# Why use vacuum permeability during derivation of $\vec{B}=\mu_0(\vec{H} + \vec{M})$

Why do we use $$\mu_0$$ during the derivation of $$\vec{B}=\mu_0(\vec{H} + \vec{M})$$ where $$\vec{M}$$ is magnetization?

The derivation given in Sadiku's Elements of Electromagnetics:
Let $$\vec{J_f}$$ be free volume current density, $$\vec{J_b}$$ be bound volume current density, \begin{align*} \nabla \, \times \left( \frac{\vec{B}}{\mu_0} \right) &= \vec{J_f} + \vec{J_b} = \vec{J} \\ &= \nabla \times \vec{H} \, + \nabla \times \vec{M} \\ &= \nabla \times (\vec{H} + \vec{M}) \\ \vec{B} &=\mu_0(\vec{H} + \vec{M}) \quad \blacksquare \end{align*}

I don't understand why we should use $$\mu_0$$ in the first place. Why don't we use $$\mu$$ instead? In free space, $$\vec{M} = 0$$ and \begin{align*} \nabla \times \vec{H} &= \vec{J_f} \\ \nabla \times \left( \frac{\vec{B}}{\mu_0} \right) &= \vec{J_f} \end{align*} then naturally we'd like to still have $$\nabla \times \vec{H} = \vec{J} = \vec{J_f} + \vec{J_b}$$ when $$\vec{M} \neq 0$$, so we could just change the $$\mu_0$$ to some constant $$\mu$$, so that \begin{align*} \nabla \times \vec{H} &= \vec{J} \\ \nabla \times \left( \frac{\vec{B}}{\mu} \right) &= \vec{J} = \vec{J_f} + \vec{J_b} \end{align*} but in the correct derivation, \begin{align*} \nabla \times \left( \frac{\vec{B}}{\mu_0} \right) &= \vec{J} = \vec{J_f} + \vec{J_b} \end{align*}

What is it that forces us to use $$\mu_0$$?

• My another question helped me to demystify this question. Basically, I was confused by the book defining $\oint {\bf H} \cdot d {\bf \ell} = I_{enc}$ while in fact it's not clearly explained in the text that $I_{enc} = I_{free}$. So after applying Stoke's theorem, I mistakenly treated $\nabla \times {\bf H} = {\bf J_f} + {\bf J_b}$ while it should be $\nabla \times {\bf H} = {\bf J_f}$ only, it's just definition and no point to add ${\bf J_b}$ to it. – endeneer May 19 '19 at 6:18

$${\bf B}=\mu_0({\bf H}+{\bf M})$$ is effectively just the definition of the auxiliary field $${\bf H}$$. $${\bf B}$$ is defined as the quantity that generates the velocity-dependent force in the Lorentz Force Law, and the magnetization $${\bf M}$$ is the magnetic moment per unit volume. $${\bf H}$$ is then defined in terms of the other two.
For the currents, the bound current $${\bf J}_{b}$$ is defined as the curl of $${\bf M}$$, and then the free current is whatever is left over, $${\bf J}_{f}={\bf J}-{\bf J}_{b}$$. There is thus no room to change the constant $$\mu_{0}$$ to something else.