78 STATIC ELECTRICITY 



Unit cells. Drawing all the unit tubes of force and all the 

 level surfaces at unit differences of potential, all the space 

 between the conductors of a system is divided up into cells. 

 Maxwell has termed these unit cells. 



The number of unit cells in a system is 2QV. In the 

 case of one conductor at potential V entirely surrounded by another 

 at zero-potential, it is evident that each tube is cut into V cells, 

 and as there are Q tubes there are in all QV cells. When there 

 are several conductors each tube does not necessarily pass from 

 the highest to the lowest level, but some may begin or end at 

 intermediate levels. 



Take any one tube passing from a conductor at level V M to one 

 at level V N . Evidently it contains V M ' V N cells, or, since it has 

 + 1 of electricity at one end and 1 at the other, we may write 

 this 



1 xV M + (-l) XV N . 



Doing this for every tube in the system, we have 



Total number of cells = 2{1 X V M +(- 1)V X J 

 = 2 each element of charge X its potential. 



If on any conductor at potential V there be both positive and 

 negative charges, say Q P and Q N , both charges will enter into 

 this expression and will contribute 



to the result. But if Q P - Q N = Q, this becomes QV. Hence the 

 total number of cells = 2QV. 



Distribution of the energy in the system. From the 

 preceding result we see that the number of unit cells is always 

 double the number of units of energy in the system. We have 

 already seen that we must suppose the energy to be in the 

 dielectric, accompanying the strain there. Bearing in mind that 

 where the cells are larger the electric strain is less, so that t lie- 

 energy is presumably less densely stored, this relation between 

 the quantity of energy and the number of cells suggests that the 

 energy is distributed at the rate of half a unit to each cell. 

 Adopting the suggestion, let us find the energy per unit volume. 



The area of the cross-section of a tube at any point being a, 

 and the distance between the two consecutive level surfaces being 

 d, we have 



volume of cell = a d 



but Ea = 47r (p. 67) 



and Ed = 1 (p. 75) 



4-7T 



whence a d = . 



