thus V is again infinite when d is nothing. There must therefore be some value of d intermediate between zero and infinity for which V is a minimum. This value is got by finding in the usual way the value of d, which makes the expression for V given in equation (1) a minimum. We find that d must satisfy the equation
1 = ε−(β−α)d {1 + (β − α)d + (β − α·d)²}.
We find by a process of trial and error that (β − α)d = 1.8 is approximately a solution of this equation; hence the distance for minimum potential is 1.8/(β − α). Since β and α are both proportional to the pressure, we see that the critical spark length varies inversely as the pressure. If we substitute this value in the expression for V we find that V, the minimum spark potential, is given by
| V = | β − α | · | 2.2 | . |
| α | ke |
Since β and α are each proportional to the pressure, the minimum potential is independent of the pressure of the gas. On this view the minimum potential depends upon the metal of which the cathode is made, since k measures the number of corpuscles emitted per unit time by the cathode when struck by positive ions carrying unit energy, and unless β bears the same ratio to α for all gases the minimum potential will also vary with the gas. The measurements which have been made of the “cathode fall of potential,” which as we shall see is equal to the minimum potential required to produce a spark, show that this quantity varies with the material of which the cathode is made and also with the nature of the gas. Since a metal plate, when bombarded by positive ions, emits corpuscles, the effect we have been considering must play a part in the discharge; it is not, however, the only effect which has to be considered, for as Townsend has shown, positive ions when moving above a certain speed ionize the gas, and cause it to emit corpuscles. It is thus necessary to take into account the ionization of the positive ions.
Let m be the number of positive ions per unit volume, and w their velocity, the number of collisions which occur in one second in one cubic centimetre of the gas will be proportional to mwp, where p is the pressure of the gas. Let the number of ions which result from these collisions be γmw; γ will be a function of p and of the strength of the electric field. Let as before n be the number of corpuscles per cubic centimetre, u their velocity, and αnu the number of ions which result in one second from the collisions between the corpuscles and the gas. The number of ions produced per second per cubic centimetre is equal to αnu + γmw; hence when things are in a steady state
| d | (nu) = αnu + γ mw, |
| dx |
and
e(nu + mw) = i,
where e is the charge on the ion and i the current through the gas. The solution of these equations when the field is uniform between the plates, is