Electric Field in the neighbourhood of a Spherical Bowl, 53 

 For 0<a: 



/COS ^r \ 



v - Bin na , / 2 \ 7r 



\ cos 2 / 



a 



^ sin (n + 1) a- , / 2 \ 7T ... 



S »» n + 1 = " COS I" t1 + 2- ' ' (/) 



For 6>a: 



^ , sin na ^ , sin(n+l)a 

 2, <2> = 2, O^ -r: 



rn n T n + 1 



cos f 



sin 



,/ Sm 2 



. 

 sm- 



(^ &/') 



By addition of (e) and (/) we see at once that 



v = v = y„ 



for 6<ot, r = a. 



The differentiation of (13) gives, for r=a, 



(-^) =-i smna + smn+1 a * — <£> 



V or / r=a Too L n + 1 J Yn 



/8V\ V ;r. , . , ,-. sinnal 



\ or J r - a 7ra o L na J ^ n 



Thus, for # > a, we have 



/8Vx = /SV' , = _V ^ 



\o> / \ o> / 7ra 



/sin 



a 



. e 

 \ sm ^ 



The mere inspection of the series (13) shows that for r = a 

 V = V, and that V vanishes at infinity. 



The function denned by the series has the stated properties, 

 and accordingly it satisfies all the conditions required for the 

 potential in the neighbourhood of a bowl charged to potential 



For 6 < a, we find, moreover, 



