  We study the classical electrodynamics of extended bodies. Currently, there
is no self-consistent dynamical theory of such bodies in the literature.
Electromagnetic energy-momentum is not conserved in the presence of charge and
some addition is required. The only somewhat suitable addition found to date
are point charges. These suffer from infinite self-energy, requiring some
renormalization procedure, and perturbative methods to account for radiation.
We review the history that has led to the understanding of these facts.
  We then investigate possible self-consistent, non-point-charge, classical
electrodynamic theories. We start with a Lagrangian consisting only of the
Ricci scalar (gravity) and the standard electromagnetic field Lagrangian, and
consider additions other than point charges and their associated interaction
Lagrangian. Including quadratic terms in the Lagrangian involving first-order
derivatives of the electromagnetic field tensor provides sufficient
stress-energy terms to allow for conservation of energy-momentum. There are
three such independent terms: a direct current-current interaction and two
curvature-mediated (non-minimally coupled), short-range interactions, one of
which changes sign under a parity transformation. These could be interpreted as
non-electromagnetic, short-range forces.
  For the simplest possible theory, with only the metric and the
electromagnetic potential 1-form as independent fields, we find a single,
stable, spherical (spin-0) solution, which due to an integrable singularity at
the solution's center, has quantized mass and charge. Its charge is smaller
than numeric error, and its mass is set by a new constant in the Lagrangian. It
has a small, central core of charge surrounded by a wave of alternatingly
charged, spherical shells, where the amplitude of the charge density wave is
inversely proportional to the radial coordinate.
