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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

\[\mu_r = 1+ \chi_m \tag{12.3.3}\label{12.3.3}\]

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