Activity: Electrostatic Potential Due to a Pair of Charges (without Series)

Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). Students then evaluate the limiting cases of the potential on the axes of symmetry.
  • group Small Group Activity schedule 30 min. build Medium whiteboards/markers/erasers, balls to represent the charges, voltmeter, coordinate axes on the ceiling, handout description Student handout (PDF)
What students learn
  • The superposition principle for the electrostatic potential;
  • How to calculate the distance formula \(\frac{1}{|\vec{r} - \vec{r}'|}\) for a simple specific geometric situation;
Electrostatic Potential from Two Charges
  • Find a formula for the electrostatic potential \(V(\vec{r})\) that is valid everywhere in space for the following two physical situations:
    • Two charges \(+Q\) and \(+Q\) placed on a line at \(z'=D\) and \(z''=-D\).
    • Two charges \(+Q\) and \(-Q\) placed on a line at \(z'=D\) and \(z''=-D\).
  • Simplify your formulas in the limiting cases of:
    • the \(x\)-axis
    • the \(z\)-axis
  • Discuss the relationship between the symmetries of the physical situations and the symmetries of the functions in these limiting cases.

Instructor's Guide

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.

Learning Outcomes