Boundary conditions on electric fields

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Section 19.2 Boundary conditions on electric fields

How does the electric field behave near a charged surface? There is no obvious reason for the electric field to be the same on both sides of the surface.

Activity 19.2.1.

Using an infinitesimally small Gaussian surface and an infinitesimally small Ampèrian loop, you should explore how the electric field changes across a surface charge density. Do not assume that the surface is flat or has any special symmetry. It will help to look separately at the component of the electric field parallel to the surface \(\EE_\parallel\) and the component perpendicular to the surface \(\EE_\perp\text{.}\)

You were asked to use an infinitesimally small Gaussian surface and an infinitesimally small Amperian loop to discover how the different components of the electric field \(\EE\) behave as they cross a surface charge density. 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 charge resides. You should be able to show that \(\EE_\parallel\) is continuous, while the discontinuity in \(\EE_\perp\) is proportional to the surface charge density (and is therefore zero for continuous media), i.e.

\begin{gather*} \EE_{\textrm{above}}-\EE_{\textrm{below}} = \frac{\sigma}{\epsilon_0} \nn . \end{gather*}

In Section 13.5, you will have found the electric field due to finite cylindrical and spherical shells of charge. You should check, that in the limit that these shells become infinitesimally thin, without changing the total charge on the shells, that the discontinuity in the electric field is of the form you found here. Notice, however, that your result from this activity, does not require the special symmetries of the previous activity.