Introduction

It may help to do the The Distance Formula (Star Trek) activity before this one. There is an alternative version of this activity Electrostatic Potential Due to a Pair of Charges (with Series) in which students find series expansions of the potential along the axes of symmetry.

Students typically know the iconic formula for the electrostatic potential of a point charge \(V=\frac{kq}{r}\). We begin this activity with a short lecture/discussion that generalizes this formula in a coordinate independent way to the situation where the source is moved away from the origin to the point \(\vec{r}^{\prime}\), \(V(\vec{r})=\frac{kq}{|\vec{r} - \vec{r}^{\prime}|}\). (A nice warm-up (SWBQ) to lead off the discussion:

Introductory SWBQ Prompt: “Write down the electrostatic potential everywhere in space due to a point charge that is not at the origin.” The lecture should also review the superposition principle. \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0}\sum_{i}\frac{q_i}{|\vec{r} - \vec{r_i}|} \]

This general, coordinate-independent formula should be left on the board for students to consult as they do this activity.

Student Conversations

• Students create an expression such as \[V(x,y,z) = \frac{Q}{ 4\pi\epsilon_0} {\left(\frac{1}{|z - D|} + \frac{1}{|z + D|}\right)}\] Each axis and charge distribution has a slightly different formula. A few groups may have trouble coordinatizing \(|\vec{r} - \vec{r}^{\prime}|\) into an expression in rectangular coordinates, but because the coordinate system is set up for them, most students are successful with this part fairly quickly.
• Many students are likely to treat this as a two-dimensional case from the start, ignoring the \(y\) axis entirely. Look for expressions like \[V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^N {q_i\over\sqrt{x ^2 + (z - z_i)^2}}. \] Prompt them to consider the larger, 3-dimensional picture.
• Most students will leave off the absolute value signs when evaluating the potential on the \(z\)-axis. If they do this, their formulas will not be correct for negative values of \(z\). The subtlety here is that \[\sqrt{a^2}=\vert a\vert\] not \[\sqrt{a^2}=a\] in contexts like this when \(\sqrt{a^2}\) is a distance and therefore necessarily positive and when \(a\) itself might be either positive or negative.