## Activity: Using Technology to Visualize Potentials

Static Fields 2022 (5 years)
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 demonstrates several different ways of plotting the potential.
What students learn
• How to represent 3-d scalar fields in several different ways;
• The symmetries of a some simple charge distributions such as a dipole and a quadrupole.
• Media
• Brainstorm several different ways in which you might visualize a scalar field in three dimensions, using graphs. You might want to use the electrostatic potential $V$ due to some simple configurations like a quadrupole of charges as examples.
• After you have brainstormed ideas of your own, open this Sage code or this Mathematica worksheet and explore some of the ways implemented there. If you have a different visualization, please bring it to the attention of the teaching team and we may incorporate it next year!

## Instructor's Guide

### Prerequisites

We find it valuable to use this activity AFTER students have done the activity Drawing Equipotential Surfaces. This pair of activities bolster students' geometric sensemaking of potentials. We have successfully used this pair of activities both before and after Electrostatic Potential Due to a Pair of Charges (with Series).

### Introduction

This activity starts with 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). (  5 minutes, can be done as a whole class discussion or in small groups.)

### Student Conversations

It is a good idea to put the least computationally confident member of each group at the keyboard. This helps to ensure that everyone is comfortable with the technology. Do not hesitate to encourage a group to change its typist if the current one is typing too quickly.

This activity can be used very effectively in a context where students are asked to brainstorm about ways in which they might graphically represent the electrostatic potential. They should be reminded to think about the fact that the electrostatic potential is a scalar field, i.e. it is a number (with appropriate units) at every point in THREE dimensional space. In a whole class setting, as the students generate ideas, the instructor projects each choice from the Mathematica worksheet for the students to examine/discuss.

A pedagogically useful representation is for each number at a point to be represented by a color. Then the students can imagine how they would try to show this “sea” of colors on a two-dimensional graph.

If students are new to using Mathematica, it is well worth showing them how to create a new line so they can enter new mathematical input. This requires moving the mouse between existing blocks; the pointer should change to a horizontal line.

### Wrap-up

Consider adding the Surfaces activity Equipotential Surfaces.

computer Mathematica Activity

30 min.

Static Fields 2022 (5 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• group Equipotential Surfaces

group 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.
• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Warm-Up

Students work in groups of three 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.

• keyboard Electrostatic potential of four point charges

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2022

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.
• computer Visualization of Quantum Probabilities for the Hydrogen Atom

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom
Central Forces 2023 (3 years) Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.
• group Electrostatic Potential Due to a Pair of Charges (with Series)

group Small Group Activity

60 min.

##### Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

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.
• group Electric Field Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three 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.

• assignment_ind Electrostatic Potential Due to a Point Charge

assignment_ind Small White Board Question

10 min.

##### Electrostatic Potential Due to a Point Charge
Static Fields 2022

Warm-Up

• group A glass of water

group Small Group Activity

30 min.

##### A glass of water
Energy and Entropy 2021 (2 years)

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

Author Information
Corinne Manogue
Learning Outcomes