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2. << The Distance Formula (Star Trek) | Ring Cycle Sequence | Electrostatic Potential Due to a Pair of Charges (with Series) >>
Write a formula for the electrostatic potential due to a point charge.
As students try to remember the formula, many will conflate potential, potential energy, force, and electric field. Their answers may have some aspects of each of these. We use this question to get the iconic equation into the students' working memory in preparation for subsequent activities. This question also be used to help student disambiguate these different physical quantities.
\[V=\frac{kq}{r}\]
\[V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\] You may want to discuss which constants to use in which contexts, e.g. \(k\) is short and easy to write, but may be conflated with other uses of \(k\) in a give problem whereas \(\frac{1}{4\pi\epsilon_0}\) assumes you are working in a particular system of units.
Possible follow-up questions to help with the disambiguation:
We use this small whiteboard question as a transition between The Distance Formula (Star Trek) activity, where students are learning about how to describe (algebraically) the geometric distance between two points, and the Electrostatic Potential Due to a Pair of Charges (with Series) activity, where students are using these results and the superposition principle to find the electrostatic potential due to two point charges.
This activity is the initial activity in the sequence Visualizing Scalar Fields addressing the representations of scalar fields in the context of electrostatics.
group Small Group Activity
30 min.
group Small Group Activity
60 min.
electrostatic potential multipole charge symmetry scalar field superposition coulomb's Law
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.accessibility_new Kinesthetic
10 min.
density charge density mass density linear density uniform idealization
Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.accessibility_new Kinesthetic
10 min.
group Small Group Activity
120 min.
computer Mathematica Activity
30 min.
group Small Group Activity
30 min.
electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula
Students work in small groups 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.
assignment Homework
Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.
A series of charges arranged in this way is called a linear quadrupole. Why?
group Small Group Activity
30 min.
coulomb's law electric field charge ring symmetry integral power series superposition
Students work in small groups 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.
keyboard Computational Activity
120 min.