260 MAGNETISM 



dH , 1 . 



M -j dv = s K j dv. 



dx % dx 



If the body is large, the total force in direction x is 



,v 



from which we may again draw the conclusion that if A: is positive 

 the body tends to move towards the position where f}3Pdv is a 

 maximum, and that if AC is negative it tends to move towards the 

 position whereyH 2 ^ is a minimum. 



Time of vibration of a small needle with M 1 very 

 small suspended in the centre of a field between the 

 poles of a magnet, the field being symmetrical about 

 the axis and the equatorial plane. Lord Kelvin has 

 shown * that the time of vibration of a needle-shaped body 



depends only on its density and magnetic 

 susceptibility, and not on the particular 

 shape or size of the needle so long as it 

 is a needle and small. This very remark- 

 able result has led to a method of de- 

 termining susceptibilities. 



First imagine a particle of volume V 9 

 density p, and susceptibility K to be 

 _ ___ somehow freed from weight and to be 

 a: suspended, under the action of magnetic 

 forces only, by a very fine fibre from the 

 central point C of the field represented 

 in Fig. 202. 



If paramagnetic, it will set at P with 

 the fibre along CP ; if diamagnetic, at Q, 

 with the fibre along CQ; and if dis- 

 Fio. 202. turbed it will vibrate, forming as it 



were a magnetic pendulum. Let us take 



axes as in the figure. We may put the intensity near C in the 

 form 



H* = H c * - A,r> + 



For putting H as a function of x and y and expanding, first powers 

 must disappear to satisfy symmetry. Powers higher than the 

 second may be neglected if we are near enough to C. The signs of 

 the terms with A and B as coefficients are determined by the fact 

 that H increases from C towards P, while it decreases from C 

 towards Q. 



1 7H2 



The magnetic force on the particle in any direction s is ^ KV j t 



Let us suppose it to be diamagnetic, so that its undisturbed 



* Electrostatics and Magnetism, Sect. 670, 



