## Power Series

Most students coming into junior level courses are familiar with the elementary cases of solving for the electrostatic potential or electric field of a given charge distribution. In their sophomore, and even junior level courses, students do not move too far beyond these elementary problems; instead they are given scenarios which involve a high amount of symmetry. Paradigms does not take this approach. Rather Paradigms has students calculate electrostatic and magnetostatic fields in non-symmetric cases. Many of the calculations involve integrals for which there is no known closed form solution. This sequence is motivated by these non-elementary problems and has students use power series approximations so that students can exploit power series ideas to visualize the electrostatic potential due to a pair of charges. The techniques students learn in this sequence can then be applied to future, more complex problems, such as the potential due to a ring of charge.

This sequence is divided into two sequences: Electrostatic Potential and $\frac{1}{|\vec{r}-\vec{r}'|}$ and Power Series Approximations. Electrostatic Potential and $\frac{1}{|\vec{r}-\vec{r}'|}$ introduces students to the notation used in calculating electrostatic potential and be skipped if students know how to calculate the distance between two points using position vectors and know the electrostatic potential due to a point charge. The Power Series Approximations sequence has two activities which are meant to be used together for students to calculate coefficients of power series in Calculating Coefficients for a Power Series and then see the accuracy of the approximations with different numbers of terms in Approximating Functions with a Power Series. These activities are followed by an example of when power series are used in a problem; here, the selected activity is to calculate the electrostatic potential due to two point charges.

### Activities: Electrostatic Potential and $\frac{1}{|\vec{r}-\vec{r}'|}$

• Recall the Electrostatic Potential due to a Point Charge (Estimated time: 5-15 minutes): This small whiteboard question has students recall the basic expression for the electrostatic potential due to a point charge. Most incoming students will be familiar with this expression, however, may not understand the geometric meaning of the equation. The question is used to begin a classroom conversation regarding what is meant by $\frac{1}{r}$.
• The Distance Between Two Points - Star Trek (Estimated time: 20-30 minutes): This kinesthetic activity has students work together to resolve a given problem using geometry which opens the class to a discussion about position vectors and how to generalize the $\frac{1}{r}$ factor to $\frac{1}{|\vec{r}-\vec{r}'|}$. Many students have had prior experience with vectors, however, few have used vectors to describe the $\frac{1}{r}$ factor found in the electrostatic potential when expressed as in many introductory level courses as $V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}$. Many problems in electricity and magnetism require evaluating and approximating $\frac{1}{|\vec{r}-\vec{r}'|}$ in various geometries therefore this concept is important for students to thoroughly understand.

### Activities: Power Series Approximations

• Calculating Coefficients for a Power Series (Estimated time: 30 minutes): This small group activity has students work out the expansion coefficients of a familiar function, $\sin(\theta)$, which gives them more experience working with power series. Many students entering middle-division physics may not recall power series approximations from math courses and may never have consciously used them within a physics context. Reviewing, and introducing to some students, the process to calculate coefficients for power series provides a foundation to use power series to approximate other expressions.
• Approximating Functions with a Power Series (Estimated time: 20 minutes): As a follow-up to Calculating Coefficients for a Power Series, this computer visualization activity stresses the importance and utility of power series expansions. Here students use their favorite computational program, Maple or Mathematica, and fit power series approximations of a given function to the actual function. This allows students to witness the power of series approximations and where the approximations are valid.
• Electrostatic Potential Due to Two Point Charges (Estimated time: 50 minutes, Wrap-Up: 30 minutes): This small group activity is a culmination of the previous activities. Students use what they learned from The Distance Between Two Points - Star Trek and apply what they know about power series approximations to find a general expression and asymptotic solution to the electrostatic potential due to two point charges. Though the charge distribution students consider is elementary, the ideas and techniques used are applied later on to more difficult scenarios such as the Ring Sequence.

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