208 Prof. A. Gray on Electric 



We can easily find another expression for this energy. 

 We calculate first the potential at a point A on the edge of 



Fig. 3. 



the disk (taken first of radius r). From the diagram (fig. 3) 

 we see that 



, 2 r sin 6 /-» - 



V A = A \ dOdr = ±or. . . . (25) 



Jo Jo 



The exhaustion of potential energy involved in adding 

 a ring of breadth dr is therefore 



Aar . 2irar dr = $7T(T 2 r 2 dr. 



Hence the whole energy exhausted in building up the 

 disk of radius a from matter formerly dispersed through 

 infinite space is 



P=:|7T(7 2 a 3 (26) 



Equating this to the former value, found as a Bessel 

 function integral, we get 



1/^)^=3^ W 



Putting a=l, we get 



i: 



j»w^-^> • • • • • ( 27 '> 



\ 2 3tt 



which is the known value of the integral. 



It may be possible to obtain more general forms of these 

 results by using a more general form of potential. 



6. We now consider the following problem. A uniform 

 thin circular disk of attracting matter has radius b. It is 



