THEORY OF THE ELECTRIC ARC 



161 



tive and negative ions. The space Aa will contain only 



negative ions and Cc only 



positive ions. Where there 



are ions of one sign only, the 



potential gradient changes rap- 



idly and the drop in potential 



in the immediate neighborhood 



of A and C may become 



very large. The mathemati- 



cal treatment of this proceeds 



easily in simple cases from the equation 



FIG. 58. 



<PV 



dx 2 dy 2 



. = 

 dz 2 



For example, when the lines at A , a, c, C represent infinite 



planes, V 2 > 



X 



> where b is the distance Aa, V 



the^change in potential through Aa, k the velocity of the 

 ions per unit electric force and i the current per unit cross 

 section. The units used are volts, amperes, centimeters 

 and seconds. 1 Thus if i equals icr 8 amperes, b equals 

 i cm., and k equals 10, then V is greater than 100 volts. 



lonization at Surface of Cathode, with Discharge at Low 

 Pressure. This discussion applies also to the case where 

 ionization is produced by impact, as with discharge through 

 a vacuum tube. But here a complication is introduced. 

 If no electrons come from C, all those between the plates 

 will soon pass to A. Let us suppose that the region Cc 

 has thus been cleared of negative ions. There will then 

 be a large drop in potential here, since the positive ions 

 will have moved into this region and there is always a large 



1 Phys. Rev., 12, 79; 1901. 



