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We shall, however, to begin with, assume that the current is so small that this cumulative effect may be neglected.

Let us now consider the rate of increase, dn/dt, in the number of corpuscles per unit volume. In consequence of the collisions, f(Xeλ)nu/λ corpuscles are produced per second; in consequence of the motion of the corpuscles, the number which leave unit volume per second is greater than those which enter it by d/dx(nu); while in a certain number of collisions a corpuscle will stick to the molecule and will thus cease to be a free corpuscle. Let the fraction of the number of collisions in which this occurs be β. Thus the gain in the number of corpuscles is f (Xeλ)nu/λ, while the loss is d/dx·(nu) + βnu/λ hence

dn f(Xeλ) nu d (nu) − βnu .
dt λ dx λ

When things are in a steady state dn/dt = 0, and we have

d (nu) = 1 (f(Xeλ) − β) nu.
dx λ

If the current is so small that the electrical charges in the gas are not able to produce any appreciable variations in the field, X will be constant and we get nu = Cεαx, where α = {f (Xeλ) − β}/λ. If we take the origin from which we measure x at the cathode, C is the value of nu at the cathode, i.e. it is the number of corpuscles emitted per unit area of the cathode per unit time; this is equal to i0/e if i0 is the quantity of negative electricity coming from unit area of the cathode per second, and e the electric charge carried by a corpuscle. Hence we have nue = i0εαx. If l is the distance between the anode and the cathode, the value of nue, when x = l, is the current passing through unit area of the gas, if we neglect the electricity carried by negatively electrified carriers other than corpuscles. Hence i = i0εα l. Thus the current between the plates increases in geometrical progression with the distance between the plates.

By measuring the variation of the current as the distance between the plates is increased, Townsend, to whom we owe much of our knowledge on this subject, determined the values of α for different values of X and for different pressures for air, hydrogen and carbonic acid gas (Phil. Mag. [6], 1, p. 198). Since λ varies inversely as the pressure, we see that α may be written in the form pφ(X/p) or α/X = F(X/p). The following are some of the values of α found by Townsend for air.

X Volts
per cm.
.17 mm.
.38 mm.
1.10 mm.
2.1 mm.
4.1 mm.
 20  .24        
 40  .65  .34      
 80 1.35 1.3   .45  .13  
120 1.8  2.0   1.1   .42  .13
160 2.1  2.8   2.0   .9   .28
200   3.4   2.8   1.6   .5 
240 2.45 3.8   4.0   2.35  .99
320 2.7  4.5   5.5   4.0  2.1  
400   5.0   6.8   6.0  3.6 
480 3.15 5.4   8.0   7.8  5.3 
560   5.8   9.3   9.4  7.1 
640 3.25 6.2  10.6  10.8  8.9 

We see from this table that for a given value of X, α for small pressures increases as the pressure increases; it attains a maximum at a particular pressure, and then diminishes as the pressure increases. The increase in the pressure increases the number of collisions, but diminishes the energy acquired by the corpuscle in the electric field, and thus diminishes the change of any one collision resulting in ionization. If we suppose the field is so strong that at some particular pressure the energy acquired by the corpuscle is well above the value required to ionize at each collision, then it is evident that increasing the number of collisions will increase the amount of ionization, and therefore α, and α cannot begin to diminish until the pressure has increased to such an extent that the mean free path of a corpuscle is so small that the energy acquired by the corpuscle from the electric field falls below the value when each collision results in ionization.

The value of p, when X is given, for which α is a maximum, is proportional to X; this follows at once from the fact that α is of the form X·F(X/p). The value of X/p for which F(X/p) is a maximum is seen from the preceding table to be about 420, when X is expressed in volts per centimetre and p in millimetres of mercury. The maximum value of F(X/p) is about 1/60. Since the current passing between two planes at a distance l apart is i0εαl or i0εXlF(X/p), and since the force between the plates is supposed to be uniform, Xl is equal to V, the potential between the plates; hence the current between the plates is i0εV·F(X/p), and the greatest value it can have is i0εV/60. Thus the ratio between the current between the plates when there is ionization and when there is none cannot be greater than εV/60, when V is measured in volts. This result is based on Townsend’s experiments with very weak currents; we must remember, however, that when the collisions are so frequent that the effects of collisions can accumulate, α may have much larger values than when the current is small. In some experiments made by J. J. Thomson with intense currents from cathodes covered with hot lime, the increase in the current when the potential difference was 60 volts, instead of being e times the current when there was no ionization, as the preceding theory indicates, was several hundred times that value, thus indicating a great increase in α with the strength of the current.

