| X = | i | ( | α | )1/2 ε −4π e²k2 qx / αi, |
| e(k1k2)1/2 | q |
and the fall of potential at the cathode is equal approximately to ∫∞0 Xdx, that is to
| i | ( | α | )1/2 | α i | . |
| e(k1k2)1/2 | q | 4πe²k2q |
The potential difference between the plates is the sum of the fall of potential in the uniform part of the discharge plus the fall at the cathode, hence
| V =( | α | )1/2 | i | (il + | i α² | 1 | ). | |
| q | ek2 | 4πe²q | √(k1k2) |
The fall of potential at the cathode is proportional to the square of the current, while the fall in the rest of the circuit is directly proportional to the current. In the case of flames or hot gases, the fall of potential at the cathode is much greater than that in the rest of the circuit, so that in such cases the current through the gas varies nearly as the square root of the potential difference. The equation we have just obtained is of the form
V = Ai + Bi²,
and H. A. Wilson has shown that a relation of this form represents the results of his experiments on the conduction of electricity through flames.
The expression for the fall of potential at the cathode is inversely proportional to q3/2, q being the number of ions produced per cubic centimetre per second close to the cathode; thus any increase in the ionization at the cathode will diminish the potential fall at the cathode, and as practically the whole potential difference between the electrodes occurs at the cathode, a diminution in the potential fall there will be much more important than a diminution in the electric force in the uniform part of the discharge, when the force is comparatively insignificant. This consideration explains a very striking phenomenon discovered many years ago by Hittorf, who found that if he put a wire carrying a bead of a volatile salt into the flame, it produced little effect upon the current, unless it were placed close to the cathode where it gave rise to an enormous increase in the current, sometimes increasing the current more than a hundredfold. The introduction of the salt increases very largely the number of ions produced, so that q is much greater for a salted flame than for a plain one. Thus Hittorf’s result coincides with the conclusions we have drawn from the theory of this class of conduction.
The fall of potential at the cathode is proportional to i − i0, where i0 is the stream of negative electricity which comes from the cathode itself, thus as i0 increases the fall of potential at the cathode diminishes and the current sent by a given potential difference through the gas increases. Now all metals give out negative particles when heated, at a rate which increases very rapidly with the temperature, but at the same temperature some metals give out more than others. If the cathode is made of a metal which emits large quantities of negative particles, (i − i0) will for a given value of i be smaller than if the metal only emitted a small number of particles; thus the cathode fall will be smaller for the metal with the greater emissitivity, and the relation between the potential difference and the current will be different in the two cases. These considerations are confirmed by experience, for it has been found that the current between electrodes immersed in a flame depends to a great extent upon the metal of which the electrodes are made. Thus Pettinelli (Acc. dei Lincei [5], v. p. 118) found that, ceteris paribus, the current between two carbon electrodes was about 500 times that between two iron ones. If one electrode was carbon and the other iron, the current when the carbon was cathode and the iron anode was more than 100 times the current when the electrodes were reversed. The emission of negative particles by some metallic oxides, notably those of calcium and barium, has been shown by Wehnelt (Ann. der Phys. 11, p. 425) to be far greater than that of any known metal, and the increase of current produced by coating the cathodes with these oxides is exceedingly large; in some cases investigated by Tufts and Stark (Physik. Zeits., 1908, 5, p. 248) the current was increased many thousand times by coating the cathode with lime. No appreciable effect is produced by putting lime on the anode.