Theory of the Electric Arc

Theory of the electric arc.---(1) Thermionic emission from the cathode. The fundamental phenomena of the arc are the cathode fall and the copious emission of electrons from the cathode. J. J. Thomson first suggested that this emission is chiefly of thermionic origin. But is thermionic emission sufficient to account for observed primary arc currents? Using the best data as to the emission from carbon and tungsten, the computed thermionic currents in the case of the carbon arc, the tungsten arc in hydrogen, and low-voltage, low-pressure arcs in various gases are found to be of the right order of magnitude. Br\"auer attempted to measure the thermionic emission directly by suddenly reducing the arc voltage to below the ionizing potential and obtained low values; but from a discussion of the effect of space charge on the current from the cathode it is shown that he certainly did not measure the thermionic emission but probably measured only the fraction released as a result of the effect on the space charge of positive thermionic emission from the anode. These facts and others all favor the thermionic theory as against the photo-electric and canal-ray theories of the origin of the electronic emission. (2) Current carried by positive ions, and cathode fall. If $i$ is the current density of electronic emission, there must be a current density of positive ions $j=\frac{i}{242{M}^{\frac{1}{2}}}$ to neutralize the space charge due to $i$, where $M$ is the atomic weight of the ions, and also an additional positive current density $J$ to maintain the excess positive space charge. From theoretical considerations it is found that $J=\frac{1.47{(10)}^{\ensuremath{-}7}{{V}_{c}}^{\frac{3}{2}}{\ensuremath{\lambda}}^{\frac{1}{2}}}{{M}^{\frac{1}{2}}{c}^{\frac{5}{2}}}$, where ${V}_{c}$ is the cathode fall in volts, $\ensuremath{\lambda}$ the electronic mean free path, and $c$ is the cathode dark space. Incidentally, since $c$ is approximately equal to $\ensuremath{\lambda}$, $J$ varies as the square of the pressure. An application of this expression to several cases, the carbon, hydrogen, and mercury arcs, leads to reasonable values for $J$. The ratio of positive to negative currents, $\frac{(j+J)}{i}$, is computed also from energy considerations at the cathode for various cases, with results that are consistent with the values obtained independently from the above equations. This is good evidence for the general correctness of the theory. (3) Ionization in the region between the electrodes of a carbon arc must be sufficient to neutralize the space charge due to the electrons. Reasons are given for concluding that neither emission from the anode nor ionization by collision will account for this ionization but that it is primarily of thermal origin. An expression for the current is derived and it is found that if the temperature is 4000\ifmmode^\circ\else\textdegree\fi{} K or over, and if the ionizing potential is about 8.6 volts or less, the thermal ionization estimated by applying Nernst's Heat Theorem, alone is sufficient. Photo-electric ionization if present would be secondary, a result of the radiation accompanying recombination. (4) The anode fall is accounted for in a qualitative way by a deficiency of positive ions near the anode due to decreased recombination. Thermionic emission from the anode may also play a part.Limitation of electronic current from a hot cathode by space charge.---The equation for the case of elastic impacts: ${i}_{e}=5.24{(10)}^{\ensuremath{-}6}(\frac{{V}^{\frac{3}{2}}}{{d}^{2}})(\frac{\ensuremath{\lambda}}{d})$ amp./${\mathrm{cm}}^{2}$, becomes ${i}_{i}=5.45{(10)}^{\ensuremath{-}6}(\frac{{V}^{\frac{3}{2}}}{{d}^{2}}){(\frac{\ensuremath{\lambda}}{d})}^{\frac{1}{2}}$ amp./${\mathrm{cm}}^{2}$, for inelastic impacts.