The problem is now reduced to one of finding the boundary conditions to be satisfied by our 

 solution. This is most easily arrived at by assuming that the field is being produced by electric 

 charges +q at z = d and -q at z = - d. The magnitude of q must be such that the potential on the sur- 

 face of the sphere of radius a surrounding the charge is Vq. Since the potential at any point in space 

 due to a point charge is given by V = — — , we see then that we must have Vq = ^ • Hence the mag- 

 nitude of the charge is given by 



q = Vo a . (5) 



We are neglecting here the interaction between the two charges since we have assumed that a « 2d, 

 For any point along the Z-axis beyond the point z = d, the potential is given by 



^ r-d r+d 

 - 2dq ,,_ ^^-l 



d^ -1 



If the quantity ( I - — g ) is expanded in a Maclaurin series, we get 



V= 2dq (^ + -^ + ^^+^3+ ), r>d. (6) 



This expression is the boundary condition for the potential when 6 = and r > d. For any value of 6 

 and r, provided r > d, the solution is 



V(r,e) = 2dq ( TiP, + 7^P3 + 7-Ip5 + ), r>d . (7) 



For any point along the Z-axis between z = and z = d, the potential is given by 



d-r d+r 

 d^ ^' d^' ■ 



Expanding again, we get 



V = -^ (I + j2 +-^+^+ ),r<d. (8) 



This expression is the boundary condition for the potential when 6 = and r < d. For any value of 6 

 and r, provided r < d, the solution is 



r,e)= ^(rP, +-gp, + -^P,+ ),r<( 



(9) 



It is easily seen that equations (7) and (9) satisfy the boundary conditions (6) and (8) simply by 



setting 6 = in the expressions Pj, P3, P5 This gives Pj = P3 = P5 ... =1. Substitution of 



equations (7) and (9) into the differential equation (I) shows that they are indeed solutions. 



We notice that the potential is given by one expression within the sphere of radius d, ind another 

 outside this sphere. Let us call the region outside the sphere of radius d the region 1, and the region 

 inside the sphere region 2. To sumnnarize our solution (setting q = Voa) we have 



I r. , d^ , d* 



V, (r,e) - 2adVo( y-gP, + 7-4P3+7-6P5+ ) volts, r>d (10) 



Vz {r,e) - ^f^ { rP, + r'P3+ r'p5 + ) volts, r<d (11) 



18 



