# Constant magnetic field attenuation by µ-metal (mu-metal)

I am interested in the magnetic field attenuation by µ-metal. Specifically shielding from earth's magnetic field $\boldsymbol{H}_E$ through closed cylindrical layers of such a metal, yielding an internal attenuated field $\boldsymbol{H}_A$.

One can buy magnetic shielding from different sources and they all offer seemingly empirical equations$^1$ to calculate the shielding factor $S$ which carries the unit dB (decibel). $$S = 20\ \log_{10}\left(\frac{\boldsymbol{H}_A}{\boldsymbol{H}_E}\right)$$ If one has a cylindrical (µ-metal) encased space, the attenuation of a DC magnetic field perpendicular to the cylinder axis is for example given through $S_i$: $$S_i = \frac{\mu}{4} - \left(1-\frac{d^2_{\mathrm{inner}}}{d^2_{\mathrm{outer}}}\right) +1\ ,$$ where $\mu$ is the magnetic permeability and $\left(d_{\mathrm{outer}}-d_{\mathrm{inner}}\right)/2$ is equal to the wall thickness of the cylinder.

Furthermore they describe the combined shielding factor of a (long) cylindrical two layer shield through $S_{\mathrm{total}}$: $$S_{\mathrm{total}} = S_1S_2\left(1-\frac{d^3_2}{d^3_1}\right)+S_1+S_2\ ,$$ where $d_1$ and $d_2$ refers to the average (I'm not quite sure about this) diameter of the first and second layer, respectively.

I have searched around and couldn't really find a source or derivation of these equations. All I can find are related to AC fields. Could somebody explain the origin of the last two equations, perhaps there is a paper/publication I've overlooked?

Thanks