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, epsilon/<(epsilon)over bar>, of the dielectric constants of the drop and the surrounding fluid. A rescaling of the equations shows that the dimensionless shape depends only an a single combination of epsilon/<(epsilon)over bar> 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 E-min for cone formation proportional to (epsilon/<(epsilon)over bar> - 1)(-5/12), at which the aspect ratio of the drop is proportional to (epsilon/<(epsilon)over bar> - 1)(1/2). Drop shapes are computed for E infinity > E-min. For E-infinity much greater than E-min the aspect ratio of the drop is proportional to E-infinity(6/7). Analogous results apply to a ferrofluid in a magnetic field.