108 Systems of Conductors [CH. iv 



indication of the difference of potential between the two pairs of quadrants. 

 This angle is most easily observed by attaching a small mirror to the fibre 

 just above the point at which it emerges from the quadrants. 



Let us suppose that when the needle has turned through an angle 6, 

 the total area A of the needle is placed so that an area 8 is inside the pair 

 of quadrants at potential Fj, and an area A S inside the pair at potential 

 F 2 . Let h be the perpendicular distance from either face of the needle to 

 the faces of the quadrants. Then the system may be regarded as two 

 parallel plate condensers of area 8, distance h, and difference of potential 

 -y F^ and two parallel plate condensers for which these quantities have the 

 values A 8, h, vV 2 . There are two condensers of each kind because 

 there are two faces, upper and lower, to the needle. The electrical energy 

 of this system is accordingly 



| 



The energy here appears as a quadratic function of the three potentials 

 concerned: it is expressed in the same form as the W v of 120. The 

 mechanical force tending to increase 0, i.e., the moment of the couple tending 



to turn the needle in the direction of increasing, is therefore -^ . Now 



dv 



in Wy the only term in the coefficients of the potentials which varies with 6 

 is 8, so that on differentiation we obtain 



W ' 4fjrh W 



If r is the radius of the needle measured from its centre, which is under 



dS 

 9(9 



r)Sf 



the line of division of the quadrants we clearly have ^ = r 2 , so that we can 



write the equation just obtained in the form 



80 





In equilibrium this couple is balanced by the torsion couple of the fibre, 

 which tends to decrease 0. This couple may be taken to be k6, where k is a 

 constant, so that the equation of equilibrium is 



r l )r' 



For small displacements of the needle, r 2 may be replaced by a 2 , the 

 radius of the needle at its centre line. Also v is generally large compared 

 with F! and F 2 . The last equation accordingly assumes the simpler form 



