  Determining the electric field (E-field) distribution on the Sun's
photosphere is essential for quantitative studies of how energy flows from the
Sun's photosphere, through the corona, and into the heliosphere. This E-field
also provides valuable input for data-driven models of the solar atmosphere and
the Sun-Earth system. We show how Faraday's Law can be used with observed
vector magnetogram time series to estimate the photospheric E-field, an
ill-posed inversion problem. Our method uses a "poloidal-toroidal
decomposition" (PTD) of the time derivative of the vector magnetic field. The
PTD solutions are not unique; the gradient of a scalar potential can be added
to the PTD E-field without affecting consistency with Faraday's Law. We present
an iterative technique to determine a potential function consistent with ideal
MHD evolution; but this E-field is also not a unique solution to Faraday's Law.
Finally, we explore a variational approach that minimizes an energy functional
to determine a unique E-field, similar to Longcope's "Minimum Energy Fit". The
PTD technique, the iterative technique, and the variational technique are used
to estimate E-fields from a pair of synthetic vector magnetograms taken from an
MHD simulation; and these E-fields are compared with the simulation's known
electric fields. These three techniques are then applied to a pair of vector
magnetograms of solar active region NOAA AR8210, to demonstrate the methods
with real data.
