In § {Activity: Boundary Conditions on Magnetic Fields}, you were asked to use an infinitesimally small Gaussian surface and an infinitesimally small Amperian loop to discover how the different components of the magnetic field $\BB$ behave as they cross a surface current. You should have chosen the surface and/or loop small enough that the field is effectively constant, except, of course, that there might be abrupt discontinuous changes where the current resides.

Given a current-carrying surface, it makes sense to ask what the component $B_\perp$ of the magnetic field is perpendicular to the surface, which is $B_\perp = \BB\cdot\nn$, where $\nn$ is the unit normal to the surface. The component parallel to the surface, $B_{\parallel}$, is more subtle, since there are an infinite number of directions parallel to the surface. However, since for a current-carrying surface there is a preferred direction in the surface, namely the direction of the current $\KK$, we can distinguish between the component in the surface and parallel to the current, called $B_{\parallel\parallel}$, and the component in the surface but perpendicular to the current called $B_{\parallel\perp}$.

You should be able to show that $B_\perp$ and $B_{\parallel\parallel}$ are continuous, while the discontinuity in $B_{\parallel\perp}$ is proportional to the (local) surface current density. You should also be able to find the proportionality constant. These conditions can be combined in the equation \begin{eqnarray*} \BB_{\hbox{above}}-\BB_{\hbox{below}} = \mu_0 \left(\KK\times \nn\right) \end{eqnarray*}

In § {Activity: Ampere's Law on Cylinders}, you will have found the magnetic field due to finite cylindrical shells of current. You should check that, in the limit that these shells become infinitesimally thin without changing the total current on the shells, the discontinuity in the magnetic field is of the form you found here. Notice, however, that your result from this activity does not require any special symmetry.