90 STATIC ELECTRICITY 



When a system of conductors is maintained each at a constant 

 potential by connection with a source of electricity, the total 

 energy supplied by the source in any displacement of the system 

 is double the increase in the electrical energy of the system. 



This may be proved as follows : 



Let Qi,Q 2 , Q 3 be the charges at potentials V r V 2 , V 8 : then the 



total energy is 



+ . . 



Let a displacement be given to the system so that the charges 

 become Q x + q v Q 2 + q v Q 3 + fc, &c - : the ener ^ is now 



whence the increase in energy is 



But the quantities q v q v q y &c., have been drawn from sources of 

 energy at constant potentials V^ V 2 , c., and therefore the energy 

 given up by these sources is 



which proves the theorem. 



Applying this to the quadrant electrometer, the sources supply 

 double the energy required for the electrical system. At first, 

 then, while the torsion couple is small, an excess of energy is 

 supplied. But the torsion couple increases, and ultimately a 

 position of equilibrium is attained when on further small displace- 

 ment the increase of torsional energy equals the increase in elec- 

 trical energy. 



To calculate the latter we assume that the motion does not 

 affect the distribution on the outer edges of the needle and under 

 the line of separation of the quadrants ; this implies that the 

 linear displacement of an edge is small compared with its distance 

 from a quadrant other than that under which it is moving. Hence 

 the motion through a small angle dO may be regarded as merely 

 transferring a part of the needle distant from both the edge and 

 the median line from the 2, % pair to the 1, 1' pair. The capacity 

 of such a part may be taken as proportional to the angle it 

 subtends. Let it be represented by CdO. Hence the charge 

 under 1,1' is increased by CdO(V n V 1 ), while that under 2, 2' 

 is decreased by CdO(V n V 2 ). The potential differences of 

 these charges from that on the needle are V n V l and V n V, 

 respectively. Then the total gain of energy is 



