32 Electrostatics Field of Force [CH. n 



where r a , r 3 , ... are the distances from (X to e 2 , e 8 , ..., so that by definition 



F=^ + ^ + - 3 + (10). 



n r 2 r 3 



39. It is now clear that the potential at any point depends only on the 

 coordinates of the point, so that the work done in bringing a small charge 

 from infinity to a point P is the same, no matter what path we choose, the 

 result assumed in 33. 



It follows that we cannot alter the amount of energy in the field by 

 moving charges about in such a way that the final state of the field is the 

 same as the original state. In other words, the Conservation of Energy is 

 true of the Electrostatic Field. 



40. Analytically, let us suppose that the charge ^ is at x^, y l} z l \ e z at 

 #2? 2/2) #2 5 an( i so on - The repulsion on a small charge e at x, y, z resulting 

 from the presence of e l at x l1 y ly 2 l is 



0,6 



ll l 



and the direction-cosines of the direction in which this force acts on the 

 charge e, are 



[(x - atf + (y - yi 

 Hence the component parallel to the axis of x is 

 _ 616 (x - ap 



etc 



' 



By adding all such components, we obtain as the component of the 

 electric intensity at x, y, z, 



TT- GI \ x ~ \) /i i \ 



JL = 2^ \*-*-)> 



and there are similar equations for Y and Z. 

 We have as the value of V at x, y, z, 



V= - [ Xt >Z (Xdx + Ydy + Zdz) 

 = _[ x ' y ' z 2 <?i {(a? - afi) dx + (y- y,) dy + (z- z,) dz] 



giving the same result as equation (10). 



