  We study the electromagnetic behaviour of dense, spherical clusters made of
hundreds of plasmonic nanoparticules under illumination by a plane wave. Using
high-precision T-matrix numerical calculations, we compute the multipolar
response of clusters up to 80 nm in radius and up to 44\% in particle volume
fraction. We then investigate whether it is possible to obtain an effective
medium description for the clusters, taking into account weak spatial
dispersion in a fully consistent way. We find that the average scattered field
as well as the average inner field of the spherical cluster can be accurately
reproduced by applying an extended Mie theory to an equivalent homogeneous
sphere characterized by three effective parameters: an electric permittivity
$\varepsilon_{\mathrm{eff}}$ and a magnetic permeability $\mu_{\mathrm{eff}}$,
associated to transverse modes, and a wavevector $k_\mathrm{L}$, associated to
a longitudinal mode in the sphere. Our results show that artificial magnetism
arises from interparticle couplings in the dense cluster, despite inclusions
not displaying any individual magnetic dipole. We also find that, although
largely overlooked in the literature on metamaterials, the presence of the
longitudinal mode is essential to accurately reproduce the fields of the
cluster, on par with the role of artificial magnetism. Our study therefore
proves that, even for high concentration in inclusions, it is possible
empirically to treat a cluster of plasmonic particles as a sphere made of a
spatially-dispersive homogeneous medium. This offers a practical solution
facilitating the computation of electromagnetic responses of such dense random
media in diverse configurations of interest for the design of metamaterials and
metasurfaces.
