Using Ampere's Law and Gauss's Law to find the electric field just above and just below an arbitrary plane with surface charge density $\sigma$.

Estimated Time: 10 - 20

• A knowledge of Ampere's and Gauss's Law

• Tabletop Whiteboard with markers
• A handout for each student

Students are given a generic plane with some constant surface charge density $\sigma $. They will be charged with finding and comparing the electric field components as they cross the surface boundary. The tools used for this particular job are the integral forms of the Maxwell's equations.

• Find the continuity (or discontinuity) conditions in the electric field parallel to the surface ($E_{\parallel}$).

• Find the continuity (or discontinuity) conditions in the electric field perpendicular to the surface ($E_{\perp}$).

• Visualizing the field – Students have trouble visualizing a “general E field” near a sheet of charge (they want it to point outwards).
• Name the thing you don't know – Students at this stage don't know that they need to give a name (i.e. assign a variable) to the thing they are trying to find. They do this in very simple 1-step, intro problems, but not in more complex problems
  • Students often can't bring themselves to start constructing Gaussian surfaces and evaluating Gauss' law on that surface with no formula (name) for the E field to start with, so it is important to emphasize that they need to “name the thing they don't know” as they flail with the idea of $E_{above}$ and $E_{below}$.
  • They will have further problems because they also won't think about the fact that each has different components. You may need to help them name $E_{above,\parallel}$ and $E_{above,\perp}$, etc.
  • Some of them will know enough to choose a rectangular coordinate system adapted to their surface.
• Gauss' Law AND Ampere's Law – This requires both not only a Gaussian surface to determine the discontinuity of $E_{\perp}$, but also “an Amperian-like loop” to show the continuity of $E_{\parallel}$. It may not be clear to some students that an Amperian-like loop can be used with electric field in $\oint{\vec{E}\cdot d\vec{l}}=0$ to determine the continuity of the tangential component of the electric field.
• Importance of both shape and size – Students do not always realize that in order to make some of the sides “go away” one has to make a careful choice about both size and shape of the Gaussian box or Amperian loop. It can be helpful to point out examples they know (infinite sheet, or cylinder) and invoke superposition to talk about how you cannot assume E points in any particular direction, only that you can conclude $E_{\perp}$ jumps across the sheet.

This activity is part of a sequence addressing Boundary Conditions in electrostatics and magnetostatics. The other activity which is included in this sequence follows.

• Magnetic Field Continuity Across a Boundary: Students use Ampere's and Gauss's laws to find the continuity conditions for the magnetic field's parallel and perpendicular components across the planar boundary carrying current density, $\vec{K}$.