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Estimated Time: 50 min; Wrap-up: 30 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. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space.
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.
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.
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 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. 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 of substituting into known series by rewriting an expression in terms of dimensionless parameters.
If students have been exposed to Taylor's theorem $f(z) = f(a) + f'(a)(z-a) + f”(a){{(z-a)^2}\over {2!}} + …$, 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.
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 most of the learning possibilities of this activity. Students will need some time just to recognize that $p = -1$; 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 $p = -1$.
MANY students make algebraic errors such as incorrectly factoring out $D$ or $x$. These should be brought to their attention quickly. Students may also have trouble dealing with the absolute value sign.
Many students are likely to treat this as a two-dimensional case from the start, ignoring the $z$ axis entirely. Look for expressions like $$V = {1\over 4\pi\epsilon_0} \sum_{i=1}^N {q_i\over\sqrt{(x - x_i)^2 + (y - y_i)^2}} $$. By the end of the wrap-up discussion, students should have considered the larger, 3-dimensional picture.
Students who are having trouble figuring out what parameter to expand in can be asked what quantity is small. Then what that means — small with respect to what. This should eventually guide them to a ratio — which is small with respect to $1$, as required.
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.
Method:
Iconic equation;
Use $|\rr-\rrp|$
Use what you know!
Why do we care about the power series expansion of an expression we have in closed form? Because in practice we don't know the closed form solution. Imagine entering a solar system and measuring the gravitational field. First, only the gravitational field due to the total mass is observable. Then, as you get closer, you start to detect a dipole field. And so on. The point is that you don't know in advance where the planets are.
The instructor should encourage students to compare and contrast the results for the eight situations. This should include careful attention to:
whether the power series is odd or even and how this relates to whether the situation is symmetric or anti-symmetric;
whether the answers “make sense” given the physical situation
and what they tell you about how the field changes along the given axis.
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.
Assuming that students have not yet been exposed to Laurent series, it should be brought to their attention that a series involving inverse powers of the variable is not a power series at all; it is a generalization of a power series called a Laurent series. 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. We have found that by introducing Laurent series in this context, students see it as no big deal.
This activity is the final activity of a sequence of activities addressing Power Series and their application to physics. The following activities are part of this sequence.
Preceding activities:
The Distance Between Two Points - Star Trek: 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 $\dfrac{1}{|\vec{r}-\vec{r}'|}$.
Calculating Coefficients for a Power Series: 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.
Approximating Functions with a Power Series: This computer visualization activity using Mathematica (or Maple) fits power series approximations of a given function to an actual function which allows students to see where approximations are valid.
This is first activity of the Ring Sequence, which uses a sequence of activities with similar geometries to help students learn how to solve a hard activity by breaking it up into several steps (A Master's Thesis about the Ring Sequence). The other activities in the sequence are:
Follow-up activities:
Electrostatic Potential Due to a Ring of Charge: In this small group activity, students aim to generalize the expression for the potential by applying what they have learned about charge densities, power series approximations, and various geometries in order to find the electrostatic potential due to a ring of charge.
Electric Field Due to a Ring of Charge: In this small group activity, students use the definition of the electric field as $\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\int_{ring}{\frac{\lambda(\vec{r'})(\vec{r}-\vec{r'})|d\vec{r'}|}{|\vec{r}-\vec{r'}|^3}} $ in order to work out the electric field due to a ring of charge.
Magnetic Field Due to a Spinning Ring of Charge: Building from their experience with the previous exercises, students in this small group activity use techniques from the previous activities to obtain an expression for the magnetic field due to a spinning ring of charge.