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\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#9, Rev. A, due Wednesday, 2014-04-09 }
\end{center}

\vspace{.1 in}

{\bf 1. Magnetic scalar potential}

\quad a) Calculate the divergence and curl of both sides of the Biot-Savart
law to obtain Amp\`ere's law $\nabla \times \bs H = \bs J$, and $\nabla \cdot
\bs B = 0$, where $\bs B = \mu_0 \bs H$.  Note the mixed symmetry: $\bs D$,
and $\bs B$ are fluxes and $\bs E$ and $\bs H$ are flows, while $\bs E$ and
$\bs B$ are the force fields (derive from potential energy/momentum) and $\bs
D$ and $\bs H$ are auxiliary `source fields', with $\rho$ and $\bs J$ as their
sources.  So the correspondence between $\bs E,\bs D$ and $\bs B, \bs H$
depends on your point of view.

\quad b) Use a Gaussian surface and an Amp\`erian loop to derive the
corresponding boundary conditions for $\bs B$ and $\bs H$, respectively, and
compare them to the electrostatic case.

\quad c) Show that a magnetic scalar potential $U$ is only defined
current-free regions ($\bs J=\bs 0$) by substituting the definition $\bs H=-
\nabla U$ into Amp\`ere's law.  Note that $U$ is related to the flow of $\bs
H$, just as the electrostatic potential $V$ is related to the flow of $\bs E$.
What is analogous equation of $-\nabla^2 V=\rho/\epsilon$ for $U$?

\quad d) There is a more subtle condition for $U$ being well defined in a
region $\mathcal R$ with holes, for example, a torus.  Re-express Amp\`ere's
law in terms of the scalar potential $U$.  What constraint is required for $U$
to be conservative (no discontinuities) in such a region $\mathcal R$?

\quad e) Express the boundary conditions of b) in terms of $U$, as we did for
the \mbox{electric potential $V$.}

\quad f) Sketch the equipotential surfaces of an infinitely long solenoid.  Is
it possible to match up $U$ continuously along the entire boundary, so that
$U_\mathrm{in} (\bs r) = U_\mathrm{out} (\bs r)$?  If not, \mbox{what is the
discontinuity?}

{\bf 2.} A {\bf cos-theta coil} is a wound longitudinally along a
cylinder instead of azimuthally like a solenoid.  We will calculate the
windings of a \mbox{uniform transverse field using the scalar potential
$U$.}

\quad a) Given a cylinder of radius $s=a$ about the $z$-axis from $z=-b$ to
$z=b$, and a constant magnetic field $\bs H=H_0 \hat{\bs x}$ inside, calculate
the normal component $H_n$ everywhere on the cylinder.

\quad b) Use the flux boundary condition a) and Laplace's equation of
Problem~1 to solve for the potential $U$ outside of an infinitely long
cylinder, $b=\infty$.

\quad c) Use the flow boundary condition (from Amp\`ere's law) to calculate
the current density on the cylinder necessary to produce these fields.  What
is the direction and magnitude of the current density $\bs K_a(\phi)$ on the
cylinder?

\quad d) Superimpose two coaxial cylinders, one of radius $a$ inside and
another of radius $A$ with current in the opposite direction.  Can the
currents be adjusted such that $\bs H=0$ outside the larger cylinder? (a
rhetorical question!)  Calculate the current densities $\bs K_a(\phi)$ and
$\bs K_A(\phi)$ on each cylinder such that $\bs H_\mathrm{out}=\bs 0$ and $\bs
H_\mathrm{in}=H_0 \,\hat{\bs x}$ on the outside and inside the two cylinders,
respectively.

\quad e) Solve parts (b)--(d) as a boundary value problem for the case of two
cylinders of finite length $-b<z<b$. Calculate and sketch the field lines of
$\bs H_\mathrm{mid}$ inside the two cylinders.  Calculate and sketch the
current on the end caps.

\quad f) Calculate the magnetic flux $\Phi_B$ flowing in and out of the inner
cylinder.  We will use this result later to calculate the inductance of such a
coil, a rare case which is exactly calculable.

Also, Griffiths 3ed[4ed] Ch.~5, \#%
13[14], %p231[239] thick wire \\
14[15], %p231[240] thick slab \\
15[16], %p231[240] two coaxial solenoids \\
18[19], %p232[240] which surface for current flux? \\
20[21], %p234[243] Ampere's law vs del . del x B = 0? \\
23[24], %p239[248] A=k phihat \\
39[41], %p247[256] Hall effect V=IR ! \\
46[47]. %p249[259] Helmholtz coil \\

\end{document}
