the neutron star. The charge density will change sign when a global current flows through the null surface where ${\bf{\Omega}}\cdot{\bf{B}}=0$. As a result, a charge-deficient region ($\rho\approx 0$) in the outer magnetosphere near the null surface will be formed. Any deviation of the charge density from $\rho_0$ results in an electric field along [**[B]{}**]{}. Cheng et al. (1986a,b, hereafter CHRI and CHRII respectively) argue that this electric field can become strong enough to accelerate $e^{\pm}$ pairs to ultra relativistic energies. These $e^{\pm}$ pairs could radiate $\gamma$-ray tangential to the curved [**[B]{}**]{} field lines. These “curvature $\gamma$-rays" are further converted into $e^{\pm}$ pairs via $\gamma + \gamma\rightarrow e^+ + e^-$. Therefore, in order to keep a steady state current flow and the charge density $\rho_0$ in the regions outside the gap, the gap will grow until it is large enough and the electric field is strong enough to maintain a copious supply of charges to the rest of the open field line region. If the gap ends in a region $\rho_0\neq 0$, charges from the surrounding region will flow in through the end. If both ends are located on the null surface, any $e^{\pm}$ produced in the gap will act to 