Spherical conducting liquid drops fission when their net electrical charge exceeds the Rayleigh limit
Q
c
≡ 4√π, where the shape becomes unstable to small amplitude perturbations which lead to two-lobed forms. We use a combination of domain perturbation and multiple timescale methods to compute the evolution of axisymmetric, inviscid oscillating drops near this limit and show that it corresponds to a transcritical bifurcation point between the families of static spherical shapes and oblate and prolate axisymmetric forms. Prolate forms exist at lower values of charge (
Q
<
Q
c
) and are unstable to small-amplitude perturbations. Finite amplitude oscillations destabilize the spheres at a value of charge below
Q
c
, with the decrease in the critical value proportional to the amplitude of the prolate component of the initial shape disturbance. Oblate static shapes exist for
Q
>
Q
c
and are stable to small axisymmetric perturbations, but are unstable to moderate-amplitude ones, where the stability boundary is given by the nonlinear stability analysis. Finite element analysis is used to calculate the static drop shapes in both the prolate and oblate families. Asymptotic analysis for the static shapes is in good agreement with the numerical calculations for even moderate-amplitude deformations of the drop.
Subcritical asymmetric Rayleigh breakup of a charged drop induced by finite amplitude perturbations in a quadrupole trap
