Why can't I use this equation for the magnetic field?

[1v1Dota2RightMeow]

Homework Statement

Homework Equations

$\oint \vec{H} \cdot d\vec{l} = I_{free,enclosed}$
$\vec{B} = \mu_0 (1+\chi _m)\vec{H}$

The Attempt at a Solution

I found the magnetic field inside to be $\vec{B} = \mu_0 (1+\chi _m)\frac{Is}{2 \pi a^2} \phi$. But why can't I use the same equation ($\vec{B} = \mu_0 (1+\chi _m)\vec{H}$) to solve for the field outside? The answer for the field outside is given as $\vec{B} = \frac{\mu_0 I}{2 \pi s}$.

 

[TSny]

But why can't I use the same equation ($\vec{B} = \mu_0 (1+\chi _m)\vec{H}$) to solve for the field outside?

You can, but what is the value of $\chi_m$ for points outside the material?

 

[1v1Dota2RightMeow]

You can, but what is the value of $\chi_m$ for points outside the material?

Lol yea I took the time to write the question, latex it up, and post, then realized a minute later the answer to my own question...

For further learning: the fact that we have to take into account the magnetic susceptibility of a material leads me to think that the material itself is either strengthening or weakening the magnetic field created by a free current passing through the material. So where do bound currents come into play?

 

[TSny]

For further learning: the fact that we have to take into account the magnetic susceptibility of a material leads me to think that the material itself is either strengthening or weakening the magnetic field created by a free current passing through the material.

Yes, that's right. The field of the free current affects the magnetization M of the material which, in turn, affects the net magnetic field. In your example, the magnetization of the material affects only the net field inside the material. The net field outside the wire is due to the free current only. This is not typical. Usually, the magnetization of the material will affect the B field outside the material as well as inside. An extreme example is a magnetized piece of iron which has no free current but still produces plenty of B field outside the material.

So where do bound currents come into play?

There are microscopic (atomic) bound currents that can orient themselves to give a nonzero, effective macroscopic bound current density J_m. https://en.wikipedia.org/wiki/Magnetization#Magnetization_in_Maxwell.27s_equations.
The net magnetic field at any point inside or outside the material is the sum of the fields of the free and bound currents.