Field and Wave Electromagnetics 2nd Edition by David Cheng

Exercise 5

The method of images can also be applied to certain magnetostatic problems. Consider a straight, thin conductor in air parallel to and at a distance d above the plane interface of a magnetic material of relative permeability µ r. A current I flows in the conductor.
a) Show that all boundary conditions are satisfied if
i) the magnetic field in the air is calculated from I and an image current I i,

and these currents are equidistant from the interface and situated in air;
ii) the magnetic field below the boundary plane is calculated from I and - I;, both at the same location. These currents are situated in an infinite magnetic material of relative permeability µ r.
b) For a long conductor carrying a current l and for µ r » 1, determine the magnetic flux density B at the point P in Figure.
Figure: A current-carrying conductor near a ferromagnetic medium