Drops with conical ends in electric and magnetic fields

Slender-body theory is used to determine the approximate static shape of a conically ended dielectric drop in an electric field. The shape and the electric-field distribution follow from solution of a second-order nonlinear ordinary differential equation that can be integrated numerically or analytically. An analytic formula is given for the dependence of the equilibrium cone angle on the ratio, 6/e, of the dielectric constants of the drop and the surrounding fluid. A rescaling of the equations shows that the dimensionless shape depends only on a single combination of e/e and the ratio of electric stresses and interfacial tension. In combination with numerical solution of the equations, the rescaling also establishes that, to within logarithmic factors, there is a critical field Em-m for cone formation proportional to (e/e- 1)~5/12, at which the aspect ratio of the drop is proportional to (e/ë- 1)1/2. Drop shapes are computed for EOO > E_min. For EOO 3≥ E_min the aspect ratio of the drop is proportional to E> . Analogous results apply to a ferrofluid in a magnetic field.