Electromagnetic induction in finite-length cylinders: an exact solution

The calculation of electromagnetic fields and currents induced in circularly cylindric metallic shells by low-frequency external fields is discussed, with focus on the effects of finite cylinder length. For an inducing field possessing rotational symmetry about an axis parallel to the shell axis, we have developed approximate analytic solutions. Further, for the degenerate case of coincident axes, we obtain an exact analytic solution. Already this coaxial solution makes manifest the complexity introduced by finite length, and yet allows one to discern the behaviour of induced field and current with varying length. We outline here the basic field-theoretic formulation and a strategy for solving the finite cylinder problem for any inducing field, which yielded the approximate paraxial solutions cited. We then present a detailed derivation of our exact coaxial solution from the basic formulation. General properties of the exact analytic solution are discussed, including its practicability, eddy-current system and cylinder-length dependence, and contrasted with recent inadequate analytical treatments for both coaxial and paraxial cases.