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per second mat 1 ** positive ions per c.c. These positive ions will proceed up to the cathode, and a certain percentage will react and bombard it. Let the chance of the ion reaching the cathode with undiminished energy be e-P*; then the energy with which it strikes the cathode is Ve, when V is the potential at x. so that the energy in the ions striking unit area of the cathode per second is

The rate of emission to will be proportional

to this energy, so that,

.-(p- (22). a

when K is a quantity that may depend on the material of which the cathode is made and on the kind of positive ions which strike against it, but will not depend on the pressure of the gas. If W be the potential difference between the anode and cathode, and / the distance by which they are separated, V may be written in the form W/ (x/e), where /(o)=o and/(i) = l. Putting x/l = y, equation (7) gives

Now both a and are proportional to the pressure p of the gas, so that / and p only occur in the combination Ip; thus in the most general case W the spark potential will be a function of Ip; this is Paschen's law, which has been shown by Carr to hold, down to very low pressures and spark lengths. When Ip is very small (23) reduces to

Thus the potential required to produce very short sparks varies in- versely as the length of the spark, so that to produce an infinitely small spark would require an infinitely large potential. The rapid increase in the spark potential as the spark length diminishes is shown by the curve in fig. 8. The spark potential will also be infinite when / is infinite so for some intermediate spark length the potential must be a minimum. We see from the form of equation (23) that if Wo is the minimum potential, .KWoa/(/3 o) is a constant, depending only on the form of the function f, and also that, if L is the spark length when the potential is a minimum, L(/3 a) is another con- stant depending also on the form of the function; if f(y) =y we get

L =, thus the critical spark length will depend upon the

/3 o

gas, but not upon the material of which the cathode is made: the minimum potential Wo is equal in this case to (j3 a.)2-2/Kea or

Wo = t r ^- Now i/Ke is the potential difference,, through L-Kea

which a positive ion must fall to get enough energy to liberate one electron from the cathode, and oL is the number of electrons pro- duced when an electron passes over the critical spark length. If

this number is n, W =. We may summarize the argu- ment as follows: if p\ is the chance of a positive ion liberating an electron from the cathode, pi, the chance of that electron making an ion in the space d, then the probability that the original posi- tive ion will be replaced by a new one is pipi, and if the process is to be regenerative p\pi must be unity.

Since K may depend on the metal against which the ion strikes as well as upon the ion itself, the minimum potential might depend upon the material of which the cathode is made. Baerwald found, however, that for many of the ordinary metals there was not much difference in the numbers of electrons they emitted when bombarded by positive ions, so that with all such metals for cathodes the criti- cal spark should be the same. There is very considerable evidence that the minimum potential required to produce a spark is equal to the cathode fall of potential when the length of discharge is much greater than the critical spark length, and Mey has shown that the cathode fall of potential is appreciably less when the cathode is made of Al, Mg, Na or K, than when it is made of Pt, Hg, Cu or Ag.

The mechanism we have hitherto considered involves the ioniza- tion of the gas between the electrodes, and no spark could pass across a vacuum. There are, however, other methods by which a discharge might pass across a vacuum. For suppose there was a stray electron between two parallel electrodes in a vacuum; then under the action of the electric field it would be driven against the anode; by the impact Rontgen radiation would be generated, which would fall on the cathode and if it were intense enough to liberate one electron from the cathode the original electron would be replaced and the passage of negative electricity from the cathode to the anode would be repeated. From these considerations it is probable that even the highest vacuum would not act as a per- fect insulator for the very intense fields.

The linear relation V Vn-\-Cilp has been obtained on the assumption that the direction of the electric force was the same in all parts of the field; this is only true when the dimensions of the

electrodes are large compared with the distance between them. The potential difference required to produce a spark of a particular length depends upon the size of the electrodes between which the spark passes, and is not a linear function of Ip, where p is the pres- sure and / the spark length, unless / is small compared with the linear dimensions of the electrodes. If these are spheres, the spark potential will depend upon their radii, and for small spheres may be considerably less than for large ones. Thus, for example, the spark potential in air for a five centimetre spark is 26,000 volts for electrodes -5 cm. in diameter, and 105,000 volts when the diameter of the electrodes is 5 centimetres.

