  Various local probes have been applied to understanding current flow through
superconducting films, which are often surprisingly inhomogeneous. Here we show
that magnetic imaging allows quantitative reconstruction of both current
density, J, and electric field, E, resolved in time and space, in a film
carrying subcritical ac current. Current reconstruction entails inversion of
the Biot-Savart law, while electric fields are reconstructed using Faraday's
law. We describe the corresponding numerical procedures, largely adapting
existing work to the case of a strip carrying ac current, but including new
methods of obtaining the complete electric field from the inductive portion
determined by Faraday's law. We also delineate the physical requirements behind
the mathematical transformations. We then apply the procedures to images of a
strip of YBa2Cu3O7 (YBCO) carrying an ac current at 400 Hz. Our scanning Hall
probe microscope produces a time-series of magnetic images of the strip with 1
micron spatial resolution and 25 microsecond time resolution. Combining the
reconstructed J and E, we obtain a complete characterization including local
critical current density, E-J curves, and power losses. This analysis has a
range of applications from fundamental studies of vortex dynamics to practical
coated conductor development.
