MECHANISMS DEPENDING ON ELECTROMAGNETISM 247

For instance, the approach can be prevented from taking place freely, as would be represented by the plain arrows of fig. 42, but by suitable guides can be made to take a slanting direction-in the direction of the feathered arrows, for instance. Or the free motion may be increased or be equalised by any of the well-known kinematic arrangements for transmission— different kinds of levers, toothed wheels, &c.; the variations in attraction may also be partially compensated by suitable springs. Another method consists in closing the circuit (fig. 42, C), more or less, by means of an iron wedge, which is moved in a direction at right angles to the centroid. By suitably choosing the section of the wedge, an approximately uniform increase of the reluctance may be obtained when the wedge is drawn out in the direction of the arrow. Where no great attraction is required, a better equalisation may be produced by never completely opening the circuit, so that the continuity of the ferromagnetic substance is never entirely broken. The plan of such an arrangement is seen in fig. 42, B, where the halves of the ring turn about a joint S, and therefore always touch in this point. The magnetic attraction will exert a pretty uniform torque on the upper half of the ring, in its rotation, as represented by the arrows.

Figs. 43, 44, and 45 represent various types of electromagnets used for the most varied purposes, the action of which

is at once seen. The range here is tolerably extensive, and can partially be regulated; the attraction is, however, not uniform. These forms are the transition to appliances in which a soft iron core is drawn into a coil, which may be either iron clad or uncovered, to the discussion of which we will now turn.

§ 160. Small Iron Sphere in a Magnetic Field.-Let us first investigate generally the mechanical forces exerted by an electromagnetic field on small ferromagnetic bodies in it. For simplicity's sake we will take the case of a small iron sphere, as it offers no preferential direction. In an external field of arbitrary distribution, its magnetisation will, by symmetry, be in the direction of the intensity of the field . From § 33, the curve of magnetisation of an iron sphere scarcely differs from a straight line through the origin of co-ordinates, the equation of which, since the demagnetising factor of a solid sphere is 4 π/3 (§ 30), is the following:

and this equation will hold with sufficient approximation up to values of something like

3

1500 C.G.S.,

that is,

5. 6000 C.G.S.

If V is the volume of the sphere, its magnetic moment may be written

(22)

M = IV = V Se

3 4 π

Now it may be shown that the mechanical force exerted by the field on a very small sphere in a particular direction-for instance, that of the X-axis-has the following component F:

If we now imagine a surface through all points of the field in which the intensity, quite apart from its direction, has a prescribed numerical value, this will form a magnetic isodynamic surface on which H., and therefore also 2, is constant. If we now consider, in the usual way, a group of such surfaces in space, which correspond to an arithmetical series of values of a constant surface parameter (§ 38), the resultant force on the sphere at each point is directed along the perpendicular N to the corresponding surface passing through the point, and amounts

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 considerations 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), z = √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 î (§§ 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

zi

In the top half of fig. 46 the function 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 Ir, according to equation (5), § 6]; at a distance x = 2 r—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 xr7 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 9 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.

π

§ 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.