48 



Electrostatics Field of Force 



[CH. n 



The same result can be seen in another way. If we start from A and 

 draw a small tube surrounding the line AB, it is clear that the cross-section 

 of the tube, no matter how small it was initially, will have become infinite 

 by the time it reaches the plane which bisects AB at right angles in fact 

 the cross-section is identical with the infinite plane. Since the product of 

 the cross-section and the normal intensity is constant throughout a tube, it 

 follows that at the point C, the intensity must vanish. 



FIG. 15. 



At a great distance R from the points A and B, the fraction 



P& + PA* 



vanishes to the order of R~ l , so that 



_ 

 dx'X' 



except for terms of the order of 1/R*. Thus at infinity the lines of force 

 become asymptotic to straight lines passing through the origin. 



Let us suppose that a line of force starts from A making an angle with 

 BA produced, and is asymptotic at infinity to a line through C which makes 

 an angle < with BA produced. By rotating this line of force about the 

 axis AB we obtain a surface which may be regarded as the boundary of 

 a bundle of tubes of force. This surface cuts off an area 



27r(l -cos0)r 2 



