"INVERSE SgKAHK" SYSTEMS 43 



Similarly the work done along BC =q ( Fv77~~7yR )> 



and so on. 



When we add up, evidently all the terms but the first and 

 l.-ist we ur twice and with opposite signs, and the total work done 



= *\SF OAA 



The \\ork done from A to 1' is, therefore, the same, whatever 

 the path from A to P. 



I \ is at an infinite distance the work done to P is q/OP. 

 That is, it is equal to the potential at P due to Q. 



The work done from A to P is equal to the difference of the 

 potentials due to q at P and A. 



In any extended system we may divide the charges into elements 

 so small that each may be regarded as at a definite point. The 

 intensity at any point in the field is the resultant of the intensities 

 due to ti it. dements, and the resolute of the intensity in 



din-rtion is t he algebraic sum of then -solutes of all theseparate 

 intensities in that direction. The work done in moving unit 

 itity along any path is, therefore, the sum of the works done 

 against th< forces due to the separate element-.. 



!! nee the work done in moving the unit from an infinite 



mcc to the point P in the field is ^-i^or is equal to the poten- 

 tial at the point. The work done in moving from 1* to Q is evidently 

 - ( -r " ~fju' or i s (M | IIa ' t() the difference of potentials at Q and P. 



The resolute of the intensity in any direction in 

 terms of potential variation. Ix-t IV Fig. .'38, be a given direc- 

 tion, and let X be the resolute of the intensity in that direction. 



P Q X 



Fio. 38. 



\ \ .! i te the potentials at P and Q, two neighbour- 

 ing I The work* done against X in ^ () j ni r from P to (^ 



I'M. >in we h;m- supposed X to act from P to Q. Then 

 \ V r =- IV 





\ 



If we denote !*<) by </ 1. then in the limit 



d\ 



