Many self-gravitating cosmic bodies are both spinning and magnetized. Examples are Earth, the Sun, black holes surrounded by highly conducting accretion disks (which hold a magnetic field on the hole), neutron stars (pulsars), and magnetic white dwarfs. As a consequence of the magnetic field’s spin-induced motion, large electric fields are produced outside the rotating body. The divergence of these electric fields must be balanced by free electric charge, which implies that the region around the body cannot be a vacuum. It is usually filled with plasma and is called a magnetosphere. MHD provides a convenient formalism for describing the structure of this magnetosphere. Magnetospheres are found around most planets and stars. Magnetospheres surrounding neutron stars and black holes are believed to be responsible for the emissions from pulsars and quasars. 

As a model of a rotating magnetosphere, consider a magnetized and infinitely conducting star, spinning with angular frequency Ω_* Suppose that the magnetic field is stationary and axisymmetric with respect to the spin axis and that the magnetosphere, like the star, is perfectly conducting.

(a) Show that the azimuthal component E_∅ of the magnetospheric electric field must vanish if the magnetic field is to be stationary. Hence show that there exists a function Ω(r) that must be parallel to Ω_* and must satisfy

Show that if the motion of the magnetosphere’s conducting fluid is simply a rotation, then its angular velocity must be Ω.
(b) Use the induction equation (magnetic-field transport law) to show that

(c) Use the boundary condition at the surface of the star to show that the magnetosphere corotates with the star (i.e., Ω = Ω_∗). This is known as Ferraro’s law of isorotation.

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E = (2x r) x B. (19.36)