No general solution of equation (8) has been found when k1 is not equal to k2, but we can get an approximation to the solution when q is constant. The equations (1), (2), (3), (4) are satisfied by the values—
n1 = n2 = (q/α)1/2
| k1n1Xe = | k1 | I, |
| k1 + k2 |
| k2n2Xe = | k2 | I, |
| k1 + k2 |
| X =( | α | )1/2 | I | . |
| q | e(k1 + k2) |
These solutions cannot, however, hold right up to the surface of the plates, for across each unit of area, at a point P, k1I/(k1 + k2)e positive ions pass in unit time, and these must all come from the region between P and the positive plate. If λ is the distance of P from this plate, this region cannot furnish more than qλ positive ions, and only this number if there are no recombinations. Hence the solution cannot hold when qλ is less than k1I/(k1 + k2)e, or where λ is less than k1I/(k1 + k2)qe.
Similarly the solution cannot hold nearer to the negative plate than the distance k2I/(k1 + k2)qe.
The force in these layers will be greater than that in the middle of the gas, and so the loss of ions by recombination will be smaller in comparison with the loss due to the removal of the ions by the current. If we assume that in these layers the loss of ions by recombination can be neglected, we can by the method of the next article find an expression for the value of the electric force at any point in the layer. This, in conjunction with the value X0 = (α/q)1/2 · I/e(k1 + k2) for the gas outside the layer, will give the value of X at any point between the plates. It follows from this investigation that if X1 and X2 are the values of X at the positive and negative plates respectively, and X0 the value of X outside the layer,
| X1 = X0(1 + | k1 | 1 | )1/2, | X2 = X0(1 + | k2 | 1 | )1/2, | ||
| k2 | ε | k1 | ε |
where ε = α/4πe(k1 + k2). Langevin found that for air at a pressure of 152 mm. ε = 0.01, at 375 mm. ε = 0.06, and at 760 mm. ε = 0.27. Thus at fairly low pressures 1/ε is large, and we have approximately