This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation ∇×V = λV , where λ is the eigenvalue of curl having smallest non-zero absolute value among such fields. It is shown that on the ball the energy-minimizers are the axially symmetric spheromak fields found by Woltjer and Chandrasekhar-Kendall, and on spherical shells they are spheromak-like fields. The geometry and topology of these minimum-energy fields, as well as of some higher-energy eigenfields, is illustrated.