2 1 8 Prof. Rutherford on Dependence of the Current through 



Let n± be the number of + ions per c. c. at a distance x 

 from lower plate. 



Let Kj K 2 be the velocities of the + and — ions for unit 

 potential-gradient. 



The current i L per sq. centim. through the gas is every- 

 where the same between the plates, and near the upper 

 plate is given by 



dV 

 where e is the charge on an ion, and -7- the potential-gradient. 



By Poisson's equation, 



We therefore have 



dx 



d 2 V . 



= K, dV d^V 



Air ' dx ' dx 2 • 



In exactly the same way it may be shown that i 2 , the current 

 when the lower plate is negatively charged, is given by 



, _ K 2 dV d 2 V 



2 ~ 47r ' dx ' dx 2 * 



Since the potential-gradient near the upper plate is the same 

 for both directions of the electric field, 



— and — 

 dx dx z 



are numerically the same for both currents at the same point. 

 Therefore 



i 2 K/ 



or the currents are directly proportional to the velocities of 

 the ions. 



Since the current is greater when the lower plate is nega- 

 tively charged, these results are in agreement with the fact, 

 first observed by Zeleny, that the negative ion moves faster 

 than the positive. 



In the experimental case, there is always a small number 

 of ions produced by the radio-active substance near the upper 

 plate. For small voltages, the current near the upper plate 

 is then not altogether carried by ions of one sign. The 

 general effect of this is to bring the currents in the two 

 directions more nearly to equality. When high voltages are 



