Numerical solutions have been obtained for the magnetic field produced by eddy currents induced in a hollow superconducting cylinder by a centered coaxial solenoid. As the dimensions of the cylinder and the solenoid are arbitrary, the solutions are useful in the design of many types of cryogenic apparatus. The results of studies of the performance of superconducting shields as a function of the length‐to‐diameter ratio, of the effectiveness of a shield in containing the field of an interior solenoid, and of the use of a superconducting cylinder to homogenize the field produced by a solenoid are discussed. The conclusions apply to all thin‐walled conducting cylinders as long as the wall thickness is greater than the skin depth at the frequency of interest.