of the ions proportional to the electric force; the second is when the hot wire is surrounded by a vacuum, and the motion of the ions is not affected by the gas.
In the first case u=K^-r-^, when K% is the mobility of the nega- tive ion, and the equation nue = l is equivalent to
%%-> (I3) -
The solution of this is
Tr) 2=8 ^ x + c -
ax / Ki
Therefore if V is the difference of potential between the anode and cathode, and / the distance between them,
(H).
If is the velocity of the negative ions at the cathode, i hence
neuo',
VbTTMo
I-i
ds).
So that, unless t is small compared with I, will be comparable with c; in this case, however, the velocity of the ion is no longer propor- tional to the electric force so that equation (13) no longer holds. Again, when the current approaches saturation, i/(I c) is large, and therefore by (15) uo will be large compared with c. For the negative ion to acquire a velocity of this magnitude the electric field would have to be so strong that sparks would pass through the gas unless the pressure were very low. Thus saturation currents from hot bodies are only obtainable at very low pressures.
Since Uo =
Comparing this with the value of 8iril/K we find, by substituting the values of K and c, that if the current is far from saturation, C will be negligible compared with Siril/K, unless /I, when / is measured in centimetres and I in milliamperes, is small compared with unity. When C can be neglected, equation (12) gives
Thus the current is proportional to the square of the potential differ- ence. A remarkable thing about this expression is that for these very small currents the intensity of the current is independent of the temperature of the wire, although, of course, the range of cur- rents over which this formula is applicable is wider the higher the temperature of the wire.
When the hot body is in a vacuum, we have, if the ions have no initial velocity,
where m is the mass and e the charge on an ion; hence the equation nue = i is equivalent to
V - V/
dx" --^ i ^ m l 2e w)>
a solution of which is
V = (9iri)5 (m/2e)lxl (18).
Hence, if V is the potential difference and / the distance between the electrodes
We see from this equation that the electric force vanishes at the cathode, and that the density of the negative electrification is pro- portional to xl ; thus it is infinite close to the cathode and dimin- ishes as the distance from the anode diminishes. The total quantity of electricity between the anode and cathode is proportional to /i 2 . We see again that for a given potential difference the current does not depend on the temperature of the hot wire ; this law only holds when the currents are less than the maximum currents which can pass between the electrodes. When the current approaches this value, the current instead of increasing as Vi becomes independent of V and the negative electricity between the electrodes diminishes as V increases. Langmuir, who has made a very complete investi- gation of the currents from hot wires, finds that the expression (7) represents, with considerable accuracy, the relation between the current and potential over a wide range in the values of the cur- rents. The curves in fig. 5 given by him represent the relation between the current and potential for wires at different tempera- tures. They illustrate the point that a colder wire, until it is approach- ing the stage of saturation, gives as large a current as a hotter one, though the hotter one, of course, has a wider range of currents.
lonization by Collision. The curve representing the relation between the currents through a gas ionized (say) by Rontgen rays and the difference of potential between the electrodes is found
to be of the form already shown in fig. i, where the ordinates represent the currents and the abscissae the potential difference. The flat part represents the state of saturation when the poten- tial difference is large enough to send all the ions produced by the rays to the electrode before they can recombine. When the poten- tial difference is still further increased we see that a stage is
moo
20OO 2200
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Temperature FIG. 5
2400
2600
reached when the current begins to increase with great rapidity with the potential difference, and reaches values much greater than could be attained by the ions produced by the Rontgen rays. Thus in addition to the ions produced by the rays there must be other ions, and some other source of ionization associated with the strong electric fields. Now the processes going on in a gas while it is conveying an electric current are: (i) the ionization of the gas by the external agent in this an electron is liberated from the molecule and the residue forms a positive ion; (2) the electron and the positive ion acquire energy under the action of the electric forces; (3) in many gases the electron finally unites with an uncharged molecule to form a negative ion. As the most noticeable change in the conditions when the intensity of the electric field increases is in the energy of the electrons and ions, it is natural to look to these as the source of the additional ionization. We have moreover direct experimental evidence that rapidly moving electrons and ions are able to ionize a gas through which they are passing. Hot wires and metals exposed to ultra-violet light yield a supply of electrons which when they leave the metal have very little energy; by applying suitable electric fields these electrons can be endowed with definite amounts of energy and can then be sent through a gas from which all extraneous ionizing agencies are shielded off. When this is done it is found that, when the energy of the electrons exceeds a certain critical value, depending upon the nature of the gas, the gas is ionized by the electrons, but no ionization occurs when the energy of the electron falls below this limit. It is convenient to measure the energy of the electron in terms of the difference of electrical potential through which the electron has to fall in order to acquire this energy. The potential difference which would give to the electron the energy at which it begins to ionize the gas is called the ionizing potential. The values of the ionizing potential have been found for several gases, as will be seen from the following table. There is, however, considerable discrepancy between the results obtained by different observers.
