must refer to the paper. Conical cores are used in some arc lamps. The moniliform core gave, as intended, a constant pull over a great range. As long as saturation is excluded, the attraction is roughly approximately proportional to about the square of the current in the coil. Bruger gives curves like those above for various currents. The phenomena for iron-clad coils, as represented in fig. 45, p. 247, are similar; the field then between the inner edges of the shell is more intense and more uniform than in a coil without shell, but outside the coil it diminishes so much the more rapidly, by which the character of the traction curve is somewhat but not materially altered. The case of coils without shells but with a lining of iron is quite different. These do not attract iron cores, but, on the contrary, repel them. Having regard to the principle discussed in § 158, this appears at once intelligible.

§ 164. Polarised Mechanisms.-The phenomena are markedly simpler if the core of the coil is not previously magnetised by the current, but has from the outset permanent magnetisation