Singular Shape of a Fluid Drop in an Electric or Magnetic Field

Beyond a threshold, electric or magnetic fields cause a dielectric or ferromagnetic fluid drop respectively to develop conical tips. We analyze the appearance of the conical tips and the associated shape transition of the drop using a local force balance as well as a global energy argument. We find that a conical interface is possible only when the dielectric constant (or permeability) of the fluid exceeds a critical value $\epsilon_c=17.59$. For a fluid with $\epsilon>\epsilon_c$, a conical interface is possible at two angles, one stable and one unstable. We calculate the critical field required to sustain a drop with stable conical tips. Such a drop is energetically favored at sufficiently high field. Our results also apply to the formation of conical dimples when a pool of fluid is placed in a normal field.