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Consider an infinitely long wire carrying a continous current $\mathbf{I}$. It is surrounded at some distance by a hollow cylinder of infinite permeability $\mu$ and zero conductivity (i.e. it can't carry any currents).

An inifitely long wire surrounded by a hollow cylinder of infinite permeability

In the empty regions surrounding the wire Ampere's Law in integral form gives

$$\oint_{\partial \Sigma} \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \int_{\Sigma} \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mathbf{I} \quad \Rightarrow \quad \mathbf{H} = \frac{I}{2 \pi r}\mathrm{i}_\varphi\,, \quad \mathbf{B} = \mu_0\mathbf{H} = \frac{\mu_0I}{2 \pi r}\mathrm{i}_\varphi$$

How do the magnetic flux density $\mathbf{B}$ and magnetic field strength $\mathbf{H}$ look like in the infinitely permeable region?


Discussion/Thoughts: Usually $\mathbf{H}$ goes to zero in infinitely permeable materials but if you apply Ampere's Law to a circle going entirely through the cylinder you'd get $0=\mathbf{I}$ which is clearly a contradiction of the law. Because the cylinder can't carry any currents there are no surface currents on the inside and outside of the cylinder to cancel out the field inside the material.

If $\mathbf{H}$ in the cylinder is not zero the question is what happens to the flux density $\mathbf{B}$. In linear, isotropic and homogeneous materials we have the relation $\mathbf{B} = \mu\mathbf{H}$. If $\mu$ is infinite that would imply the flux density would be infinite too (unless $\mathbf{H}$ is $0 in which case it would be undefined)?

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From the Ampere law $H$ field in the cylinder is finite and its value is independent of the cylinder permeability.

The more magnetizable the body is, the more it gets magnetized by external field and the higher the B field inside gets.

In the limit $\mu \to \infty$, magnitude of magnetic induction $B \to \infty$. An extreme assumption leads to an extreme conclusion.

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Usually, we say that the magnetic excitation tends towards 0 when the permeability tends towards infinity. The magnetic field remaining finite. But this is not true when the domain of high permeability is not simply connected and embraces a non-zero current. Here, we therefore have $\vec{H}=\frac{I}{2\pi r}\vec{e_\varphi}$ and the magnetic field tends to infinity when the permeability becomes infinite.

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