56 Electrostatics Field of Force [OH. n 



From a study of the section of the equipotentials as shewn in fig. 21, it is 

 easy to construct the complete surfaces. We see that each equipotential for 

 which V has a very high value consists of three small spheres surrounding the 

 points A, B, C. For smaller values of V, which must, however, be greater 



3'04 



than , each equipotential still consists of three closed surfaces surround- 

 ing A, B, C, but these surfaces are no longer spherical, each one bulging out 

 towards the point D. As V decreases, the surfaces continue to swell out, 



3'04 

 until, when V = , the surfaces touch one another simultaneously, in a 



Cb 



way which will readily be understood on examining the section of this equi- 

 potential as shewn in fig. 21. It will be seen that this equipotential is 



shaped like a flower of three petals from which the centre has been cut away. 



g 



As V decreases further the surfaces continue to swell, and when V=-, the 



a 



space at the centre becomes filled up. For still smaller values of V the 

 equipotentials are closed singly-connected surfaces, which finally become 

 spheres at infinity corresponding to the potential V = 0. 



The sections of the equipotentials by a plane through DA perpendicular 

 to the plane ABC are shewn in fig. 22. 



SPECIAL PROPERTIES OF EQUIPOTENTIALS AND LINES OF FORCE. 



The Equipotentials and Lines of Force at infinity. 

 67. In | 40, we obtained the general equation 



If r denotes the distance of x, y, z from the origin, and TI the distance of 

 #i> 2/i > #1, from the origin, we may write this in the form 



T7" S? 6 1 



[r 2 - 2 tej + y^ + zz-d + n 2 ]^ 



At a great distance from the origin this may be expanded in descending 

 powers of the distance, in the form 



f yy, + zz, , 3 (xx, + yy, + zz^ 1 r? j 



- , 



L ~r^~ 2 " r* 2 



1 2c 



The term of order - is 1 . 

 r r 



The term of order is %e 1 (xx l + yy l + zz-^. 



