  A two-dimensional electrodynamical model is used to study particle
acceleration in the outer magnetosphere of a pulsar. The charge depletion from
the Goldreich-Julian charge density causes a large electric field along the
magnetic field lines. The charge particles are accelerated by the electric
field and emit $\gamma$-rays via the curvature process. Some of the emitted
$\gamma$-rays may collide with $X$-ray photons to make new pairs, which are
accelerated again on the different field lines in the gap and proceed similar
processes. We simulate the pair creation cascade in the meridional plane using
the pair creation mean-free path, in which the $X$-ray photon number density is
proportional to inverse square of radial distance. With the space charge
density determined by the pair creation simulation, we solve the electric
structure of the outer gap in the meridional plane and calculate the curvature
spectrum.
  Because the two-dimensional model can link both gap width along the magnetic
field line and trans-field thickness with the spectral cut-off energy and flux,
we can diagnose both the current through the gap and inclination angle between
the rotational and magnetic axes. We apply the theory to the Vela pulsar. By
comparing the results with the $EGRET$ data, we rule out any cases that have a
large particle injection at the outer boundary. We also suggest the inclination
angle of $\alpha_{inc}\geq65^{\circ}$. The present model predicts the outer gap
starting from near the conventional null charge surface for the Vela pulsar.
