Editing Electromagnetic shielding (section)

==Mathematical model==

Suppose that we have a spherical shell of a (linear and isotropic) diamagnetic material with [[relative magnetic permeability|relative permeability]] {{nowrap|<math>\mu_\text{r}</math>,}} with inner radius <math>a</math> and outer radius {{nowrap|<math>b</math>.}} We then put this object in a constant magnetic field:
<math display="block">\mathbf{H}_0 = H_0 \hat\mathbf{z} = H_0 \cos(\theta) \hat\mathbf{r} - H_0 \sin(\theta) \hat\boldsymbol{\theta}</math>
Since there are no currents in this problem except for possible bound currents on the boundaries of the diamagnetic material, then we can define a magnetic scalar potential that satisfies [[Laplace's equation]]:
<math display="block">\begin{align} \mathbf{H} &= -\nabla \Phi_{M} \\ \nabla^{2} \Phi_{M} &= 0 \end{align}</math>
where
<math display="block">\mathbf{B} = \mu_\text{r}\mathbf{H}</math>
In this particular problem there is azimuthal symmetry so we can write down that the solution to Laplace's equation in spherical coordinates is:
<math display="block">\Phi_{M} = \sum_{\ell=0}^\infty \left(A_{\ell}r^{\ell}+\frac{B_{\ell}}{r^{\ell+1}}\right) P_{\ell}(\cos\theta)</math>
After matching the boundary conditions
<math display="block">\begin{align}\left(\mathbf{H}_2 - \mathbf{H}_1\right)\times\hat\mathbf{n}&=0\\\left(\mathbf{B}_2 - \mathbf{B}_1\right) \cdot \hat\mathbf{n} &=0 \end{align}</math>
at the boundaries (where <math>\hat{n}</math> is a [[unit vector]] that is normal to the surface pointing from side 1 to side 2), then we find that the magnetic field inside the cavity in the spherical shell is:
<math display="block">\mathbf{H}_\text{in}=\eta\mathbf{H}_{0}</math>
where <math>\eta</math> is an [[attenuation coefficient]] that depends on the thickness of the diamagnetic material and the magnetic permeability of the material:
<math display="block">\eta = \frac{9\mu_\text{r}}{\left(2\mu_\text{r} + 1\right) \left(\mu_\text{r} + 2\right) - 2\left(\frac{a}{b}\right)^3 \left(\mu_\text{r} - 1\right)^2}</math>
This coefficient describes the effectiveness of this material in shielding the external magnetic field from the cavity that it surrounds. Notice that this coefficient appropriately goes to 1 (no shielding) in the limit that <math>\mu_\text{r} \to 1</math>. In the limit that <math>\mu_\text{r} \to \infty</math> this coefficient goes to 0 (perfect shielding). When <math>\mu_\text{r} \gg 1</math>, then the attenuation coefficient takes on the simpler form:
<math display="block">\eta = \frac{9}{2 \left(1 - \frac{a^3}{b^3}\right) \mu_\text{r}}</math> 
which shows that the magnetic field decreases like {{nowrap|<math>\mu_\text{r}^{-1}</math>.}}<ref>{{cite book |first=John David |last=Jackson |title=Classical Electrodynamics |edition=third |at=Section 5.12 |date=10 August 1998 |isbn=978-0471309321 <!-- hardcover -->}}</ref>