44 STATIC ELECTRICITY 



Equipotential surfaces. We have seen that 

 assign to each point in a field a definite number expressing js 

 potential. Let us suppose a surface drawn through all points 

 having the same number indicating their potential. Such 

 is termed an equipotential surface or a level surface. Since it is 

 all at one potential, no work is done in moving unit quantity in a 

 path lying on the surface. The intensity must, therefore, be 

 normal to the surface. In other words, it cuts the lines of t 

 everywhere at right angles. If we draw a series of cquipotcntial 

 surfaces with unit difference of potential between successj\ , incur 

 of the series, their distances apart show the magnitude of the 

 intensity everywhere. For if I is the intensity at any point on 

 one surface and d the distance of the point from the next 

 face, Id is the work done on the unit in going from one Kir! 

 to the next. But this is by supposition unit work, then Id = 1 

 or I = I Id. 



The energy of a system in terms of the charges and 

 potentials. If there is a charge q^ at a point A. the work dour 

 in bringing up unit charge to a point B against the force <1> 

 g l is ft/AB. If instead of unit charge we bring up charge q r tin- 

 work done is g 1 g 2 /A'R. 



Let us suppose that we have a system consisting of el. 

 charges ft at A, q ^ at B, q 3 at ( . and so on, and let us put AH 

 AC = r 13 , BC = 7- 23 , and so on. Imagine that initially q } on! 

 in position, and that all the other eh t .it inlin 



away and apart from each other. Bring q up to B. The uork 

 done is-ii^ ^ow i )r j n g g^ U p j. o Q an( j t j u> ^^ ( j 0||r js 

 r i2 



Ms + Ml. 



Continue this process till the system is built up, and e\ ident 

 shall have the product of each pair of charges divid.-d b\ their 

 distance apart occurring once and once only. The total \\ork <! 



or the potential energy of the system, is therefore 1 here we 



sum for all pairs of elementary char 



Now suppose that all the charges are in position except 0,, which 

 is at an infinite distance. The work done in bringing n. into 

 position is 



H 



where V, is the potential at A due to the ix-l of Ih, 



