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\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#10, Rev. A, due Wednesday, 2014-04-16 }
\end{center}

\vspace{.3 in}

{\bf 1. Magnetic shielding} --- $\mu$-metal is a special Ni/Fe alloy which has
high magnetic permeability $\mu_r$ = 80,000--100,000.  It is used as shielding
to reduce the ambient magnetic field, just as a {\it Faraday cage} shields
electrostatic and RF fields.

\quad a) Given an infinitely long $\mu$-metal cylindrical shell of relative
permeability $\mu_r$, radius $R$, and thickness $d$, calculate the shielding
factor $H_\mathrm{ext}/H_\mathrm{int}$, in the limit $d\ll R$.  In other
words, if the cylinder is placed along the $z$-axis in a uniform external
field $\bs H = H_\mathrm{ext} \hat x$, how many times smaller is the resulting
uniform field $\bs H = H_\mathrm{int} \hat x$ inside the cylinder?

The simplest way to solve this 3-region boundary value problem is backwards,
from the inside out
[\href{http://www.pa.uky.edu/~crawford/phy311_sp14/priv/sumner_jpd_20_1095_20.pdf}
{J. Phys. D: Appl. Phys. 20 (1987) 1095-1101}] Start by assuming a constant
field $H_\mathrm{int} \bh x$ inside the cylinder.  What is $U(s,\phi)$ in this
region?  Apply the two continuity boundary conditions at the inside surface of
the cylinder to calculate $U(s,\phi)$ inside the $\mu$-metal itself.  Apply
the boundary conditions once more on the exterior boundary to calculate
$U(x,\phi)$ outside the cylinder.

\quad b) Calculate the thickness $d$ as a function of the external field $H_0$
needed to avoid saturation of the $\mu$-metal?  The magnetic must be less than
$B_\mathrm{sat} \approx 1~$T everywhere inside the cylinder.

\quad c) Compare the shielding factor of two concentric shields versus one
that is twice as thick.  You may assume that the sheilds are separated enough
that the total shielding factor is just the product of the individual factors.

Also, Griffiths 3ed[4ed] Ch.~5 \#%
33[34], %p246[255] dipole field \\
59[61], %p253[264] delta fn magnetic dipole field \\
Ch.~6 \#%
9[9],  %p265[277] M in a bar magnet \\
12[12], %p272[282] inf cylinder M=ks zhat \\
13[13], %p272[282] sphere,needle,disk \\
14[14], %p274[284] prob9, sketch MBH, compare PDE \\
23[25], %p283[293] electric/magnetic parallel  \\
26[27]. %p283[294] angle of bent field line \\

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