Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/44

Since the direction of h₃ is arbitrary, we must have

${\begin{matrix}R={\frac {3m_{1}m_{2}}{r^{4}}}(m_{12}-5\lambda _{1}\lambda _{2}),\\H_{1}={\frac {3m_{1}m_{2}}{r^{4}}}\lambda _{2},\quad H_{2}={\frac {3m_{1}m_{2}}{r^{4}}}\lambda _{1}.\end{matrix}}$

(12)

The force R is a repulsion, tending to increase r; H_l and H₂ act on the second magnet in the directions of the axes of the first and second magnet respectively.

This analysis of the forces acting between two small magnets was first given in terms of the Quaternion Analysis by Professor Tait in the Quarterly Math. Journ. for Jan. 1860. See also his work on Quaternions, Art. 414.

Particular Positions.

388.] (1) If λ₁ and λ₂ are each equal to 1, that is, if the axes of the magnets are in one straight line and in the same direction, μ₁₂ = 1, and the force between the magnets is a repulsion

$R+H_{1}+H_{2}=-{\frac {6m_{1}m_{2}}{r^{4}}}.$

(13)

The negative sign indicates that the force is an attraction.

(2) If λ₁ and λ₂ are zero, and μ₁₂ unity, the axes of the magnets are parallel to each other and perpendicular to r, and the force is a repulsion

${\frac {3m_{1}m_{2}}{r^{4}}}.$

(14)

In neither of these cases is there any couple.

(3) If

$\lambda _{1}=1{\text{ and }}\lambda _{2}=0,{\text{ then }}\mu _{12}=0.\,$

(15)

The force on the second magnet will be ${\frac {3m_{1}m_{3}}{r^{4}}}$ in the direction of its axis, and the couple will be ${\frac {2m_{1}m_{2}}{r^{4}}}$, tending to turn it parallel to the first magnet. This is equivalent to a single force ${\frac {3m_{1}m_{2}}{r^{4}}}$ acting parallel to the direction of the axis of the second magnet, and cutting r at a point two-thirds of its length from m₂.

Fig. 1.

Thus in the figure (1) two magnets are made to float on water, m₂