Analytical models of finite thin disks in a magnetic field

Analytical models of axially symmetric thin disks of finite extension in the presence of magnetic field are presented based on the well-known Morgan-Morgan solutions. The source of the magnetic field is constructed separating the equation corresponding to the Ampere's law of electrodynamics in spheroidal oblate coordinates. This produces two associated Legendre equations of first order for the magnetic potential and hence that can be expressed as a series of associated Legendre functions of the same order. The discontinuity of its normal derivate across the disk allows us to interpret the source of the magnetic field as a ringlike current distribution extended on all the plane of the disk. We also study the circular speed curves or rotation curve for equatorial circular orbits of charged test particles both inside and outside the disk. The stability of the orbits is analyzed for radial perturbation using an extension of the Rayleigh criterion.