70 



Conductors and Condensers 



[CH. Ill 



sphere, from the axis of the cylinder, and from the plane, respectively, we 

 have found that 



outside the sphere, R is proportional to , 



outside the cylinder, R is proportional to - , 

 outside the plane, R is constant. 



From the point of view of tubes of force, these results are obvious enough 

 deductions from the theorem that the intensity varies inversely as the cross- 

 section of a tube of force. The lines of force from a sphere meet in a point, 

 the centre of the sphere, so that the tubes of force are cones, with cross- 

 section proportional to the square of the distance from the vertex. The 

 lines of force from a cylinder all meet a line, the axis of the cylinder, at right 

 angles, so that the tubes of force are wedges, with cross-section proportional 

 to the distance from the edge. And the lines of force from a plane all meet 

 the plane at right angles, so that the tubes of force are prisms, of which the 

 cross-section is constant. 



78. We may also examine the results from the point of view which 

 regards the electric intensity as the resultant of the attractions or repulsions 

 from different elements of the charged surface. 



Let us first consider the charged plane. Let P, P f be two points at 

 distances r, r' from the plane, and let Q be the 

 foot of the perpendicular from either on to the 

 plane. If P is near to Q, it will be seen that 

 almost the whole of the intensity at P is due 

 to the charges in the immediate neighbourhood 

 of Q. The more distant parts contribute forces 

 which make angles with QP nearly equal to a 

 right angle, and after being resolved along QP 

 these forces hardly contribute anything to the 

 resultant intensity at P. 



Owing to the greater distance of the point P', 

 the forces from given elements of the plane are 

 smaller at P' than at P, but have to be resolved 

 through a smaller angle. The forces from the 

 regions near Q are greatly diminished from the 

 former cause and are hardly affected by the latter. 

 The forces from remote regions are hardly affected 

 by the former circumstance, but their effect is 

 greatly increased by the latter. Thus on moving FIG. 29. 



