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). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
1. << Series Notation 2 | Power Series Sequence (E&M) |
2. << Magnetic Field Due to a Spinning Ring of Charge | E&M Ring Cycle Sequence |
Find a third order approximation to the electrostatic potential \(V(\vec{r})\) for one of the following physical situations.
- Two charges \(+Q\) and \(+Q\) are placed on a line at \(z'=D\) and \(z''=-D\) respectively.
- On the \(x\)-axis for \(|x| \ll D\)?
- On the \(z\)-axis for \(|z| \ll D\)?
- On the \(x\)-axis for \(|x| \gg D\)?
- On the \(z\)-axis for \(|z| \gg D\)?
- Two charges \(+Q\) and \(-Q\) are placed on a line at \(z'=+D\) and \(z''=-D\) respectively.
- On the \(x\)-axis for \(|x| \ll D\)?
- On the \(z\)-axis for \(|z| \ll D\)?
- On the \(x\)-axis for \(|x| \gg D\)?
- On the \(z\)-axis for and \(|z| \gg D\)?
Work out your problem by brainstorming together on a big whiteboard and also answer the following questions:
- For what values of \(\vec{r}\) does your series converge?
- For what values of \(\vec{r}\) is your approximation a good one?
- Which direction would a test charge move under the influence of this electric potential?
If your group gets done early, go on to another problem. The fourth problem in each set is the most challenging, and the most interesting.
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. \[ V(\vec{r})=\sum_{i}\frac{kq_i}{|\vec{r} - \vec{r_i}|} \]
This general, coordinate-independent formula should be left on the board for students to consult as they do this activity.
If time allows, each of the eight groups should have an opportunity to present their results to the class. The instructor should encourage students to compare and contrast the results for the eight situations. This should include careful attention to:
End with a discussion that extracts from the different examples the overall method: