41 o Note 20: capacity of a thick disk 



We have now to multiply the charge of an element of the upper disk into 

 the potential due to the lower disk, and integrate for the whole surface of the 

 upper disk, 



i 2 Jo \2 / 



= A 1 A t a-*(?- jT'tan-iy^) (10) 



Between the limits of integration we may write with a sufficient degree of 

 approximation, , /h\l\ r h\i\~ 1 



tan -1 y = y = - -ji + (-J I ]/'+(") I ( IJ ) 



At the centre of the disk p = i and 

 At the circumference, 



y = - , which agrees with (9). 



p = o and y = (-) + o(-) by (9), 

 whereas the equation (n) gives 



f b \* L b 



rrU) +** 



so that when b is very small compared with a, the value of y cannot differ greatly 

 from that given by equation (n). Hence we may write the expression (10) 



12 



The corresponding quantity for the action of the upper disk on itself is got 

 by putting A^ = A% and 6 = 0, and is 



^2 2 - 1 f- (13) 



In the actual case A = A z = J, where E is the whole charge, and the 

 capacity is 



or, since in our approximation we have neglected f-J , our result may be ex- 

 pressed with sufficient accuracy in the form 



(15) 



showing that the capacity of two disks very near together is equal to that of 

 an infinitely thin disk of somewhat larger radius. 



If the space between the two disks is filled up, so as to form a disk of 

 sensible thickness, there will be a certain charge on the curved surface, but at 

 the same time the charge on the inner sides of the disks will disappear, and that 

 on the outer sides near the edges will be diminished, so that the capacity of 

 a disk of sensible thickness is very little greater than that given by (15). 



