152 The State of the Medium in the Electrostatic Field [OH. vi 



Thus at any point in any medium the displacement has magnitude and 

 direction. The displacement, then, is a vector, and its component in any 

 direction may be measured by the total quantity of electricity per unit area 

 which has crossed a small area perpendicular to this direction, the quantity 

 being measured from a time at which no electric intensity was acting. 



172. Suppose, now, that an electric field is gradually brought into 

 existence, the field at any instant being exactly similar to the final field 

 except that the intensity at each point is less than the final intensity in 

 some definite ratio K. Let the displacement be c times the intensity, so 

 that when the intensity at any point is /cR, the displacement is c/cR. The 

 direction of this displacement is along the lines of force, so that the 

 electricity may be regarded as moving through the tubes of force : the lines 

 of force become identical now with the current-lines of a stream, to which 

 they have already been compared. 



Let us consider a small element of volume cut off by two adjacent 

 equipotentials and a tube of force. Let the cross section of the tube of 

 force be co, and the normal distance between the equipotentials where they 

 meet the tube of force be ds, so that the element under 

 consideration is of volume cods. On increasing the intensity 

 from /cR to (K + d/c) R, there is an increase of displacement 

 from c/cR to c (K + die) R, and therefore an additional dis- 

 placement of electricity of amount cRdic per unit area. 



Thus of the electricity originally inside the small element 

 of volume, a quantity cRcodrc flows out across one of the 

 bounding equipotentials, whilst an equal quantity flows in 

 across the other. Let T, T be the potentials of these 

 surfaces, then the whole work done in displacing the electricity originally 

 inside the element of volume cods, is exactly the work of transferring a 

 quantity cRdfc of electricity from potential K to potential V z . It is 

 therefore cRco (V 2 - V[) d/c and, since V 2 -V 1 = /cRds, this may be written as 

 cR*codstcd/c. Thus as the intensity is increased from to R } the total work 

 spent in displacing the electricity in the element of volume cods 



= I cR* (cods) /cd/c ^cR 2 . cods. 



Jo 



This work, on Maxwell's theory, is simply the energy stored up in the 



D2 



element of volume cods of the medium, and is therefore equal to cods. 



Thus c must be taken equal to - , and the displacement at any point is 

 measured by 



47T* 



-ds- 



