Page:EB1911 - Volume 17.djvu/343

Rh If H0 is constant, the force will be zero; if H0 is variable, the sphere will tend to move in the direction in which H0 varies most rapidly. The coefficient / (1 + ) is positive for ferromagnetic and paramagnetic substances, which will therefore tend to move from weaker to stronger parts of the field; for all known diamagnetic substances it is negative, and these will tend to move from stronger to weaker parts. For small bodies other than spheres the coefficient will be different, but its sign will always be negative for diamagnetic substances and positive for others; hence the forces acting on any small body will be in the same directions as in the case of a sphere.

Directing Couple acting on an Elongated Body.—In a non-uniform field every volume-element of the body tends to move towards regions of greater or less force according as the substance is paramagnetic or diamagnetic, and the behaviour of the whole mass will be determined chiefly by the tendency of its constituent elements. For this reason a thin bar suspended at its centre of gravity between a pair of magnetic poles will, if paramagnetic, set itself along the line joining the poles, where the field is strongest, and if diamagnetic, transversely to the line. These are the “axial” and “equatorial” positions of Faraday. It can be shown that in a uniform field an elongated piece of any non-crystalline material is in stable equilibrium only when its length is parallel to the lines of force; for diamagnetic substances, however, the directing couple is exceedingly small, and it would hardly be possible to obtain a uniform field of sufficient strength to show the effect experimentally.

Relative Magnetization.—A substance of which the real susceptibility is will, when surrounded by a medium having the susceptibility ′, behave towards a magnet as if its susceptibility were a = ( − ′) / (1 + 4′). Since 1 + 4′ can never be negative, the apparent susceptibility a will be positive or negative according as is greater or less than ′. Thus, for example, a tube containing a weak solution of an iron salt will appear to be diamagnetic if it is immersed in a stronger solution of iron, though in air it is paramagnetic.

Circular Magnetization.—An electric current i flowing uniformly through a cylindrical wire whose radius is a produces inside the wire a magnetic field of which the lines of force are concentric circles around the axis of the wire. At a point whose distance from the axis of the wire is r the tangential magnetic force is

H = 2ir/a2 (39)

it therefore varies directly as the distance from the axis, where it is zero. If the wire consists of a ferromagnetic metal, it will become “circularly” magnetized by the field, the lines of magnetization being, like the lines of force, concentric circles. So long as the wire (supposed isotropic) is free from torsional stress, there will be no external evidence of magnetism.

Magnetic Shielding.—The action of a hollow magnetized shell on a point inside it is always opposed to that of the external magnetizing force, the resultant interior field being therefore weaker than the field outside. Hence any apparatus, such as a galvanometer, may be partially shielded from extraneous magnetic action by enclosing it in an iron case. If a hollow sphere of which the outer radius is R and the inner radius r is placed in a uniform field H0, the field inside will also be uniform and in the same direction as H0, and its value will be approximately

(40)

For a cylinder placed with its axis at right angles to the lines of force,

(41)

These expressions show that the thicker the screen and the greater its permeability, the more effectual will be the shielding action. Since can never be infinite, complete shielding is not possible.

Magneto-Crystallic Phenomenon.—In anisotropic bodies, such as crystals, the direction of the magnetization does not in general coincide with that of the magnetic force. There are, however, always three principal axes at right angles to one another along which the magnetization and the force have the same direction. If each of these axes successively is placed parallel to the lines of force in a uniform field H, we shall have

I1 = 1H, &emsp; I2 = 2H, &emsp; I3 = 3H,

the three susceptibilities being in general unequal, though in some cases two of them may have the same value. For crystalline bodies the value of (+ or −) is nearly always small and constant, the magnetization being therefore independent of the form of the body and proportional to the force. Hence, whatever the position of the body, if the field be resolved into three components parallel to the principal axes of the crystal, the actual magnetization will be the resultant of the three magnetizations along the axes. The body (or each element of it) will tend to set itself with its axis of greatest susceptibility parallel to the lines of force, while, if the field is not uniform, each volume-element will also tend to move towards places of greater or smaller force (according as the substance is paramagnetic or diamagnetic), the tendency being a maximum when the axis of greatest susceptibility is parallel to the field, and a minimum when it is perpendicular to it. The phenomena may therefore be exceedingly complicated.

3.

Magnetic Moment.—The moment M of a magnet may be determined in many ways, the most accurate being that of C. F. Gauss, which gives the value not only of M, but also that of H, the horizontal component of the earth’s force. The product MH is first determined by suspending the magnet horizontally, and causing it to vibrate in small arcs. If A is the moment of inertia of the magnet, and t the time of a complete vibration, MH = 42A / t2 (torsion being neglected). The ratio M/H is then found by one of the magnetometric methods which in their simplest forms are described below. Equation (44) shows that as a first approximation.

M / H = (d&#8202;2 − l&#8202;2) tan /2d,

where l is half the length of the magnet, which is placed in the “broadside-on” position as regards a small suspended magnetic needle, d the distance between the centre of the magnet and the needle, and the angle through which the needle is deflected by the magnet. We get therefore

M2 = MH × M/H = 22A (d&#8202;2 − l&#8202;2)2 tan /t&#8202;2d (42)

H2 = MH × H/M = 82Ad / $($t&#8202;2 (d&#8202;2 − l&#8202;2)2 tan $)$. (43)

When a high degree of accuracy is required, the experiments and calculations are less simple, and various corrections are applied. The moment of a magnet may also be deduced from a measurement of the couple exerted on the magnet by a uniform field H. Thus if the magnet is suspended horizontally by a fine wire, which, when the magnetic axis points north and south, is free from torsion, and if is the angle through which the upper end of the wire must be twisted to make the magnet point east and west, then MH = C, or M = C/H, where C is the torsional couple for 1°. A bifilar suspension is sometimes used instead of a single wire. If P is the weight of the magnet, l the length of each of the two threads, 2a the distance between their upper points of attachment, and 2b that between the lower points, then, approximately, MH = P(ab/l) sin. It is often sufficient to find the ratio of the moment of one magnet to that of another. If two magnets having moments M, M′ are arranged at right angles to each other upon a horizontal support which is free to rotate, their resultant R will set itself in the magnetic meridian. Let be the angle which the standard magnet M makes with the meridian, then M′/R = sin, and M/R = cos , whence M′ = M tan.

A convenient and rapid method of estimating a magnetic moment has been devised by H. Armagnat. The magnet is laid on a table with its north pole pointing northwards, A compass having a very short needle is placed on the line which bisects the axis of the magnet at right angles, and is moved until a neutral point is found where the force due to the earth’s field H is balanced by that due to the magnet. If 2l is the distance between the poles m and −m, d the distance from either pole to a point P on the line AB (fig. 5), we have for the resultant force at P

R = −2 cos × m / d&#8202;2 = −2lm / d&#8202;3 = -M / d&#8202;3.

When P is the neutral point, H is equal and opposite to R; therefore M = Hd&#8202;3, or the moment is numerically equal to the cube of the distance from the neutral point to a pole, multiplied by the