Page:Elementary Text-book of Physics (Anthony, 1897).djvu/279

§ 242] (2) The force due to a short bar magnet in any direction may be determined by determining the rate of change of its potential in that direction. It is not, however, important to determine this force in the general case: it will be sufficient to determine it for points in the line of the axis of the magnet.

Let the length of the magnet $$NS$$ (Pig. 75) be represented by $$2l$$ and the distance from its centre to the point $$P$$ by $$r.$$ Then the force at $$P$$ due to the pole at $$N,$$ and directed away from the magnet, is $$\frac{m}{(r - l)^2}\cdot$$ and the force due to the pole at $$S,$$ and directed toward the magnet, is $$\frac{m}{(r + l)^2}\cdot$$ Now we may write $$\frac{m}{(r - l)^2} = \frac{m}{r^2 - 2lr} = m \left( \frac{1}{r^2} + \frac{2l}{r^2} \right),$$ since $$l$$ is very small in comparison with $$r,$$ and similarly $$\frac{}{} = m \left( \frac{1}{r^2} - \frac{2l}{r^3} \right)\cdot$$ The force at $$P$$ due to the magnet and directed away from it is, therefore, (3) In the construction of apparatus used in the measuring of magnetic quantities it is important to know the moment of couple set up by one magnet on another. We will determine this for the particular case in which both the magnets are small in comparison with the distance between their centres, and in which the centre of one is situated on the prolongation of the axis of the other. We will call the magnet, the axis of which lies in the line joining the centres, the first magnet, and the other the second magnet, and will examine the couple exerted on the second magnet by the first. Under the limitations made as to the size of the magnets, we may assume that the forces exerted by the first magnet on the poles of the second are the same as if the poles of the second magnet lay in the prolongation of the axis of the first magnet, and that they are the same for any position of the second magnet (Fig. 76).