A steady current is flowing parallel to the axis through an infinitely long cylindrical shell of inner radius $a$ and outer radius $b$. Choose one or more of the current densities given below: (In each case, $\alpha$ and $k$ are constants with appropriate units.)

1. $\vert \vec J\vert=\alpha\, r^3$.
2. $\vert \vec J\vert=\alpha\, {\sin{kr}\over r}$.
3. $\vert \vec J\vert=\alpha\, e^{kr^2}$.
4. $\vert \vec J\vert=\alpha\, {e^{kr}\over r}$.

For each chosen density, answer each of the following questions:

1. Find the total current flowing through the wire.
2. Use Ampere's Law and symmetry arguments to find the magnetic field at each of the three radii below:
   1. $r_1>b$
   2. $a<r_2<b$
   3. $r_3<a$
3. What dimensions do $\alpha$ and $k$ have?
4. For $\alpha=1$, $k=1$, sketch the magnitude of the magnetic field as a function of $r$.