  The maximum current that can be transported across a vacuum diode is limited
by forces arising due to space charge. In a planar diode configuration, the
space charge limited (SCL) current density from a planar emitting patch is
given by the Child-Langmuir (CL) law $J_{CL} \sim V_g^{3/2}/D^2$ where $V_g$ is
the potential difference across the diode and $D$ is the separation between the
anode and cathode. We show here analytically using the nonlinear line charge
model that for a curved emitter in a planar diode configuration, the limiting
current obeys the scaling relationship $J_{SCL} \sim \gamma_a V_g^{3/2}/D^2$
where $\gamma_a$ is the apex field enhancement factor of the curved emitter.
For an emitter with large height ($h$) to apex radius of curvature ($R_a$)
ratio, the limiting current far exceeds the planar value. The result is
verified using the particle-in-cell code PASUPAT for two curved emitters
shapes.
