46 Electrostatics Field of Force [OH. n 



the surface density of electrification is cr', v being of course negative, and 

 that the intensity in the direction of the lines of force is R. Then, as in 



equation (21), 



R'dS' = - *Tr<r'dS f , 



since the outward intensity is now R'. 



Since R, R are the intensities at two points in the same tube of force 

 at which the normal cross-sections are dS, dS', it follows from the theorem 



of 56, that 



RdS = RdS' 



and hence, on comparing the values just found for RdS and R'dS', that 



adS = cr dS' ' . 



Since crdS and cr'dS' are respectively the charges of electricity from which 

 the tube begins and on which it terminates, we see that : 



The negative charge of electricity on which a tube of force terminates is 

 numerically equal to the positive charge from which it starts. 



If we close the ends of the tube of force by two small caps inside the 

 conductors, as in fig. 14, we have a closed surface such that the normal 

 intensity vanishes at every point. Thus, by Gauss' Theorem, the total 

 charge inside must vanish, giving the result at once. 



59. The numerical value of either of the charges at the ends of a 

 tube of force may conveniently be spoken of as the strength of the tube. A 

 tube of unit strength is spoken of by many writers as a unit tube of force. 



The strength of a tube of force is adS in the notation already used, and 

 this, by Coulomb's Law, is equal to j RdS where R is the intensity at the 



end dS of the tube. By the theorem of 56, RdS is equal to J^o^ where 

 R lt &>! are the intensity and cross-section at any point of the tube. Hence 

 R l a) l = 4-7T times the strength of the tube. It follows that : 



The intensity at any point is equal to 4?r times the aggregate strength per 

 unit area of the tubes which cross a plane drawn at right angles to the 

 direction of the intensity. 



In terms of unit tubes of force, we may say that the intensity is 4?r 

 times the number of unit tubes per unit area which cross a plane drawn at 

 right angles to the intensity. 



The conception of tubes of force is due to Faraday : indeed it formed 

 almost his only instrument for picturing to himself the phenomena of the 

 Electric Field. It will be found that a number of theorems connected with 

 the electric field become almost obvious when interpreted with the help of 

 the conception of tubes of force. For instance we proved on p. 37 that 



