  (abridged) Pulsar activity and its related radiation mechanism are usually
explained by invoking some plasma processes occurring inside the magnetosphere.
Despite many detailed local investigations, the global electrodynamics around
those neutron stars remains poorly described. Better understanding of these
compact objects requires a deep and accurate knowledge of their immediate
electromagnetic surrounding within the magnetosphere and its link to the
relativistic pulsar wind.
  The aim of this work is to present accurate solutions to the nearly
stationary force-free pulsar magnetosphere and its link to the striped wind,
for various spin periods and arbitrary inclination. To this end, the
time-dependent Maxwell equations are solved in spherical geometry in the
force-free approximation using a vector spherical harmonic expansion of the
electromagnetic field. An exact analytical enforcement of the divergenceless of
the magnetic part is obtained by a projection method. Special care has been
given to design an algorithm able to look deeply into the magnetosphere with
physically realistic ratios of stellar $R_*$ to light-cylinder $\rlight$
radius. We checked our code against several analytical solutions, like the
Deutsch vacuum rotator solution and the Michel monopole field. We also retrieve
energy losses comparable to the magneto-dipole radiation formula and consistent
with previous similar works. Finally, for arbitrary obliquity, we give an
expression for the total electric charge of the system. It does not vanish
except for the perpendicular rotator. This is due to the often ignored point
charge located at the centre of the neutron star. It is questionable if such
solutions with huge electric charges could exist in reality except for
configurations close to an orthogonal rotator. The charge spread over the
stellar crust is not a tunable parameter as is often hypothesized.
