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\section*{Electrostatic Potential for 2 Discrete Charges}
\section*{Instructor Guide}

Keywords: Upper-division, E and M, Electrostatic Potential,
Symmetry, Discrete Charges
\subsection*{Brief overview of the activity}

Students work in small groups to create power series expansions for
the electrostatic potential due to two electric charges separated by
a distance D.
\subsection*{This activity brings together student understanding of:}
\begin{enumerate}
\item Electrostatic potential
\item The physical and geometric meaning of $1\over {\rr - \rrp}$
\item Superposition
\item Power series expansion
\end{enumerate}
\subsection*{Student prerequisite skills}

Before starting this activity, through traditional lecture or the
optional activities linked below, students need to acquire
understandings of:
\begin{enumerate}
\item Electrostatic potential, $V(\rr) = {1\over 4\pi\epsilon_0}
{q\over|\rr-\rrp|}$. \texttt{Link to electrostatic potential
activity}.

\item The physical and geometric meaning of $1\over {\rr - \rrp}$.
\texttt{Link to position vector activity}.

\item Superposition, $V(\rr) = {1\over 4\pi\epsilon_0} \sum_{i=1}^N
{q_i\over|\rr-\rr_i|}$. \texttt{Link to superposition activity}.

\item Conceptual understanding of power series expansion and knowledge
of the 4-10 most common power series formulas (or students should
know where to find them in a reference book) including $(1 + z)^p =
1 + pz + {p(p-1)\over 2!}{z^2} + ...$  \texttt{Link to power series
activities}.
\end{enumerate}
\subsection*{Props}
\begin{itemize}
\item Balls to represent point charges

\item Voltmeter Coordinate system (e.g. with straws or Tinkertoys)

\item Poster-sized whiteboards

\item Markers

\item Whiteboards around room. \texttt{Link to room set-up}.
\end{itemize}

\section*{The activity - Allow 50 minutes}

\subsection*{Overview}

Students should be given or have been reminded of the formula
$V(\rr) = {1\over 4\pi\epsilon_0} {q\over|\rr-\rrp|}$ and be
assigned to work in groups of three on the Electrostatic Potential -
Discrete Charges Worksheet. This activity is designed for eight
groups, but can be used with as few as two groups. If working with
only two groups, have each group do two of the first four problems
on the worksheet. If there are more groups do more examples or have
each group just do one problem. Students do their work collectively
with markers on a poster-sized sheet of whiteboard at their tables.
\texttt{Link to worked solutions for power series expansions}.

\subsection*{What the students will be challenged by and how to facilitate their
learning}
\begin{enumerate}
\item Students are unlikely to start with the general case and work
toward the specific as in Eq.2 and Eq. 3 in the solutions. Instead
they are likely to treat this as a two-dimensional case from the
start and ignore the $z$ axis entirely and start with something like
\begin{equation}
V = {1\over 4\pi\epsilon_0} \sum_{i=1}^N {q_i\over\sqrt{(x - x_i)^2
+ (y - y_i)^2}}
\end{equation}
Although this is not problematic for obtaining a solution to this
problem, it is problematic for visualizing a 3-dimensional field
intersecting with a particular plane or axis. Frequently students
are initially trying to find a formula they can plug things into to
get an answer, or at least are trying to only see what is needed to
obtain the required solution. By the end of the wrap-up and final
whole class discussions, students should at least have considered
the 3-dimensional case and be seeing their case as an example of a
larger picture.

\item 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. The students will definitely spend
several minutes thinking about this and working through it. However,
because the coordinate system is set up for them, most students are
successful with this part fairly quickly. Some may have trouble
turning $|\rr - \rrp|$ into rectangular coordinates or have problems
with correct signs when applying the superposition principle. If
students get stuck here, help should be given fairly quickly.

