Control of the Space Charge
Author: J.B. Hoag
The effects due to the space charge can be shown in another fashion. Let the temperature of the filament of the tube in Fig. 10 A be held constant and let the voltage on the plate be increased.

Fig. 10 E. The plate voltage V changes the plate current i. Fixed filament temperatures, T_{2} and T_{1}. (From E. & N. P.) 
As shown in Fig. 10 E, the spacechargelimited current increases along the line AD and eventually equals the saturation value shown by the horizontal line, upper right of the figure. With a hotter filament the dotted curve ADT_{2} will be obtained. The relationship between the applied voltage V and the resulting current i is expressed by the following equation: i = BV^{3/2}, where B is a constant. This is variously known as the threehalvespower law, the Child law, and also as the Langmuir law. In actual tubes, the exponent in this equation varies between twohalves and fivehalves. This is because of the drop of voltage between one end of the filament and the other, and because of the initial velocities with which the electrons leave the filament.
