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186 ions at the place fixed by the coordinate x; Ui and U 2 the velocities of these ions. The volume density of the electrification in the gas, if it is entirely due to the ions, is (i rti)e when e is the charge on an ion, hence

= If i is the current through unit area of the gas i = e(niU\-\-ntfii) (2). Hence from (i) and (2) we have

dX dx

n\e-

+i

dx

(3), (4).

When things are in a steady state, neglecting any loss of ions by diffusion we have

(5), (6),

where q is the number of ions produced per second in a cub. cm. of gas, and a is the coefficient of recombination; if K\, K 2 are the mobilities of the positive and negative ions respectively, then

M 1 =^,X, 2 = A' 2 X. From equations (i), (5) and (6) we get

dX

= *e q -an l n

and, substituting the values of i and n 2, we get /i i \ ( a

(7)-

, dX --

No general solution of this equation has been obtained, but when i is small compared with the saturation current qle, an approximate solution is represented by the graph in fig. 2.

FIG. 2

The force is practically constant, and equal to

except close to the electrode, where it increases; and as the mobil- ity of the negative ion is greater than that of the positive the increase in the force will be greater at the cathode than at the anode. As the potential difference between the electrodes increases, and the cur- rent approaches more nearly the saturation value, the flat part of the graph diminishes, and the graph for X takes the form given in fig. 3- When the potential difference is so large that the current is

FIG. 3

nearly saturated, X is very approximately constant from one elec- trode to another.

In one extremely important case, that in which the negative ions are electrons and have a mobility which may be regarded as infinite in comparison with that of the positive ions, equation (7) admits of integration : for by putting Ki/K 2 = o in equation (8) it becomes dX oire K?fiX Siri

TX-+ ZK; (8) -

If, as is more convenient in this case, x is the distance from the cathode instead of from the anode, as we have hitherto assumed, the solution of this equation is

gKle 1 01

The second term on the right-hand side diminishes very rapidly as x increases and soon gets negligible, so that we see that the elec- tric force will be constant except in the immediate neighbourhood of the cathode. To find the value close to the cathode we must find the value of C in equation (9). We have from equation (7)

jr T& ; TS\ I = I w o.n\n2)dx (10). e dx (AI +A 2 ) J o Jo

The right-hand side of this equation is the excess of ionization over recombination in the region between the cathode and x; it must therefore be equal to the excess of number of the negative ions passing through the gas at x; it must therefore be equal to (t it,)/e where m is the amount of negative electricity, emitted by unit area of the cathode in unit time. Putting this value for the right-hand side of equation (10) we find approximately, since KI is small com- pared with KZ,

01(1-10)

=

qKiKt K.Z Substituting this value for C, we find

This distribution of force is represented by the graph in fig. 4; the force at some distance from the cathode is equal to

Keg

and is thus proportional to the current ; the force at the cathode itself is { KZ(L La)l Kii^ times greater than this. The fall of potential be-

tween the electrodes is made up of two parts, one arising from the con- stant force; as this force is proportional to i, this part of the potential fall will be proportional to d when / is the distance between the elec- trodes, and may be represented by Ai/ when A is a constant; the other part of the potential fall is that which occurs close to the cathode. We find from equation (11) that this is proportional to i"

FIG. 4

Distance from Cathcde

and does not depend upon /. Thus, if V is the potential difference between the electrodes when A and B are constants V = Ai/+Bi 2 (12).

H. A. Wilson has shown that an equation of this type represents the relation between the current and potential difference for con- duction through flames. In many cases the drop of potential at the cathode is much greater than the fall in the rest of the circuit ; when this is so we see that the current is proportional to the square root of the potential difference. The value of B increases with the pres- sure and decreases with the amount of the ionization.

Current from Hot Wires. A case of great importance from its industrial application in hot wire valves is one where all the ions are negative and are emitted from the cathode. Metal wires raised to incandescence emit electrons, and if they are used as cathodes can transmit across a vacuum or gas at a low pressure very consider- able currents. No currents will pass if they are used as anodes.

Take the hot cathode as the origin from which * is measured; let V be the potential at the point x, n the density of the negative ions at this point, and i the current through unit area. If o is the velocity of the negative ion, we have

nwe=iand

d#

There are two cases to be considered; the first is when the hot wire is surrounded by gas of sufficient density to make the velocity