Elementary model of pulsar magnetosphere

Main study material for this topic is the textbook by Vladimir Karas, in particular, Section 7. You may also check my hand-notes for some derivations. The textbook is rather straghtforward; nevertheless, let me point out several notes:

• force-free approximation is used to derive electric field componensts at the surface of the neutron star (which is simply treated as a conductive sphere)
• derivation of the magnetic flux at infinity is intrinsically inconsistent. In spite of that it provides reasonably good approximation. The inconsistency lies in that, by means of qualitative arguments, we suppose that the magnetic field lines depart from the dipole shape close to / beyond the light cylynder. This consideration motivates us to expect non-zero magnetic flux at infinity. However, in order to derive that magnetic flux quantitaively, we consider the undisturbed dipole magnetic field
• estimate of the azimuthal component of the magnetic field at large distances, $B_{\phi }$, is not explicitly given in the textbook. You may have a look at sketch of the field line as viewed from top (along the rotation axis) at page labelled EP19 of my hand notes. Imagine that the spiral rotates. Charged particles are supposed to be fixed at the magnetic field line, but they are allowed to move freely along it. Let's consider that the charged particle stays at fixed $\phi$, i.e., the field line slides below it and pushes it to larger radii as the spiral rotates. The ratio of $B_r$ and $B_{\phi }$ then comes from that the spiral rotates at angular velocity $\Omega$ and the particle moves radially with velocity equal to the speed of light