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===== Potential Due to a Pair of Charges: Instructor's Guide =====
* Electrostatic potential due to a point charge
* Distance between source and field point
* Superposition principle
==== Main Ideas ====
==== Students' Task ====
//Estimated Time: 10 min; Wrap-up: 5 min//
Students work in small groups to find the electrostatic potential due to two electric charges separated by a distance $D$. Different groups are assigned different arrangements of charges and different regions of space to consider.
==== Prerequisite Knowledge ====
* Understanding of the geometric and analytic meaning of $|\Vec r - \Vec r' |$. (This may be more far more difficult for your students than you expect. You might want to try our <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=vfstartrek"> displacement vector activity</a> </html>.)
==== Props/Equipment ====
* Balls to represent point charges
* [[Props:start#voltmeter|Voltmeter]]
* [[Props:start#coordinate axes|Coordinate Axes]]
* [[Props:start#whiteboards|Table top whiteboards]] and markers
* [[Props:start#whiteboards|Wall-mounted whiteboards]]
* A handout for each group
==== Activity: Introduction ====
Students typically know the electrostatic formula $V=\frac{kq}{r}$. 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 $r'$, $V=\frac{kq}{|\Vec r - \Vec r'|}$. The lecture should also review the superposition principle. The general, coordinate-independent formula should be left on the board for them to consult as they do this activity.
A nice warm-up (SWBQ) to lead off the discussion: Write down the electrostatic potential everywhere in space due to a point charge that is not at the origin.
==== Activity: Student Conversations ====
* 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.
* This could be a good time to clarify the notation $V(\rr)$.
* Students may use the iconic equation to give an answer along the lines of $V=k(\frac{q_1}{r_1}+\frac{q_2}{r_2})$. But this doesn't provide enough information to describe $r_1$ and $r_2$, a good example of the need to "unpack" the iconic equation.
* 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. Some students may have trouble turning $|\Vec r - \Vec r'|$ into rectangular coordinates, but because the coordinate system is set up for them, most students are successful with this part fairly quickly.
* Students may use the iconic equation $V = \frac{kq}{r}$ and tried to force $V(r)$ as the $f(z)$ from the power series expression $f(z) = \sum_{n=0}^{\infty} C_n (z-a)^n$. Students then plug and chug and tried to find the coefficient of the fourth order term because the students know that $c_n = \frac{1}{n!} \frac{d^{(n)} f}{dz^{(n)}} (z)\bigg \rvert_{z=a}$. This can be avoided by reminding the students that there are two charges and they need to use superposition principle.
* Once students arrived at $V(x) = {Q\over 4\pi\epsilon_0} {\left({1\over{|D - x|}} - { 1\over{|D + x|}}\right)}$, students want to have a common denominator and then do the power series approximation to that combine expression, i.e. $\frac{1}{D-x}-\frac{1}{D+x} = \frac{2x}{D^2 -x^2}$. Let them know doing the approximation to each term is easier than combining the two terms into one.
==== Activity: Wrap-up ====
**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.
==== Extensions ====
This activity is part one of a longer activity,
* <html>
<a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=vfvpoints">Electrostatic Potential Due to Two Point Charges</a>
</html>,
in which students do a series expansion of the expressions that they develop here.
This activity works particularly well when [[:whitepapers:sequences:emsequence:start|sequenced]] with other activities. This is the first activity of the Ring Sequence (calculating fields due to charge distributions).