Townsend has shown that we can deduce from the values of α the mean free path of a corpuscle. For if the ionization is due to the collisions with the corpuscles, then unless one collision detaches more than one corpuscle the maximum number of corpuscles produced will be equal to the number of collisions. When each collision results in the production of a corpuscle, α = 1/λ and is independent of the strength of the electric field. Hence we see that the value of α, when it is independent of the electric field, is equal to the reciprocal of the free path. Thus from the table we infer that at a pressure of 17 mm. the mean free path is 1/325 cm.; hence at 1 mm. the mean free path of a corpuscle is 1/19 cm. Townsend has shown that this value of the mean free path agrees well with the value 1/21 cm. deduced from the kinetic theory of gases for a corpuscle moving through air. By measuring the values of α for hydrogen and carbonic acid gas Townsend and Kirby (Phil. Mag. [6], 1, p. 630) showed that the mean free paths for corpuscles in these gases are respectively 1/11.5 and 1/29 cm. at a pressure of 1 mm. These results again agree well with the values given by the kinetic theory of gases.

If the number of positive ions per unit volume is m and v is the velocity, we have nue + mve = i, where i is the current through unit area of the gas. Since nue = i0εnx and i = i0εnl, when l is the distance between the plates, we see that

nu / mv = εnx / (εnl − εnx),
n = v · εnx .
m u εneεnx

Since v/u is a very small quantity we see that n will be less than m except when εnlεnx is small, i.e. except close to the anode. Thus there will be an excess of positive electricity from the cathode almost up to the anode, while close to the anode there will be an excess of negative. This distribution of electricity will make the electric force diminish from the cathode to the place where there is as much positive as negative electricity, where it will have its minimum value, and then increase up to the anode.

The expression i = i0εαl applies to the case when there is no source of ionization in the gas other than the collisions; if in addition to this there is a source of uniform ionization producing q ions per cubic centimetre, we can easily show that

i = i0εαl + qe (eαl − 1).

With regard to the minimum energy which must be possessed by a corpuscle to enable it to produce ions by collision, Townsend (loc. cit.) came to the conclusion that to ionize air the corpuscle must possess an amount of energy equal to that acquired by the fall of its charge through a potential difference of about 2 volts. This is also the value arrived at by H. A. Wilson by entirely different considerations. Stark, however, gives 17 volts as the minimum for ionization. The energy depends upon the nature of the gas; recent experiments by Dawes and Gill and Pedduck (Phil. Mag., Aug. 1908) have shown that it is smaller for helium than for air, hydrogen, or carbonic acid gas.

If there is no external source of ionization and no emission of corpuscles from the cathode, then it is evident that even if some corpuscles happened to be present in the gas when the electric field were applied, we could not get a permanent current by the aid of collisions made by these corpuscles. For under the electric field, the corpuscles would be driven from the cathode to the anode, and in a short time all the corpuscles originally present in the gas and those produced by them would be driven from the gas against the anode, and if there was no source from which fresh corpuscles could be introduced into the gas the current would cease. The current, however, could be maintained indefinitely if the positive ions in their journey back to the cathode also produced ions by collisions, for then we should have a kind of regenerative process by which the supply of corpuscles could be continually renewed. To maintain the current it is not necessary that the ionization resulting from the positive ions should be anything like as great as that from the negative, as the investigation given below shows a very small amount of ionization by the positive ions will suffice to maintain the current. The existence of ionization by collision with positive ions has been proved by Townsend. Another method by which the current could be and is maintained is by the anode emitting corpuscles under the impact of the positive ions driven against it by the electric field.

J. J. Thomson has shown by direct experiment that positively