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## Potential Due to a Pair of Charges: Instructor's Guide

- Electrostatic potential due to a point charge
- Distance between source and field point
- Superposition principle

### Main Ideas

### Students' Task

*Estimated Time: 10 min; Wrap-up: 5 min*

Students work in small groups to find the electrostatic potential due to two electric charges separated by a distance $D$. Different groups are assigned different arrangements of charges and different regions of space to consider.

### Prerequisite Knowledge

- Understanding of the geometric and analytic meaning of $|\Vec r - \Vec r' |$. (This may be more far more difficult for your students than you expect. You might want to try our displacement vector activity .)

### Props/Equipment

- Balls to represent point charges
- Table top whiteboards and markers
- A handout for each group

### Activity: Introduction

Students typically know the electrostatic formula $V=\frac{kq}{r}$. 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 $r'$, $V=\frac{kq}{|\Vec r - \Vec r'|}$. The lecture should also review the superposition principle. The general, coordinate-independent formula should be left on the board for them to consult as they do this activity.

A nice warm-up (SWBQ) to lead off the discussion: Write down the electrostatic potential everywhere in space due to a point charge that is not at the origin.

### Activity: Student Conversations

- Note: two of the eight cases on the worksheet are trivial (the potential on the $y$-axis is zero for the $+Q$ and $-Q$ situation). Once these groups have established the correct answer and can justify it, they should be directed to work on one of the other six questions.
- This could be a good time to clarify the notation $V(\rr)$.
- Students may use the iconic equation to give an answer along the lines of $V=k(\frac{q_1}{r_1}+\frac{q_2}{r_2})$. But this doesn't provide enough information to describe $r_1$ and $r_2$, a good example of the need to “unpack” the iconic equation.
- As an intermediate step, students will create an expression such as $V(x,y,z) = {Q\over 4\pi\epsilon_0} {\left({1\over{|D - x|}} + { 1\over{|D + x|}}\right)}$; each situation has a slightly different formula. Some students may have trouble turning $|\Vec r - \Vec r'|$ into rectangular coordinates, but because the coordinate system is set up for them, most students are successful with this part fairly quickly.
- Students may use the iconic equation $V = \frac{kq}{r}$ and tried to force $V(r)$ as the $f(z)$ from the power series expression $f(z) = \sum_{n=0}^{\infty} C_n (z-a)^n$. Students then plug and chug and tried to find the coefficient of the fourth order term because the students know that $c_n = \frac{1}{n!} \frac{d^{(n)} f}{dz^{(n)}} (z)\bigg \rvert_{z=a}$. This can be avoided by reminding the students that there are two charges and they need to use superposition principle.
- Once students arrived at $V(x) = {Q\over 4\pi\epsilon_0} {\left({1\over{|D - x|}} - { 1\over{|D + x|}}\right)}$, students want to have a common denominator and then do the power series approximation to that combine expression, i.e. $\frac{1}{D-x}-\frac{1}{D+x} = \frac{2x}{D^2 -x^2}$. Let them know doing the approximation to each term is easier than combining the two terms into one.

### Activity: Wrap-up

**Consideration of the 3-dimensional case** Most students will have thought about this problem entirely within two dimensions. They should be asked to consider points with a non-zero z component. Envisioning the three-dimensional potential field will help students towards the types of thinking they will need to apply to future problems.

### Extensions

This activity is part one of a longer activity,

in which students do a series expansion of the expressions that they develop here.

This activity works particularly well when sequenced with other activities. This is the first activity of the Ring Sequence (calculating fields due to charge distributions).