  We investigate the electrodynamics of an outer gap in the meridional plane of
the aligned-rotator. The charge depletion from the Goldreich-Julian charge
density causes a large electric field along the magnetic field line. The
electrons or the positrons are accelerated by the field-aligned electric field
and radiate the $\gamma$-rays tangentially to the local magnetic field line.
Some of such $\gamma$-rays collide with $X$-rays to materialize as the
electron-positron pairs on different field lines from the field line on which
they were emitted. As a result, the electric field structure is expected to
change across the field lines. Including these trans-field effects, we solve
the formation of the electric field self-consistently with the curvature
radiation and the pair creation processes. The $\gamma$-ray emission and the
pair creation are treated by use of Monte Carlo technique. We demonstrate that
the distribution of the electric field along the field lines is affected by
both the gap geometry and the external currents coming into the gap through the
boundaries. In the electrodynamical model, it has been known that the solution
disappears if the current density carried by the electron-positron pairs
produced in the gap exceeds a critical value. We show that the critical current
density is significantly increased when the trans-field structure is taken into
account. We also find that the location of the inner boundary of the gap shifts
toward the stellar surface from the conventional null surface as the current
density increases. The reason for the shift is derived from the stability
condition of the inner boundary. We also argue that the ideal-MHD condition
holds outside of the gap only when the low energy particles coexist with the
high energy particles migrating from the gap.
