A Basic Theory of the Mercury Cathode Spot

Most writers about the cathode spot have concluded that it is too complicated for valid conclusions. There is presented here a simple theory, which can be called basic, since it is founded on the single assumption that the electron emission is field emission. It follows immediately from this assumption that the positive ion current density must be about 5×10⁵ A/cm² to supply the necessary field of 2.7×10⁷ V/cm; and this, with an allowance of 20 electrons per ion for the two‐stage ionization, determines an electron current density of 10⁷ A/cm². This large ion current and the resultant heating of the spot precludes an energy balance, which would require temperatures and pressures far above the critical to dissipate the heat by conduction and evaporation. Such a balance would never be reached, however, because of a mechanical balance between the forces of surface tension and vapor pressure, which limits the pressure and temperature to much lower values; namely a pressure of about 1 atm and the temperature of the order of 400°C. The balance is given by the simple equation $$\text{p=(2γ/r)}{\mathrm{dyn/cm}}^2,$$ where γ is the surface tension in dyn/cm, and r the radius of the spot in cm. A table is given for the calculated pressure and temperature for different currents. This maximum temperature determines the velocity of (random) motion of the spot, which is just its diameter divided by the time to heat a new spot to this temperature. The calculated velocities are included in the table. Retrograde motion of the spot in a magnetic field is explained by a drift of slow ``residual'' electrons parallel to the surface in the ``correct'' direction, thus weakening the electric field drawing ions to the surface. A theory of drift of slow electrons by Tonks, makes possible a calculation of the drift angle, which determines the change from retrograde to ``correct'' motion as the magnetic field and vapor pressure are changed. The calculated values are in good agreement with experiment. There are no measurements of pressure on a free spot, but the theory predicts correctly the height to which mercury droplets are thrown, assuming the erupted droplets to be the same diameter as the spot. Observed pressures for anchored spots are much higher, but these depend on an entirely different mechanism. The observed velocities of retrograde motion are much less than those calculated; but it is pointed out that the motion may be largely random, rather than in a straight line, which would make agreement with the theory possible. Evidence is given that the limiting current of 30 A per spot is due to the IR potential drop in the mercury, which leaves insufficient energy for ionization if this current is exceeded. A few values of pressure and temperature are calculated for other low‐boiling‐point metal arcs, assuming that the same theory applies to them.