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Pulsar activity and its related radiation mechanism are usually explained by invoking some plasma processes occurring inside the magnetosphere, be it polar caps, outer/slot gaps or the transition region between the quasi‐static magnetic dipole regime and the wave zone, like the striped wind. Despite many detailed local investigations, the global electrodynamics around those neutron stars remains poorly described with only little quantitative studies on the largest scales, i.e. of several light‐cylinder radii  r  L . A 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. This is compulsory to make any reliable predictions about the whole electric circuit, energy losses, sites of particle acceleration and the possibly associated emission mechanisms.  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 divergencelessness of the magnetic part is obtained by a projection method. Special care has been given to designing an algorithm able to look deeply into the magnetosphere with physically realistic ratios of stellar  R  *  to light‐cylinder  r  L  radius. However, currently available computational resources allow us only to set  R  * / r  L = 10 −1  corresponding to pulsars with a period of 2 ms. The spherical geometry permits a proper and mathematically well‐posed imposition of self‐consistent physical boundary conditions on the stellar crust. 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 magnetodipole 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 often hypothesized.

The pulsar force‐free magnetosphere linked to its striped wind: time‐dependent pseudo‐spectral simulations