  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.
