  We study a geometric variational problem arising from modeling
two-dimensional charged drops of a perfectly conducting liquid in the presence
of an external potential. We characterize the semicontinuous envelope of the
energy in terms of a parameter measuring the relative strength of the Coulomb
interaction. As a consequence, when the potential is confining and the Coulomb
repulsion strength is below a critical value, we show existence and partial
regularity of volume-constrained minimizers. We also derive the Euler--Lagrange
equation satisfied by regular critical points, expressing the first variation
of the Coulombic energy in terms of the normal $\frac12$-derivative of the
capacitary potential.
