Power Series Sequence (E&M)

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.

1. Activity: Calculating Coefficients for a Power Series

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.

2. Activity: Approximating Functions with Power Series
Students use 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 a power series expansion in a previous activity, Calculating Coefficients for a Power Series.
3. Homework Problem: Memorize Power Series

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?

4. Homework Problem: Power Series Practice Unpublished
  1. Calculate the \(n=0, 1, 2, 3, 4\) coefficients of the power series for \(\cos{z}\) expanded around \(z=\pi\). Using these coefficients, find a power series approximation for this function.
  2. Plot both the original function and your approximation.
  3. For what values of \(z\) is your approximation good?
5. Homework Problem: Series Convergence

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.

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.

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).

6. Activity: Electrostatic Potential Due to a Pair of Charges (with 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). 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.
7. Homework Problem: Linear Quadrupole (w/ series)

Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

  1. Find the electrostatic potential at a point \(P\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole.
  2. Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.
  3. A series of charges arranged in this way is called a linear quadrupole. Why?

8. Activity: Electrostatic Potential Due to a Ring of Charge

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.

9. Activity: Electric Field Due to a Ring of Charge

Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.

10. Activity: Magnetic Vector Potential Due to a Spinning Charged Ring

Students work in groups of three to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) 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.

11. Activity: Magnetic Field Due to a Spinning Ring of Charge

Students work in groups of three to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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.

12. Homework Problem: Series Notation 1

Write out the first four nonzero terms in the series:

  1. \[\sum\limits_{n=0}^\infty \frac{1}{n!}\]

  2. \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
  3. \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}

13. Homework Problem: Series Notation 2

Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

  1. \[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]

  2. \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]