  We consider the voltage structure in the open-field circuit and outer
magnetosphere of a magnetar. The standard polar-cap model for radio pulsars is
modified significantly when the polar magnetic field exceeds 1.8x10^{14} G.
Pairs are created by accelerated particles via resonant scattering of thermal
X-rays, followed by the nearly instantaneous conversion of the scattered photon
to a pair. A surface gap is then efficiently screened by e+- creation, which
regulates the voltage in the inner part of the circuit to ~10^9 V. We also
examine the electrostatic gap structure that can form when the magnetic field
is somewhat weaker, and deduce a voltage 10-30 times larger over a range of
surface temperatures. We examine carefully how the flow of charge back to the
star above the gap depends on the magnitude of the current that is extracted
from the surface of the star, on the curvature of the magnetic field lines, and
on resonant drag. The rates of different channels of pair creation are
determined self-consistently, including the non-resonant scattering of X-rays,
and collisions between gamma rays and X-rays. We find that the electrostatic
gap solution has too small a voltage to sustain the observed pulsed radio
output of magnetars unless i) the magnetic axis is nearly aligned with the
rotation axis and the light of sight; or ii) the gap is present on the closed
as well as the open magnetic field lines. Several properties of the radio
magnetars -- their rapid variability, broad pulses, and unusually hard radio
spectra -- are consistent with a third possibility, that the current in the
outer magnetosphere is strongly variable, and a very high rate of pair creation
is sustained by a turbulent cascade.
