Rewriting the Ampere's law in term of free currents only

[Riemann9471]

Homework Statement

Prove that Ampere's law ( ∮ B dl = mu_0 x enclosed current ) can be rewritten in terms of free currents only like this : ∮ (1/mu_r) B dl = mu_0 x free enclosed current where mu_r is the relative permeability of the material.

Relevant Equations

Ampere's law : ∮ B dl = mu_0 x enclosed current
Ampere's law rewritten : ∮ (1/mu_r) B dl = mu_0 x free enclosed current

This is not really the assignment of my homework ( my assignment require me to find the magnetic field inside a small air gap on a toroide magnet wrapped with N turns of a wire that carry a current I ) . I'm at some point in the solution where I kind of need to use the rewritten Ampere's law to continue but I can't use it unless I prove it because it has not been proven in class. I couldn't find the proof anywhere so I was hoping to find a little help here !

 

[hutchphd]

You can work through this:

https://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/lecture-notes/lecture29.pdf

 

[rude man]

Not sure what the problem is here, but

Ampere's law is actually $\oint \vec H \cdot d \vec l = I$
and B relates to H by $B = \mu H = \mu_0 \mu_r H$
so there you go. I think.