190 Methods for the Solution of Special Problems [CH. vm 



As it ought, this gives a constant potential G over the surface of the 

 sphere. 



The lines of force of the uniform field F disturbed by the presence of a 

 doublet of strength Fa 3 are shewn in fig. 63. On obliterating all the lines 

 of force inside a sphere of radius a, we obtain fig. 64, which accordingly 

 shews the lines of force when a sphere of radius a is placed in a field of 

 intensity F. These figures are taken from Thomson's Reprint of Papers on 

 Electrostatics and Magnetism (pp. 488, 489)*. 



218. Line of no electrification. The theory of lines of no electrification 

 has already been briefly given in 98. We have seen that on any conductor 

 on which the total charge is zero, and which is not entirely screened from 

 an electric field, there must be some points at which the surface-density a- 

 is positive, and some points at which it is negative. The regions in which a- 

 is positive and those in which a is negative must be separated by a line or 

 system of lines on the conductor, at every point of which cr = 0. These lines 

 are known as lines of no electrification. 



If R is the resultant intensity, we have at any point on a line of no 

 electrification, 



R = 47TO- - 0, 



so that every point of a line of no electrification is a point of equilibrium. 

 At such a point the equipotential intersects itself, and there are two or more 

 lines of force. 



If the conductor possesses a single tangent plane at a point on a line of 

 no electrification, then one sheet of the equipotential through this point will 

 be the conductor itself: by the theorem of 69, the second sheet must 

 intersect the conductor at right angles. 



These results are illustrated in the field of fig. 64. Clearly the line of no 

 electrification on the sphere is the great circle in a plane perpendicular to 

 the direction of the field. The equipotential which intersects itself along 

 the line of no electrification (V=C) consists of the sphere itself and the 

 plane containing the line of no electrification. Indeed, from formula (126), 



TT 



it is obvious that the potential is equal to C, either when 6 = , or 

 when r d. 



The intersection of the lines of force along the line of no electrification 

 is shewn clearly in fig. 64. 



* I am indebted to Lord Kelvin for permission to use these figures. 



