V = Ai + Bi².
Mie (Ann. der. Phys., 1904, 13, P. 857) has by the method of successive approximations obtained solutions of equation (8) (i.) when the current is only a small fraction of the saturation current, (ii.) when the current is nearly saturated. The results of his investigations are represented in fig. 12, which represents the distribution of electric force along the path of the current for various values of the current expressed as fractions of the saturation current. It will be seen that until the current amounts to about one-fifth of the maximum current, the type of solution is the one just indicated, i.e. the electric force is constant except in the neighbourhood of the electrodes when it increases rapidly.
Though we are unable to obtain a general solution of the equation (8), there are some very important special cases in which that equation can be solved without difficulty. We shall consider two of these, the first being that when the current is saturated. In this case there is no loss of ions by recombination, so that using the same notation as before we have
| d | (n1k1X) = q, |
| dx |
| d | (n2k2X) = -q. |
| dx |
The solutions of which if q is constant are
| n1k1X = qx, |
| n2k2X = I/e − qx = q(l − x), |
if l is the distance between the plates, and x = 0 at the positive electrode. Since
dX/dx = 4π(n1 − n2)e,
we get