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king vitamin

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## Homework Statement

Consider a uniform [tex]B_{0}[/tex] in the z direction, into which a hollow sphere is placed. If the sphere is made of a material with a high permeability ([tex]B = \mu \ H[/tex]), the magnetic field inside will be greatly reduced. Calculate the magnetic field inside the sphere and the maximum field in the metal.

(my professor talks at length about the method of solving using laplace's eqn., etc)

You will find the shielding improves with [tex]\mu[/tex], but the maximum field in the metal also increases.

## The Attempt at a Solution

I've already done all the grunt work - that is, I've calculated the field in all three regions (outside the sphere, inside the sphere, and in the spherical shell). The problem is actually similar to an example in Jackson (page 203), which gives expressions for the field inside and outside which agree with what I obtained, and conclude that the magnetic field inside the sphere decreases as [tex]1/\mu[/tex].

The problem is the second part of the problem - that the maximum magnetic field inside the metal increases with [tex]\mu[/tex]. The expression I obtain for the magnetic field inside the shell (with a being the inner radius, b the outer radius, and [tex]\mu_{r}=\frac{\mu}{\mu_{0}}[/tex]) is as follows:

[tex]\begin{equation}

\frac{3 \ \mu_{r} \ B_{0}}{(\mu_{r}+2)(2\mu_{r}+1) - 2\frac{a^{3}}{b^{3}}(\mu_{r}-1)^{2}} \ \{ ( \ (2 \mu_{r}+1)-\frac{2 a^{3} (\mu_{r}-1)}{r^{3}})cos \theta \mathbf{\hat{r}} - ( \ (2 \mu_{r} + 1) +\frac{a^3(\mu_{r}-1)}{r^{3}})sin \theta \mathbf{\hat{\theta}} \}

\end{equation}

[/tex]

To maximize this, I put in [tex]\theta = 0[/tex] and [tex]r = b[/tex], and the magnitude of the magnetic field (just the r component) is:

[tex]\begin{equation}

\frac{3 \ \mu_{r} \ B_{0} \ \{(2 \mu{r} + 1) - 2 \frac{a^{3}}{b^{3}} (\mu_{r} - 1) \} }{(\mu_{r}+2)(2\mu_{r}+1) - 2\frac{a^{3}}{b^{3}}(\mu_{r}-1)^{2}}

\end{equation}

[/tex]

But as [tex]\mu[/tex] gets large, this doesn't increase, it converges to 3B. Did I not find the maximum of the field in the metal correctly? Am I missing the question somehow? Maybe I took the gradient wrong? (I'm pretty certain I did the "potential" part of the problem correctly)

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