Page:Elementary Text-book of Physics (Anthony, 1897).djvu/278

264 Let the angle $$PON$$ equal $$\theta$$ and draw the perpendiculars $$NQ$$ and $$OR$$ to $$PS.$$ Then, in the limit, if $$SN$$ is very small in comparison with $$OP,$$ we have $$PN = r - \Delta r$$ and $$PS = r + \Delta r,$$ where $$\Delta r$$ is a small length equal to $$SR = l. \cos \theta.$$ The potential at $$P$$ due to the pole at $$N$$ is $$\frac{m}{r - \Delta r} = m \left( \frac{1}{r} + \frac{\Delta r}{r^2} \right),$$ since $$\Delta r$$ is very small in comparison with $$r.$$ Similarly the potential at $$P$$ due to the pole at $$S$$ is $$-\frac{m}{r + \Delta r} = -m \left( \frac{1}{r} - \frac{\Delta r}{r^2} \right) \cdot$$ The potential at $$P$$ due to the magnet is therefore where $$M$$ is the magnetic moment of the magnet. We may consider the magnetic moment as projected upon the line $$r$$ by multiplication by $$\cos \theta;$$ the formula shows that the potential at any point due to a short magnet is equal to the projection of the magnetic moment upon the line joining the centre of the magnet with the point, divided by the square of the length of that line.

The maximum value of the potential due to the magnet, for a given value of $$r,$$ is $$\frac{M}{R^2},$$ where $$R$$ represents the assigned value of $$r.$$ If we set $$\frac{M}{R} = \frac{M. \cos \theta}{r^2}$$ we obtain $$r^2 = R^2 \cos \theta$$ as the equation of the equipotential surfaces at a considerable distance from the small magnet. When $$R = \infty,$$ it determines an equipotential surface of zero potential, for which, for every finite value of $$r,$$ we have $$\cos \theta = 0,$$ and $$\theta = \frac{\pi}{2}\cdot$$ The plane passing through the centre of the magnet and perpendicular to its axis is therefore an equipotential surface of zero potential. Since $$r = 0$$ whenever $$\cos \theta = 0,$$ whatever be the value of $$R,$$ all the other equipotential surfaces pass through the point $$O;$$ they are in general ovoid surfaces surrounding the poles. The lines of force of the magnet arise at the north pole and pass perpendicularly through all these surfaces to the south pole.