A Theory of Liquid Surface Rupture by a Uniform Electric Field

Abstract

Surface distortion and rupture permits field emission from liquid surfaces at field strengths less than those effective for equally smooth solid surfaces. An approximate mathematical theory of the rupture of a plane liquid surface in a uniform electric field has been developed. The relation between initial distortion, rupture time, and field strength has been calculated for fields large compared to that which just renders the surface unstable. This critical field is ${E}_{m}=2{\ensuremath{\pi}}^{\frac{1}{2}}{(\ensuremath{\rho}gT)}^{\frac{1}{4}}=53$ kv ${\mathrm{cm}}^{\ensuremath{-}1}$ for mercury, where $\ensuremath{\rho}$ is density and $T$ is surface tension. Typically, the theory shows that a hump initially 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ cm high and of diameter 9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ cm will lead to rupture in 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ sec. in a field of ${10}^{6}$ v ${\mathrm{cm}}^{\ensuremath{-}1}$. Relative to initial humps in the surface whose linear dimensions vary inversely as the square of the field, the time to rupture varies inversely as the cube of the field strength. This calculation shows that a lowered sparking potential to liquid mercury can be ascribed to surface rupture and shows that it is possible that surface rupture plays a part in Beams' low field emission from liquid mercury. Possible application of the theory to the high field condition at the cathode spot of the Hg arc is not clear.