Some applications of an Electron Beam Ion Source (EBIS) require intensities of highly charged ions significantly greater than those which have been achieved in present EBIS sources. For example, the ion source for the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) must be capable of generating 3x10{sup 9} ions of Au{sup 35+} or 2 x 10{sup 9} ions of U{sup 45+} per pulse [1]. In this case, if the fraction of ions of interest is 20% of the total ion space charge, the total extracted charge is {approx}{approx} 5 x 10{sup 11}. It is also desirable to extract these ions in a 10 ps pulse to allow single turn injection into the first synchrotron. Requirements for an EBIS which could meet the needs of the LHC at CERN are similar ({approx} 1.5 x 10{sup 9} ions of Pb{sup 54+} in 5.5 {micro}s). This charge yield is about an order of magnitude greater than that achieved in existing EBIS sources, and is what is meant here by ''high current''. This also implies, then, an EBIS with a high electron beam current. The scope of problems in a high current EBIS is broad, and includes generating a sufficient total …