58-62] Tubes of Force 47 



when a number of charged bodies are placed inside a hollow conductor, they 

 induce on its inner surface a charge equal and opposite to the sum of all 

 their charges. This may now be regarded as a special case of the obvious 

 theorem that the total charge associated with the beginnings and termi- 

 nations of any number of tubes of force, none of which pass to infinity, must 

 be nil. 



EXAMPLES OF FIELDS OF FORCE. 



60. It will be of advantage to study a few particular fields of electric 

 force by means of drawing their lines of force and equipotential surfaces. 



I. Two Equal Point Charges. 



61. Let A, B be two equal point charges, say at the points x = a, + a. 

 The equations of the lines of force which are in the plane of x, y are 

 easily found to be 



a 

 where P is the point x, y. 



This equation admits of integration in the form 



x -t- a x a 



(23). 



From this equation the lines of force can be drawn, and will be found to lie 

 as in fig. 15. 



62. There are, however, only a few cases in which the differential 

 equations of the lines of force can be integrated, and it is frequently simplest 

 to obtain the properties of the lines of force directly from the differential 

 equation. The following treatment illustrates the method of treating lines 

 of force without integrating the differential equation. 



From equation (22) we see that obvious lines of force are 

 (i) y = 0, g| = 0, giving the axis AB ; 



(ii) x = 0, PA = PB, ^- = co , giving the line which bisects AB at 



right angles. 

 These lines intersect at C, the middle point of AB. At this point, then. 



P has two values, and since ^ = -^ , it follows that we must have X = 

 dx dx X 



Y= 0. In other words, the point C is a point of equilibrium, as is otherwise 

 obvious. 



7 





ft 



