cos θ − cos θ′ = constant.
(9)
 | |
| Fig. 2. | Fig. 3. |
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 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 = − | δV | = | M | (3 cos² θ − 1), |
| δx | r³ |
| Y = − | δV | = | M | (3 sin² θ cos θ). |
| δy | r³ |
(11)