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.
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Electrostatic Potential from Two ChargesFind 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:
group Small Group Activity
30 min.
format_list_numbered Sequence
The first three activities provide an active-engagement version of the canonical mathematical and geometric fundamentals for power series. The subsequent activities apply these ideas to physical situations that are appropriate for an upper-division electromagnetism course, using concepts, terminology, and techniques that are common among physicists, but not often taught in mathematics courses. In particular students use the memorized formula for the binomial expansion to evaluate various electrostatic and magnetostatic field in regions of high symmetry. By factoring out a physical quantity which is large compared to another physical quantity, they manipulate the formulas for these fields into a form where memorized formulas apply. The results for the different regions of high symmetry are compared and contrasted. A few homework problems that emphasize the meaning of series notation are included.
Note: The first two activities are also included in Power Series Sequence (Mechanics) and can be skipped in E&M if already taught in Mechanics.
assignment_ind Small White Board Question
10 min.
group Small Group Activity
30 min.
Taylor series power series approximation
This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:
\[c_n={1\over n!}\, f^{(n)}(z_0)\]
After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a \(\sin\theta\) (around both the origin and \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.
assignment Homework
Recall that, if you take an infinite number of terms, the series for \(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent representations of the same thing for all real numbers \(z\), (in fact, for all complex numbers \(z\)). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of \(z\). The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”
Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!
Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”
computer Computer Simulation
30 min.
Taylor series power series approximation
Students use prepared Sage code or a prepared Mathematica notebook to plot \(\sin\theta\) simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.group Small Group Activity
30 min.
electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula
Students work in groups of three to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
assignment Homework
Look up and memorize the power series to fourth order for \(e^z\), \(\sin z\), \(\cos z\), \((1+z)^p\) and \(\ln(1+z)\). For what values of \(z\) do these series converge?
assignment Homework
assignment Homework