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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). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.

Computational Activity

120 min.

Electrostatic potential of four point charges
Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
  • electrostatic potential python
    Found in: Computational Physics Lab II course(s) Found in: Computational integrating charge distributions sequence(s)

The electrostatic potential due to a point charge at the origin is given by: \begin{equation*} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation*}

  1. (2pts) Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
  2. (2pts) Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
  3. (2pts) Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)

Consider the fields at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\).

  1. Write down an expression for the electrostatic potential \(V(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
  2. Write down an expression for the electric field \(\vec{E}(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
  3. Working in rectangular coordinates, compute the gradient of \(V\).
  4. Write several sentences comparing your answers to the last two questions.

  • Found in: Gradient Sequence sequence(s)

Small White Board Question

10 min.

Electrostatic Potential Due to a Point Charge

Write the equation for the electrostatic potential due to a point charge.

Instructor's Guide

Prerequisite Knowledge

Students will usually have seen the electrostatic potential due to a point charge in their introductory course, but may have trouble recalling it.

Whole-Class Conversations

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.

Correct answers you're likely to see

\[V=\frac{kq}{r}\]

\[V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\] 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.

Incorrect answers you're likely to see

  • Two charges instead of one \[\cancel{V=\frac{kq_{1}q_{2}}{r}}\]
  • Distance squared in the denominator \[\cancel{V=\frac{kq}{r^2}}\]
  • Vector values \[\cancel{V=\frac{kq\, \hat r}{r}}\]

Possible follow-up questions to help with the disambiguation:

  • Relationship between potential and potential energy \(U = qV\)
  • Which function is the derivative of the other: \(1/r\) or \(1/r^2\)?
  • Which physical quantity (potential or electric field, potential energy or force) is the derivative of the other?
  • What is the electrostatic potential conceptually?
  • Which function falls off faster: \(1/r\) or \(1/r^2\)?
  • What are the dimensions of potential? Units?
  • Where is the zero of potential?

Wrap-up

  • This could be a good time to refer to the (correct) expression for the potential as an iconic equation, which will need to be further interpreted (”unpacked”) in particular physical situations. This is where the course is going next.
  • This SWBQ can also serve to help students learn about recall as a cognitive activity. While parts of the equations that students write may be incorrect, many other parts will be correct. Let the way in which you manage the class discussion model for the students how a professional goes about quickly disambiguating several different choices. And TELL the students that this is what you are doing. Deliberately invoke their metacognition.
  • Many students may not know that the electrostatic potential that we are talking about in this activity is the same quantity as what a voltmeter reads, in principle, but not in practice. You may need to talk about how a voltmeter actually works, rather than idealizing it. It helps to have a voltmeter with leads as a prop. Students often want to know about the “ground” lead. We often tie a long string to it (to symbolize making a really long wire) and send the TA out of the room with the string, “headed off to infinity” while discussing the importance of setting the zero of potential. The extra minute or two of humerous byplay gives the importance of the zero of potential a chance to sink in.

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.

  • Found in: Static Fields course(s) Found in: Warm-Up, E&M Ring Cycle Sequence sequence(s)

In this unit, you will explore the electrostatic potential \(V(\vec{r})\) due to one or more discrete charges and the gravitational potential \(\Phi(\vec{r})\) due to one or more discrete masses. How does the potential vary in space? How do equipotential surfaces and the superposition principle help you answer these questions graphically? How does the value of the potential fall-off as you move away from the charges? How do power series approximations help you answer these questions algebraically?

Key Activities/Problems

At the end of this unit, you should be able to:

  • Describe the important similarities and differences between the electrostatic potential and the gravitational potential.
  • Sketch the potential due to a small number of discrete charges or masses, showing important regions of interest and qualitatively depicting the correct spacing between equipotential surfaces (or curves).
  • Compute power and Laurent series expansions from a real-world problem using simple, memorized power series.
  • Truncate a series properly at a given order by keeping all the terms up to that order and none of the terms of higher order.
  • Discuss in detail the relationship between the graphical and algebraic representations of the potentials.

  • Found in: Static Fields course(s)
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.

Kinesthetic

10 min.

Acting Out Current Density
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

Kinesthetic

10 min.

Acting Out the Gradient
Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
  • gradient vector fields electrostatics
    Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Gradient Sequence sequence(s)

Small Group Activity

30 min.

Visualization of Curl
Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
  • Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence sequence(s)

Small Group Activity

30 min.

Visualization of Divergence
Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
  • Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Flux Sequence sequence(s)

Mathematica Activity

30 min.

Using Technology to Visualize Potentials
Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.

Small Group Activity

60 min.

Visualizing Plane Waves

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.

Small Group Activity

30 min.

The Hillside
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
  • Found in: Vector Calculus I course(s) Found in: Gradient Sequence, Workshop Presentations 2023 sequence(s)

Computer Simulation

30 min.

Visualizing Flux through a Cube
Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.
  • Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Geometry of Vector Fields Sequence, Flux Sequence sequence(s)

Small Group Activity

30 min.

DELETE Navigating a Hill
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
  • Found in: Static Fields, AIMS Maxwell course(s)

Kinesthetic

10 min.

Acting Out Charge Densities
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 we demonstrate the answers with coins under a doc cam.

Small Group Activity

30 min.

The Hill
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
  • Gradient
    Found in: Vector Calculus II, Vector Calculus I, Surfaces/Bridge Workshop course(s) Found in: Gradient Sequence sequence(s)

Small Group Activity

120 min.

Equipotential Surfaces
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.