The penetration of a magnetic field into a cylindrical plasma of a density that varies both radially and axially is studied. The magnetic field penetrates rapidly due to the Hall field, along constantnr2lines (nis the dimensionless plasma density andris the dimensionless radial coordinate). For a plasma that conducts between two cylindrical electrodes, it is shown that there is magnetic field penetration for both positive and negative polarity cases as long as there is penetration along the electrodes. The magnetic field evolution is found, analytically and numerically, for different time behaviors of the magnetic field at the boundaries. Ion velocities are also calculated.