# H-Field in a region with magnetic susceptibility from an infinite line charge

## Homework Statement

Pollack, Stump - Electromagnetism. Prob. 9.23

A long straight wire carrying current 1 is parallel to the z axis and passes through
the point (a, 0, 0) . The region x > 0 is vacuum, and the region x < 0 is a material
with magnetic susceptibility Xm .
Show that the H-field in the region x < 0 is the same as H that would be produced
by a current 211(2 + Xm) in the wire with the material everywhere; and the H-field
in the region x > 0 is the same as H that would be produced by the combination of
the current 1 in the wire and a current Xm 1 I (2 + Xm) along the line parallel to the z
axis through (-a, 0, 0), with vacuum everywhere. (Hint: Appeal to the uniqueness
theorem. What are the boundary conditions?)

## Homework Equations

H = B/μ0 - M

M = χH

∫H dl = I (free, enclosed)

## The Attempt at a Solution

Ok, so my first thought was to find the H-field in both regions using ampere's law, but won't I have to find the magnetization to be able to find it in the region with the magnetic susceptibility? What is tripping me up is figuring out to solve it with 2 separate regions. Any initial help to get me started would be great. Thanks