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

Static Fields 2023 (5 years)
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). This activity can be paired with activity 29 to find 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 description Student handout (PDF)
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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 $z''=-D$, respectively.
• 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.

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