Passive end body plasma contactors have been operationally validated in space and have been shown to provide a simple, effective, and robust means of current collection at the positive terminal of an electrodynamic tether system. A grid sphere has been suggested as a possible end body since it potentially has distinct advantages compared to a solid sphere, including a lower neutral dynamic drag and a higher current-to-mass ratio. This paper estimates the maximum current collected by a grid sphere, taking into account its orbital motion and ion production inside the grid sphere. We first review the data from the tethered satellite system (TSS 1) and the TSS 1R flights, formulate a model for current collection by a solid sphere, and suggest how to incorporate it into the grid sphere current collection estimate. Then we calculate the potential distribution inside the grid sphere and the potential distribution outside the solid sphere for the same system parameters. Finally, we estimate the maximum current collected by a grid sphere depending on its transparency.