Total Magnetic Flux Density Law Problem

[Saikat]

We know Total Magnetic Flux B = B_0 + B_m

Where, B_0 is the external field and B_m is the field inside a material.

Now, we get,

B = B_0 + μ_0*M (M is the magnetization)
My question is -

Do I always have to use μ_0 ? If yes then Why?

The material isn't free space, is it?& also

B = μ_0*(H + M) , where H is the Magnetic Field Intensity/Strength

Same problem here. Do I always have to use μ_0 here too? Why not only μ ??

The main problem is we know Magnetic Field inside a material is B_m = μ_0*M

Why we are using μ_0 here while the material isn't free space!?

This problem is the main reason of those 2 questions I asked before.

Please help..

 

[Simon Bridge]

The answer to your question is "no". Sometimes the material has a susceptability different from that of a vacuum. The definitions of the terms should answer your questions for you.

 

[Dale]

B = μ_0*(H + M) , where H is the Magnetic Field Intensity/Strength

Same problem here. Do I always have to use μ_0 here too? Why not only μ ??

This equation is the definition of H in SI units. The H field is defined as the quantity that makes that equation true.

In that equation we always use $\mu_0$. The purpose of $\mu_0$ is simply to convert the SI units of M and H into the SI units of B. The purpose is not to describe the material, that is done by M.

 

[Simon Bridge]

... ah, yes: a niggly feeling had been building over that one.
Dalespam is correct

You'll see:
$\vec B = \mu_0(\vec H+\vec M)$
$\vec B = \mu \vec H$
... which can lead to confusion.