  A general axion-electrodynamic formalism is presented on the phenomenological
level when the environment is dielectric (permittivity and permeability assumed
to be constants). Thereafter, a strong and uniform magnetic field is considered
in the $z$ direction, the field region having the form of a long material
cylinder (which corresponds to the haloscope setup). If the axion amplitude
depends on time only, the axions give rise to an oscillating electric current
in the $z$ direction. We estimate the magnitudes of the azimuthal magnetic
fields and the accompanying Joule heating in the cylinder, taking the cylinder
to have ordinary dissipative properties. We evaluate and calculate the electric
current and the heat production separately, without using the effective
approximation, both when there is a strong magnetic field and when there is a
strong electric one, showing that with the magnetic field there is a heat
production, while with the electric field there is not.
  The heat generation that we consider, is a nontrivial effect as it is
generated by the electrically neutral axions, and has obvious consequences for
axion thermodynamics. The heat production can moreover have an additional
advantage, since the effect is accumulative and so grows with time. The
boundary conditions (in a classical sense) are explained and the use of them in
a quantum mechanical context is discussed. This point is nontrivial,
accentuated in particular in connection with the Casimir effect. For comparison
purposes, we present finally some results for heat dissipation taken from the
theory of viscous cosmology.
