Prerequisites

Activity: Ampère's Law on Cylinders

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 $r$.