  We show that artificial magnetism of periodic dielectric or metal/dielectric
structures has limitations and is subject to at least two "uncertainty
principles". First, the stronger the magnetic response (the deviation of the
effective permeability tensor from identity), the less accurate ("certain") the
predictions of any homogeneous model. Second, if the magnetic response is
strong, then homogenization cannot accurately reproduce the transmission and
reflection parameters and, simultaneously, power dissipation in the material.
These principles are general and not confined to any particular method of
homogenization. Our theoretical analysis is supplemented with a numerical
example: a hexahedral lattice of cylindrical air holes in a dielectric host.
Even though this case is highly isotropic, which might be thought as conducive
to homogenization, the uncertainty principles remain valid.
