Skip to main content

Section 18.4 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 \(b\text{.}\) Choose one or more of the current densities given below: (In each case, \(\alpha\) and \(k\) are constants with appropriate units.)

  1. \(\vert \vec J\vert=\alpha\, s^3\text{.}\)

  2. \(\vert \vec J\vert=\alpha\, {\sin{ks}\over s}\text{.}\)

  3. \(\vert \vec J\vert=\alpha\, e^{ks^2}\text{.}\)

  4. \(\vert \vec J\vert=\alpha\, {e^{ks}\over s}\text{.}\)

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

  1. Find the total current flowing through the wire.

  2. Use Ampère's Law and symmetry arguments to find the magnetic field at each of the three radii below:

    1. \(\displaystyle s_1>b\)

    2. \(\displaystyle a\lt s_2\lt b\)

    3. \(\displaystyle s_3\lt a\)

  3. What dimensions do \(\alpha\) and \(k\) have?

  4. For \(\alpha=1\text{,}\) \(k=1\text{,}\) sketch the magnitude of the magnetic field as a function of \(s\text{.}\)