Capacity of Solids under Varying Fields. 83 



for the upper side of the plate, and &=e for the under side, 

 where e' and e are the quantities of electricity upon unit area 

 of the respective sides. Whence we Lave for the capacities 

 c 1 and c of unit areas of the two sides, c , = e , V = <r'V and 



G G 



c = eY = crV, and a'= ^, and a = ^ . S uhstitute these values 

 for a' and a in (3), and 



Cd. 2 -G% W 



Now, if the dielectric is as large as the plates, and these 

 are not too far apart, the field may be taken as uniform over 

 an area, A, equal to that of one of the small plates, M or N. 



Hence, . 



^-7^ o (5) 



:7Tl 



and _ A 



C ~±7T(d 2 + d 3 ) { } 



Substitute these values of c' and c in (4) and 



d 



r + d± -f d 



K=_-I ■ (7) 



a 2 c/ 4 



d 2 + d 3 a\ 

 K 

 Reduced, 



4 + d A + d 



*=J^kr (8) 



dd 2 —d z d 4i 



d 4 may be taken =0, and then 



K =S w 



We shall now determine the best thickness to give to the 

 dielectric so that errors made in adjusting the plate N will 

 produce the smallest error in deducing the value of K. The 

 errors made in measuring d and d\ may be neglected in 

 comparison with the error made in setting the plate N. 

 Now, taking the case where c? 4 = 0, and where each pair of 

 large plates are the same distance apart, 



K =«7(fe) ("» 



G2 



