ELECTRICAL LEVEL OR POTENTIAL 11 



an operation, the charges and potentials of the conductors will be 

 all reduced in the same ratio, so that the points representing them 

 on the diagram will all travel in straight lines towards O. The 

 figure may be regarded as merely shrinking in towards O. 



Now consider the work done by the small charge removed from 

 the first conductor while its state changes from A' to A." The 

 quantity removed is evidently MN. The potential at the 

 beginning of the removal is A'M, and at the end it is A"N. Then 

 the work done lies between 



MN. A'M and MN. A"N. 



But one of these is greater and the other less than the area of the 

 slip A MNA , and the less the charge MN, the more nearly are they 

 equal to each other, ultimately differing negligibly. Then the 

 work done must ultimately be represented by A'MNA". 



The total work done in discharging the first conductor is the 

 sum of all such slips or is represented by the area of the triangle 

 AEO, which equaU 



AE X EO Q 1 V 1 



2 a 



Similarly for the other conductors. Whence 



the total energy = ^ + 9& + . . . = 2 ^. 



When all the charges have been thus brought to the same level 

 all the electrical energy has been transformed and the system 

 is exhausted. 



It i> perhaps suflicieiiily evident that the quantity of energy 

 obtained must be independent of the arbitrary zero of potential 

 choM-ii. but we may also deduce this from the expression for the 

 total. For i t \\ e had chosen another zero at potential U relatively to 

 tin liiM, then the potentials would have been V, U, V 2 U,c., 

 and the total energy would have been expressed by 



for Q 1 + Q, + . . . = 0. 



I ;> to be noted that if any charge is on the original zero-level 

 surface, it must be taken into account in expressing the energy in 

 terms of the levels referred to the new zero. 



