# Dependency of $\mu_r$ vs $B$ of ferromagnetic material

How $\mu_r$ will change in a specimen of ferromagnetic material as the flux density is increased from $0$ to $2.5T$ ? [Given: Saturation Magnetisation of the ferromagnetic material is $8000$ Gauss]

My Thoughts:
We know ferromagnetic material obey hysteresis curve
and $10^4$ Gauss $=1T$
$\therefore 8000$ Gauss $=0.8T$
so,

We also know: $$B= \mu_0(H+M)$$ so from the hysteresis curve,
$1)$ As $B$ is increased from $0T$ to a certain value ($<2.5T$), such that $M=0.8T$; $H$ and $M$ both will increase
Let us denote this certain value as $B_c$
$2)$ And as $B$ is further increased from $B_c$ to $2.5T$, $M$ will saturates and changes in $B$ will be reflected in $H$ ;thus only $H$ will increase now

we also know: $$M= \chi_M \times H$$ where $\chi_M=$ magentic susceptibility of the material $$\implies M= (\mu_r - 1) \times H$$ $$\implies H= \frac{M}{\mu_r -1}$$
so when $M$ saturates and $H$ is increasing to $\infty$ ; then $\mu_r$ is tending towards $1$
and we know for ferromagnetic materials: $\mu_r>>1$
Thus we conclude in case$2$ (as $B$ is further increased from $B_c$ to $2.5T$) ; $\mu_r$ decreases

Now i am not getting any conclusion about $\mu_r$ in case$1$? so any hints or suggestions please.....

• The permeability of a material changes with the amount of magnetic flux forced through it. If you have a sheet steel the curve will be higher – Bardeen Feb 27 '18 at 8:02
• You didn't get my question;i mean whether $\mu_r$ will increase throughout $0T$ to $2.5T$;or $\mu_r$ will increase first and then decrease ;or $\mu_r$ will decrease first and then increase or $\mu_r$ will only decrease – Suresh Feb 27 '18 at 10:32
• It wil increase the stay constant just like the letter S – Bardeen Feb 27 '18 at 11:14
• Increase 0 ----> 2,5 T very slow increase 2,5T ----> inf (magnetic saturation) – Bardeen Feb 27 '18 at 11:26
• In a ferromagnetic (or any magnetic) material $B$ can never saturate, instead what saturates is $M$, where $B=\mu_0 (H+M)$. While $\mu_r$ is not a very useful concept for ferromagnets, but if you must use it then $\mu_r \rightarrow 1$ as $H \rightarrow \infty$ – hyportnex Feb 27 '18 at 14:23