IDEA OF POTENTIAL. 185 



is the potential at any point [r, 6] outside of a metal cylinder 

 placed in a region which but for the presence of the cylinder 

 would be a uniform field of intensity e at right angles to the axis 

 of the cylinder. The moment M of the line doublet must be 

 such as to give a value of zero to the quantity in the parenthesis 

 when the radius of the cylinder R is substituted for r. That is, 

 we must have : 



eR - ^ = o 

 or 



(6) 



U 



Substituting the value of M from (6) in (5), we get: 



sfl (7) 



which expresses the potential at each point (r, 6) near a 

 cylinder of radius R placed in a region which but for the 

 presence of the cylinder would be a uniform electric field of 

 intensity e at right angles to the axis of the cylinder, as shown 

 in Fig. 123. The line doublet is the image in the metal cylinder 

 of the distant charges which produce the uniform field. 



It is evident from Fig. 123 that the presence of the metal 

 cylinder causes a concentration of electrical stress, and the 

 maximum stress is at a (or b} where r = R and B = o. But 

 the electric field at any point is the potential gradient at that 

 point. Therefore the field intensity at a, being evidently 

 parallel to the #-axis, is found by differentiating V in equation 

 (7) with respect to r and placing 6 = o and r = R in the 

 result, because r is parallel to x when 6 is zero. In this way 

 we get: 



J field intensity at ) , . 



1 a in Fig. 123 f ~ 2 * 



That is, the field intensity at a in Fig. 123 is twice as great as 

 the intensity e of the original uniform field. 



