Let us consider a neutron star with a surface magnetic field $ B_0 $, rotating as a rigid body with angular velocity $ \Omega $. We assume that the rotation axis is aligned with the magnetic field. For high enough values of the magnetic field and the rotation angular velocity such an object cannot be in vacuum. This is because the induced electric field will be strong enough to overcome the gravitational pull. The neutron star is assumed to be an ideal conductor, so the induced electric field is

$ E \approx B_0 \Omega R /c $

where $ R $ is the radius of the neutron star and $ c $ is the speed of light. The ratio between the gravitational and electrostatic force is

$ \frac{G M}{R^2} / q \frac{B_0 \Omega R}{c} \approx \frac{\sqrt{G M / R^3}}{\Omega} \frac{\sqrt{M c^2/R^3}}{B_0} \frac{\sqrt{G} m_p}{q} $

where $ q $ is the elementary charge, $ M $ is the mass of the neutron star and $ G $ is the universal constant of gravity. The first term is the ratio between the breakup angular velocity and the actual angular velocity, the second term is the square root of the ratio between the rest mass energy and the magnetic energy, and the last term is a measure of ratio between the strength of electric and gravitational forces on the proton. For a typical neutron star the first two terms are larger than unity, but the third term is tiny ([1]$ 10^{-18} $) and hence the entire expression is smaller than one.

A neutron star is therefore surrounded by co-rotating plasma. Outside the neutron star the magnetic field is roughly given by

$ B \approx B_0 R^3/r^3 $

The induced electric field is

$ E \approx \Omega r B/c $

and the corresponding charge density is

$ \rho \approx \frac{E}{r} \approx \Omega B /c $