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electrified particles when they strike against a metal plate cause the metal to emit corpuscles (J. J. Thomson, Proc. Camb. Phil. Soc. 13, p. 212; Austin, Phys. Rev. 22, p. 312). If we assume that the number of corpuscles emitted by the plate in one second is proportional to the energy in the positive ions which strike the plate in that second, we can readily find an expression for the difference of potential which will maintain without any external ionization a current of electricity through the gas. As this investigation brings into prominence many of the most important features of the electric discharge, we shall consider it in some detail.

Let us suppose that the electrodes are parallel plates of metal at right angles to the axis of x, and that at the cathode x = 0 and at the anode x = d, d being thus the distance between the plates. Let us also suppose that the current of electricity flowing between the plates is so small that the electrification between the plates due to the accumulation of ions is not sufficient to disturb appreciably the electric field, which we regard as uniform between the plates, the electric force being equal to V/d, where V is the potential difference between the plates. The number of positive ions produced per second in a layer of gas between the planes x and x + dx is αnu·dx. Here n is the number of corpuscles per unit volume, α the coefficient of ionization (for strong electric field α = 1/λ′, where λ′ is the mean free path of a corpuscle), and u the velocity of a corpuscle parallel to x. We have seen that nu = i0εαx, where i0 is the number of corpuscles emitted per second by unit area of the cathode. Thus the number of positive ions produced in the layer is αi0εαx dx. If these went straight to the cathode without a collision, each of them would have received an amount of kinetic energy Vex/d when they struck the cathode, and the energy of the group of ions would be Vex/d·αi0εdx dx. The positive ions will, however, collide with the molecules of the gas through which they are passing, and this will diminish the energy they possess when they reach the cathode.

The diminution in the energy will increase in geometrical proportion with the length of path travelled by the ion and will thus be proportional to εβx, β will be proportional to the number of collisions and will thus be proportional to the pressure of the gas. Thus the kinetic energy possessed by the ions when they reach the cathode will be

εβx·V(ex/d) · αi0εαxdx,

and E, the total amount of energy in the positive ions which reach the cathode in unit time, will be given by the equation

E = d0 εβx · V(ex/d) · α i0εαx dx = Veα i0 d0 ε−(βα)x x dx
= Veα i0 { 1 ε−(βα)d{ 1 + d }}  (1).
d (βα)2 (βα)2 (βα)

If the number of corpuscles emitted by the cathode in unit time is proportional to this energy we have i0 = kE, where k is a constant; hence by equation (1) we have

V = (βα)2 · d ,
ke α I


I = 1 − ε−(βα)d (1 + d (βα)).

Since both β and α are proportional to the pressure, I and (βα)2d/α are both functions of pd, the product of the pressure and the spark length, hence we see that V is expressed by an equation of the form

V = 1 ƒ (pd)   (2),

where ƒ(pd) denotes a function of pd, and neither p nor d enter into the expression for V except in this product. Thus the potential difference required to produce discharge is constant as long as the product of the pressure and spark length remains constant; in other words, the spark potential is constant as long as the mass of the gas between the electrodes is constant. Thus, for example, if we halve the pressure the same potential difference will produce a spark of twice the length. This law, which was discovered by Paschen for fairly long sparks (Annalen, 37, p. 79), and has been shown by Carr (Phil. Trans., 1903) to hold for short ones, is one of the most important properties of the electric discharge.

We see from the expression for V that when (βα)d is very large

V = (βα)2d/keα.

Thus V becomes infinite when d is infinite. Again when (βα)d is very small we find

V = 1/keαd;

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−(βα)d {1 + (βα)d + (βα·d)2}.

We find by a process of trial and error that (βα)d = 1.8 is approximately a solution of this equation; hence the distance for minimum potential is 1.8/(βα). 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 = βα · 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 + γmw,


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

enu = Cε(αγ)xγi / (αγ),  emw = −Cε(αγ)x + αi / (αγ),

where C is a constant of integration. If there is no emission of positive ions from the anode enu = i, when x = d. Determining C from this condition we find

enu = i {αε (αγ) (xd)γ },  emw = αi {1 − ε (αγ) (xd) }.
αγ αγ

If the cathode did not emit any corpuscles owing to the bombardment by positive ions, the condition that the charge should be maintained is that there should be enough positive ions at the cathode to carry the current i.e. that emw = i; when x = 0, the condition gives

i { αε−(αγ)dγ} = 0


ε αd/α = ε γd/γ.

Since α and γ are both of the form pƒ(X/p) and X = V/d, we see that V will be a function of pd, in agreement with Paschen’s law. If we take into account both the ionization of the gas and the emission of corpuscles by the metal we can easily show that

αγε(αγ)d = kαVe [ 1 ε−(β+γα)d{ 1 + d }],
αγ d (β + γα)2 (β + γα)2 β + γα

where k and β have the same meaning as in the previous investigation. When d is large, ε(αγ)d is also large; hence in order that the left-hand side of this equation should not be negative γ must be less than α/ε(αγ)d; as this diminishes as d increases we see that when the sparks are very long discharge will take place, practically as soon as γ has a finite value, i.e. as soon as the positive ions begin to produce fresh ions by their collisions.

In the preceding investigation we have supposed that the electric field between the plates was uniform; if it were not uniform we could get discharges produced by very much smaller differences of potential than are necessary in a uniform field. For to maintain the discharge it is not necessary that the positive ions should act as ionizers all along their path; it is sufficient that they should do so in the neighbourhood of cathode. Thus if we have a strong field close to the cathode we might still get