Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity “Electric Potential of a Parallel Plate Capacitor” before this activity.

Students should know that

objects with like charge repel and opposite charge attract,

object tend to move toward lower energy configurations

The potential energy of a charged particle is related to its charge: \(U=qV\)

The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

Students work in groups of three to use Coulomb's Law
\[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\]
to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

Consider a thin charged rod of length \(L\) standing along the \(z\)-axis
with the bottom end on the \(xy\)-plane.
The charge density \(\lambda\) is constant.
Find the electric field at the point \((0,0,2L)\).

Find the electric field around a finite, uniformly charged, straight
rod, at a point a distance \(s\) straight out from the midpoint,
starting from Coulomb's Law.

Find the electric field around an infinite, uniformly charged,
straight rod, starting from the result for a finite rod.