There is an equation: dB/dt - curl( v x B) = - curl ( L curl B) for the magnetic field B in MagnetoHydroDynamic domains. For the boundary conditions we have curl B = -mu j (j is the electric current density)
the answer depends on the Problem you are solving. But typically if it is for far field conditions you can apply some Kind of "open BC"....with d2B/dx2=0 or dB/dx=0. But once agin it depends on the Problem and on the scale.
no references no text only experience.
Well you can find plenty of references in the field of Astrophysical MHD.
Boundary conditions are obtained by integration of equations over a small domain containing boundary. You can find example of such calculation in available book on electrodynamics. In MHD some small terms are neglected.
Seems you are searching solution with resistive conductors presented somewhere. In this cases it is better to estimate initially whether you conductor thin or thick. If characteristic time of your process is much larger than characteristic skin time of conductor, you need not solve problem on current distribution inside conductor and take it uniform.
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