394 Permanent Magnetism [CH. xi 



11. Prove that there are four positions in which a given bar magnet may be placed 

 so as to destroy the earth's control of a compass-needle, so that the needle can point 

 indifferently in all directions. If the bar is short compared with its distance from the 

 needle, shew that one pair of these positions are about l times more distant than the 

 other pair. 



12. Three small magnets, each of magnetic moment /i, are fixed at the angular points 

 of an equilateral triangle ABC, so that their north poles lie in the directions AC, AB, BC 

 respectively. Another small magnet, moment //, is placed at the centre of the triangle, 

 and is free to move about its centre. Prove that the period of a small oscillation is the 

 same as that of a pendulum of length /6 3 ^/V351/u/i', where b is the length of a side of the 

 triangle, and / the moment of inertia of the movable magnet about its centre. 



13. Three magnetic particles of equal moments are placed at the corners of an 

 equilateral triangle, and can turn about those points so as to point in any direction in 

 the plane of the triangle. Prove that there are four and only four positions of equilibrium 

 such that the angles, measured in the same sense of rotation, between the axes of the 

 magnets and the bisectors of the corresponding angles of the triangle are equal. Also 

 prove that the two symmetrical positions are unstable. 



14. Four small equal magnets are placed at the corners of a square, and oscillate 

 under the actions they exert on each other. Prove that the times of vibration of the 

 principal oscillations are 



2<r J- MW I* 



27T 



27T 



H 







where m is the magnetic moment, and J/F the moment of inertia, of a magnet, and d is a 

 side of the square. 



15. A system of magnets lies entirely in one plane and it is found that when the 

 axis of a small needle travels round a contour in the plane that contains no magnetic 

 poles, the needle turns completely round. Prove that the contour contains at least one 

 equilibrium point. 



16. Prove that the potential of a body uniformly magnetised with intensity 7 is, at 

 any external point, the same as that due to a complex magnetic shell coinciding with the 

 surface of the body and of strength Ix, where x is a coordinate measured parallel to the 

 direction of magnetisation. 



17. A sphere of hard steel is magnetised uniformly in a constant direction and a 

 magnetic particle is held at an external point with the axis of the particle parallel to the 

 direction of magnetisation of the sphere. Find the couples acting on the sphere and on 

 the particle. 



18. A spherical magnetic shell of radius a is normally magnetised so that its strength 

 at any point is S it where S t is a spherical surface harmonic of positive order i. Shew 

 that the potential at a distance r from the centre is 



