Various conditions near the end of a semi-infinite, perfectly conducting pipe with a perfectly conducting flange are critically examined with regard to their effects on the interior field when the pipe is exposed to a low-frequency magnetic field. This is accomplished by solving several pertinent boundary-value problems. The results of this investigation indicate that (1) the effect of removing the flange is about 34 %,(2) the effect of inserting a resistive cap is to yield a "shielding ratio" similar to that in other low-frequency shielding problems, and (3) the effect of an extremely narrow annular slot in an otherwise perfectly conducting cap is surprisingly large.