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

[FrancisD]

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

 

[mark726]

!
Thank you for your post regarding Pollack and Stump's Electromagnetism problem 9.23. I am happy to assist you in finding a solution to this problem.

To begin, let us first consider the region x < 0, which is the material with magnetic susceptibility Xm. We can use the uniqueness theorem to show that the H-field in this region is the same as the H-field produced by a current 211(2 + Xm) in the wire with the material everywhere. This is because the boundary conditions for the H-field at the interface between the vacuum and the material are the same in both cases.

Next, let us consider the region x > 0, which is the vacuum. The H-field in this region is the same as the H-field 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. This is also due to the uniqueness theorem, as the boundary conditions at the interface between the vacuum and the material are satisfied by this combination of currents.

To solve this problem, you can use the equations H = B/μ0 - M and M = χH, along with Ampere's law ∫H dl = I (free, enclosed). Using these equations and applying the uniqueness theorem, you should be able to show that the H-field in both regions is the same as described in the problem statement.

I hope this helps you in solving this problem. Please let me know if you have any further questions or need any additional clarification.