1 84 ADVANCED ELECTRICITY AND MAGNETISM. 



but Ar = AJC-COS 8 (see Fig. 124), and M = (?-A#; therefore 

 this expression for the potential at p in Fig. 124 reduces to: 



If the positive and negative line charges are interchanged in 



Fig. 124 the algebraic sign of the potential V" will be changed 

 and we will have: 



F-- *=* (3) 



These expressions [equations (2) and (3)] for the potential of a 

 line doublet give V" = o when 6 = 90 if r is not zero. 

 That is, the potential along the ^-axis is zero except at the origin 

 in Fig. 123 when a horizontal doublet is placed at 0. 



The potential in a region where a line doublet is placed and which 

 but for the presence of the doublet would be a uniform electric field 

 at right angles to the doublet. This potential is found by adding 

 the above potentials as expressed by equations (i) and (2), or 

 (i) and (3); and for x in equation (i) we may write r cos 0. 

 Thus we get for the combined potential 



v .( e , .BM\ COS() ,.) 



v \ er i cos {4; 



If the negative sign is chosen (which means that the line doublet 

 is the reverse of what is shown in Fig. 124), then for a certain 

 value of r the quantity in the parenthesis will be zero. Therefore 

 V will be zero for this value of r whatever the value of may be. 

 That is to say one of the constant potential surfaces surrounding 

 the line doublet placed in a uniform field will be the surface of a 

 circular cylinder, and this particular cylindrical surface may be 

 replaced by a metal cylinder without altering the field distribu- 

 tion. Therefore 



/ "RKJ \ 



cos (5) 



