  We calculate the screening charge density distribution due to a point charge,
such as that of a positive muon ($\mu^+$), placed between the planes of a
highly anisotropic layered metal. In underdoped hole cuprates the screening
charge converts the charge density in the metallic-plane unit cells in the
vicinity of the $\mu^+$ to nearly its value in the insulating state. The
current-loop ordered state observed by polarized neutron diffraction then
vanishes in such cells, and also in nearby cells over a distance of order the
intrinsic correlation length of the loop-ordered state. This in turn strongly
suppresses the loop-current field at the $\mu^+$ site. We estimate this
suppressed field in underdoped YBa$_2$Cu$_3$O$_{6+x}$ and
La$_{2-x}$Sr$_x$CuO$_4$, and find consistency with the observed 0.2--0.3 G
field in the former case and the observed upper bound of $\sim$0.2 G in the
latter case. This resolves the controversy between the neutron diffraction and
$\mu$SR experiments. The screening calculation also has relevance for the
effect of other charge impurities in the cuprates, such as the dopants
themselves.
