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# 12.3: Magnetization and Susceptibility

The $$H$$-field inside a long solenoid is $$nI$$. If there is a vacuum inside the solenoid, the B-field is $$\mu_o H = \mu_o nI$$. If we now place an iron rod of permeability $$\mu$$ inside the solenoid, this doesn't change $$H$$, which remains $$nI$$. The B-field, however, is now $$B=\mu H$$. This is greater than $$\mu_oH$$, and we can write

$B = \mu_o(H+M) \tag{12.3.1}\label{12.3.1}$

The quantity $$M$$ is called the magnetization of the material. In SI units it is expressed in A m-1. We see that there are two components to $$B$$. There is the $$\mu_o H = \mu_o nI$$, which is the externally imposed field, and the component $$\mu_oM$$, originating as a result of something that has happened within the material.

Note

It might have occurred to you that you would have preferred to define the magnetization from

$B = \mu_0H + M$

so that the magnetization would be the excess of $$B$$ over $$\mu_0H$$. The equation $$B = \mu_0H + M$$, would be analogous to the familiar

$D= \epsilon_oE + P$

and the magnetization would then be expressed in tesla rather than in A m-1. This viewpoint does indeed have much to commend it, but so does

$B = \mu_o (H+M).$

The latter is the recommended definition in the SI approach, and that is what we shall use here.

The ratio of the magnetization $$M$$ ("the result") to $$H$$ ("the cause"), which is obviously a measure of how susceptible the material is to becoming magnetized, is called the magnetic susceptibility $$\chi_m$$ of the material:

$M = \chi_m H. \tag{12.3.2}\label{12.3.2}$

On combining this with Equation $$\ref{12.3.1}$$ and $$B = mH$$, we readily see that the magnetic susceptibility (which is dimensionless) is related to the relative permeability $$\mu_r = \mu/\mu_o$$ by

$\mu_r = 1+ \chi_m \tag{12.3.3}\label{12.3.3}$