# Electromagnetism : solenoid and magnetic material inside

I have an infinite solenoid with a current I, and $n$ turns per unit length.

I know that the magnetic field inside will be :

$\vec{B}=\mu_0 n I \vec{U_z}$.

Imagine that I put a magnetic material inside.

I thus have : $\vec{B}=\mu_0(\vec{H}+\vec{M})$

Will I have : $\vec{H}=n I \vec{U_z}$ or $\vec{B}=\mu_0 n I \vec{U_z}$.

Another way to express the question. The first three steps are always true :

• 1: The solenoïd creates a field
• 2: This field goes in the magnetic material, the latter produces a "response" $\vec{M}$
• 3: This response changes the global magnetic field.

But is the last step this one :

• 4: As the global magnetic field is changed, the current also because $\vec{B}=\mu_0 n I \vec{U_z}$ is always true here.

OR this one

• 4: The global magnetic field is changed, but as the current injected has'nt changed, I will not have $\vec{B}=\mu_0 n I \vec{U_z}$ but $\vec{H}=n I \vec{U_z}$ will still be valid.

 : In fact I think I understand the fact that the "good" relations will always be $\vec{H}=n I \vec{U_z}$ because by definition $H$ represents the magnetic field induced by "free" currents (I don't know if it is the good word in english). And the only free current here is the one inside the solenoid.

But how do we know that the material will not change this current. Indeed as it modifies the $B$ field, it could change by induction the current in the solenoid ?

Thank you.

The derivation of the magnetic field is from Ampere's law: $$\oint B\cdot dl = \mu_0 I$$ But when inserting a magnetic material, this is including the "bounded current". The magnetic field of the wire will make the little magnetic dipoles point to same direction, enhancing (or weakening) the net magnetic field.
What stays the same is $$\oint H\cdot dl = I_{free}$$ so $H$ is the one here which remains the same and $B$ is changing.