Consider a rectangular loop of wire of width a and height b that is at a distance d from a very long wire carrying a steady current I.

a) Calculate the external inductance of the rectangular loop due to the wire. Does the answer change if the rectangular loop has current flowing through it? What happens if the rectangular loop is made up of, say, N coils. Hint: What's the magnetic field of an infinite long wire?

b) The loop and the wire are now immersed halfway in a medium of relative permeability r normal to the surface of the medium. Find the external inductance of the rectangular loop due to the wire in this new setting.