an angle ε with the normal, the normal component of the magnetization,
I cos ε, is called the surface density of the magnetism, and is generally denoted by σ.
Potential and Magnetic Force.—The magnetic potential at any point in a magnetic field is the work which would be done against the magnetic forces in bringing a unit pole to that point from the boundary of the field. The line through the given point along which the potential decreases most rapidly is the direction of the resultant magnetic force, and the rate of decrease of the potential in any direction is equal to the component of the force in that direction. If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then
| −δVδn＝F,−δVδx＝X,−δVδy＝Y,−δVδz＝Z. | (6) |
Surfaces for which the potential is constant are called equipotential surfaces. The resultant magnetic force at every point of such a surface is in the direction of the normal (n) to the surface; every line of force therefore cuts the equipotential surfaces at right angles. The potential due to a single pole of strength m at the distance r from the pole is
| V＝m/r, | (7) |
the equipotential surfaces being spheres of which the pole is the centre and the lines of force radii. The potential due to a thin magnet at a point whose distance from the two poles respectively is r and r ′ is
| V＝m(l/r＝l/r ′). | (8) |
When V is constant, this equation represents an equipotential surface.
The equipotential surfaces are two series of ovoids surrounding the two poles respectively, and separated by a plane at zero potential passing perpendicularly through the middle of the axis. If r and r ′ make angles θ and θ′ with the axis, it is easily shown that the equation to a line of force is
| cos θ−cos′ θ＝constant. | (9) |
At the point where a line of force intersects the perpendicular bisector of the axis r＝r ′＝r0, say, and cos θ − cos θ′ obviously＝l/r0, l being
| |
| Fig. 2. | Fig. 3. |
the distance between the poles; l/r0 is therefore the value of the constant in (9) for the line in question. Fig. 2 shows the lines of force and the plane sections of the equipotential surfaces for a thin magnet with poles concentrated at its ends. The potential due to a small magnet of moment M, at a point whose distance from the centre of the magnet is r, is
| V＝M cos θ/r2, | (10) |
where θ is the angle between r and the axis of the magnet. Denoting the force at P (see fig. 3) by F, and its components parallel to the co-ordinate axes by X and Y, we have
| X＝δVδx＝Mr 2(3 cos2θ−1), Y＝δVδy＝Mr 3(3 sin θ cos θ. |
(11) |
If Fr is the force along r and Ft that along t at right angles to r,
| Fr＝X cosθ+Y sinθ＝Mr 3 2 cos θ, | (12) |
| Ft＝−X sinθ+Y cosθ＝Mr 3 sin θ. | (13) |
For the resultant force at P,
| F＝√(Fr2 + Ft2)＝Mr 3 √(3 cos2 θ + 1). | (14) |
The direction of F is given by the following construction: Trisect OP at C, so that OC＝OP/3; draw CD at right angles to OP, to cut the axis produced in D; then DP will be the direction of the force at P. For a point in the axis OX, θ＝0; therefore cos θ＝1, and the point D coincides with C; the magnitude of the force is, from (14),
| Fx＝2M / r 3, | (15) |
its direction being along the axis OX. For a point in the line OY bisecting the magnet perpendicularly, θ＝π/2 therefore cos θ＝0, and the point D is at an infinite distance. The magnitude of the force is in this case
| Fy＝M / r 3, | (16) |
and its direction is parallel to the axis of the magnet. Although the above useful formulae, (10) to (15), are true only for an infinitely small magnet, they may be practically applied whenever the distance r is considerable compared with the length of the magnet.
Fig. 4.
Couples and Forces between Magnets.—If a small magnet of moment M is placed in the sensibly uniform field H due to a distant magnet, the couple tending to turn the small magnet upon an axis at right angles to the magnet and to the force is
| MH sin θ, | (17) |
where θ is the angle between the axis of the magnet and the direction of the force. In fig. 4 S′N′ is a small magnet of moment M′, and SN a distant fixed magnet of moment M; the axes of SN and S′N′ make angles of θ and φ respectively with the line through their middle points. It can be deduced from (17), (12) and (13) that the couple on S′N′ due to SN, and tending to increase φ, is
| MM′ (sin θ cos φ − 2 sin φ cos θ) / r 3. | (18) |
This vanishes if sin θ cos φ＝2 sin φ cos θ, i.e. if tan φ＝12 tan θ, S′N′ being then along a line of force, a result which explains the construction given above for finding the direction of the force F in (14). If the axis of SN produced passes through the centre of S′N′, θ＝0, and the couple becomes
| 2MM′ sin φ/r 3, | (19) |
tending to diminish φ; this is called the “end on” position. If the centre of S′N′ is on the perpendicular bisector of SN, θ＝12π, and the couple will be
| MM′ cos φ/r 3, | (20) |
tending to increase φ; this is the “broadside on” position. These two positions are sometimes called the first and second (or A and B) principal positions of Gauss. The components X, Y, parallel and perpendicular to r, of the force between the two magnets SN and S′N′ are
| X＝3MM′ (sin θ sin φ − 2 cos θ cos φ) / r 4, | (21) |
| Y＝3MM′ (sin θ cos φ + sin φ cos θ) / r 4. | (22) |
It will be seen that, whereas the couple varies inversely as the cube of the distance, the force varies inversely as the fourth power.
Distributions of Magnetism.—A magnet may be regarded as consisting of an infinite number of elementary magnets, each having a pair of poles and a definite magnetic moment. If a series of such elements, all equally and longitudinally magnetized, were placed end to end with their unlike poles in contact, the external action of the filament thus formed would be reduced to that of the two extreme poles. The same would be the case if the magnetization of the filament varied inversely as the area of its cross-section a in different parts. Such a filament is called a simple magnetic solenoid, and the product aI is called the strength of the solenoid. A magnet which consists entirely of such solenoids, having their ends either upon the surface or closed upon themselves, is called a solenoidal magnet, and the magnetism is said to be distributed solenoidally; there is no free magnetism in its interior. If the constituent solenoids are parallel and of equal strength, the magnet is also uniformly magnetized. A thin sheet of magnetic matter magnetized normally to its surface in such a manner that the magnetization at any place is inversely proportional to the thickness h of the sheet at that place is called a magnetic shell; the constant product hI is the strength of the shell and is generally denoted by Φ or φ. The potential at any point due to a magnetic shell is the product of its strength into the solid angle ω subtended by its edge at the given point, or V＝Φω. For a given strength, therefore, the potential depends solely upon the boundary of the shell, and the potential outside a closed shell is everywhere zero. A magnet which can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, is called a lamellar magnet, and the magnetism is said to be distributed lamellarly. A magnet consisting of a series of plane shells of equal strength arranged at right angles to the direction of magnetization will be uniformly magnetized.
It can be shown that uniform magnetization is possible only when the form of the body is ellipsoidal. (Maxwell, Electricity and Magnetism, II., § 437). The cases of greatest practical importance are those of a sphere (which is an ellipsoid with three equal axes) and an ovoid or prolate ellipsoid of revolution. The potential due to a uniformly magnetized sphere of radius a for an external point at a distance r from the centre is
| V＝43πa3I cos θ/r 2, | (23) |
θ being the inclination of r to the magnetic axis. Since 43πa3I is the moment of the sphere (＝volume × magnetization), it appears from (10) that the magnetized sphere produces the same external effect as a very small magnet of equal moment placed at its centre and magnetized in the same direction; the resultant force therefore is the same as in (14). The force in the interior is uniform, opposite
