64 IONIZATION THEORY OF GASES [CH. 
intense ionization it is very small near the active surface. The 
potential gradient varies slightly according as the active plate is. 
charged positively or negatively. 
47. Variation of current with voltage for surface ion- 
ization. Some very interesting results, giving the variation of the 
current with voltage, are observed when the ionization is intense, 
and confined to the space near the surface of one of two parallel 
plates between which the current is measured. 
The theory of this subject has been worked out independently 
by Child? and Rutherford’. Let V be the potential difference 
between two parallel plates at a distance d apart. Suppose that 
the ionization is confined to a thin layer near the surface of the 
plate A (see Fig. 1) which is charged positively. When the electric 
field is acting, there is a distribution of positive ions between the 
plates A and B. 
Let »,=number of positive ions per unit volume at a distance 
x from the plate 4, 
K, = mobility of the positive ions, 
e= charge on an ion. 
The current 7, per square centimetre through the gas is 
constant for all values of 2, and is given by 
dV 
(Kenn a 
By Poisson’s equation 
Bae = 47rne. 
dx? 
2 KG dV d?V 
Then Fel reli pet ee 
Integrating (d a _ 8142 for 
Ge) BOGEG 
: ‘2 p dV 
where A is a constant. Now A is equal to the value of — when 
da 
1 Phys. Rev. Vol. 12, 1901. 
° Phil. Mag. p. 210, 1901; Phys. Rev. Vol. 13, 1901. 