In this connexion it may be noted that, if the electric field is sufficiently intense at any place to produce there a local supply of ions, these may redistribute themselves between the electrodes, and by their electrostatic action produce a change in the distribu- tion of the electric force more favourable to the passage of the spark than that prior to the production of the ions. To illustrate this, take the very simple case when the electrodes are two parallel plates: if there are any ions available these may distribute them- selves so that the force between the plates is no longer uniform. Thus let us suppose that there are enough positive ions to congre- gate round the cathode in sufficient numbers to produce within the distance of the " critical spark length " or thickness of the cathode dark space a difference of potential equal to the minimum spark potential. This would ensure that from close to the cathode there was a continual emission of electrons, and even though the electric field from this place to the anode was too feeble to give an electron enough energy to ionize the gas, the electrons coming from the cathode would be able to carry a small current, though this part of the discharge might not be luminous. The ions here would be all of one sign, so that the electric force will increase up to the anode. If the current is gradually increased, the place where the electric. force will just rise to the value necessary to make the electrons ionize will be close to the anode. When this occurs a supply of positive ions will start from the anode and move towards the cath- ode, accompanied by luminosity close to the anode and very faint luminosity through the rest of the tube. The introduction of the positive ions into the region between the anode and cathode will diminish the retarding effect of the negative space charge which existed in this region, so that the current will increase. This increase in current will again increase the ionization at the anode, and thus the supply of positive ions. In this way there might be a supply of electrons coming from the cathode, and of positive ions from close to the anode, which will maintain the current in spite of the fact that between these places there was a region where the electric force was below that required to produce ionization by collision, and the potential difference between the electrodes less than that cal- culated on the supposition that the electric force was uniform from one to the other. We should expect from these considerations that, if the electric force at any point were intense enough to produce ionization by collision, some discharge would take place.

Russell (Phil. Mag. 6. xi., p. 237) states that the results of the different experiments made on the potential difference required to produce sparks of various lengths between spherical electrodes of various radii are in good agreement with the rule that the discharge takes place in air at atmospheric pressure if the electric force at any point in the field before discharge begins is as great as 37,000 volts per centimetre. This value agrees well with that required to make electrons produce in air at atmospheric pressure other ions by collisions.

The curious lag observed by Warburg between the application of the potential difference and the passage of the spark, which may amount in extreme cases to several. seconds, e.g. when the applied potential is only a very little greater than that required to produce the spark, is naturally explained as the time necessary for the ions to distribute themselves so as to produce the distribution of poten- tial required for the discharge.

The discharge of electricity from points affords a good illus- tration of the preceding considerations. Suppose that the elec- trodes are a needle point and a plane. When the discharge first begins the only place where any light is to be seen is close to the point ; the current between the electrodes is very small ; as the poten- tial difference increases a stage is reached where light begins to appear close to the points, the space between the point and plate being quite dark. This stage is marked by a large increase in the current. With further increase in current the luminosity extends into the gas and ultimately stretches from one electrode to another.

The potential required to start the discharge is less where the point is negative than where it is positive. This is what might be expected, for to maintain the discharge from the negative point there must be (l) ionization of the gas by the outgoing electrons, and (2) liberation of electrons by the incoming positive ions, while when the point is positive there must be (l) ionization of the gas by outgoing positive ions, and (2) liberation of positive ions by the impact of incoming electrons; as the process is not the same as for the negative point we should expect that there would be a differ- ence between the potentials. It is not only the potential difference which is affected but the type of discharge. This can be shown by allowing the point discharge to pass in the neighbourhood of a photographic plate. Beautiful figures are found on developing the