\item Students will take the equation from part 1 and develop a 4th
order power series expansion. About 20 minutes will be needed for
this portion of the activity. Almost all students will struggle with
creating the power series (actually a Laurent series in some cases,
but students do not need to be familiar with this concept before the
beginning of the activity. The difference between power series and
Laurent series emerges naturally in the wrap-up). Although our
students have some experience with power series from mathematics
courses, they have never before had the chance of employing the
common physics strategy substituting into known series by rewriting
an expression in terms of dimensionless parameters. Depending on the
exact nature of prior instruction, students may encounter different
challenges.
\end{enumerate}
\begin{itemize}
\item 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.

\item If students have been exposed to Taylor's theorem $f(z) = f(a) +
f'(a)(z-a) + f"(a){{(z-a)^2}\over {2!}} + …$ and have not been told
to use a known power series expansion, they will probably first
attempt to apply this basic formula to this situation. This will
rapidly lead to an algebraic mess. In general, we let students ``get
stuck'' at this stage for about five minutes before suggesting that
they try a known power series expansion. We don't tell them which
one, but they rapidly rule out formulas for trigonometric functions
and other functions that clearly don't apply.

\item Once students are aware that $(1 + z)^p = 1 + pz + {p(p-1)\over
2!}{z^2} + ...$ is the expansion they need to be using, they still
face a substantial challenge. It is not immediately obvious to them
how an expression such as $1\over {|x-D|}$ can be transformed to the
form $(1 + z)^p$. Simply giving students the answer at this point
will defeat much of the learning possibilities of this activity.
Students will need some time just to recognize that $p = -1$, but
they will need much more time to determine if $x$ or $D$ is the
smaller amount and recognize that by factoring out D they can have
an expression that starts looking like $(1 + z)^p$, with $z =
{x\over D}$ (or $D\over x$ or…) and $p = -1$. \texttt{Link to
student language and conceptual problems regarding "factoring out"
terms}.

Students should not be allowed to stay entirely stuck for too long,
but they must be given time to struggle with the problem in order
for the learning to ``stick'' and be useful in future problems.
Students should be given substantial time (about 20 minutes) to
grapple with this portion of the problem. Some students may make
algebraic errors such as they fincorrectly factoring out $D$ or $x$.
These should be brought to their attention quickly. Students may
also have trouble dealing with the absolute value sign. Questioning
strategies should be used to ensure understanding if this is the
case.

\item Once the correct power series expansion for each term is
established, students will then need to work through the algebra to
add the two power series together. Frequently students will make
sign errors. Most student mistakes on this portion are careless
algebra errors and help can be given liberally as needed.
\end{itemize}
\subsection*{Debriefing, Whole-Class Discussion, Wrap-up and
Follow-up}

\subsubsection*{Group sharing}
Each of the eight groups should have an opportunity to present their
results to the class such that everyone can see their work. If
facilities permit, this is ideally done on large whiteboards around
the room.
\subsubsection*{Compare and contrast}
The instructor should encourage students to compare and contrast the
results for the eight situations. This should include careful
attention to: 1) whether the power series is odd or even and how
this relates to whether the situation is symmetric or anti-symmetric
2) whether the answers ``make sense'' given the physical situation
and what they tell you about moving in the + or - direction on the
given axis
\subsubsection*{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.
\subsubsection*{Laurent Series}
Assuming that students have not yet been exposed to Laurent series,
it should be brought to their attention that a ``power series'' with
$1\over x$ factors is called a Laurent series. We have found that by
introducing Laurent series this way, students see it as no big deal
and have sufficient understanding, but if introduced before this
activity they are often intimidated and confused.
\subsubsection*{Suggested homework}
Determine the general case for  $V(\rr) = {1\over 4\pi\epsilon_0}
\sum_{i=1}^N {q\over|\rr-\rrp|}$in rectangular coordinates - Answer
- $V(x,y,z) = {1\over 4\pi\epsilon_0} \sum_{i=1}^N {q_i\over\sqrt{(x
- x_i)^2 + (y - y_i)^2 + (z - z_i)^2}}$
\subsubsection*{\texttt{Link to equipotential surfaces activity.}}
\subsubsection*{\texttt{Link to "Visualizing voltage" Maple worksheet.}}

\vfill
\leftline{\it by Corinne Manogue}
\leftline{\copyright DATE Corinne A. Manogue}
\end{document}