Activity: Pineapples and Pumpkins

AIMS Maxwell AIMS 21 Static Fields Winter 2021

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

  • groups Whole Class Activity schedule 10 min.
  • group Scalar Surface and Volume Elements

    group Small Group Activity

    30 min.

    Scalar Surface and Volume Elements
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    Integration Sequence

    Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

    This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

  • group Vector Surface and Volume Elements

    group Small Group Activity

    30 min.

    Vector Surface and Volume Elements
    AIMS Maxwell AIMS 21

    Integration Sequence

    Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

    This activity is identical to Scalar Surface and Volume Elements except uses a more sophisticated vector approach to find surface, and volume elements.

  • group Visualization of Divergence

    group Small Group Activity

    30 min.

    Visualization of Divergence
    AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021 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 definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.
  • group Acting Out Flux

    group Small Group Activity

    5 min.

    Acting Out Flux
    AIMS Maxwell AIMS 21

    flux electrostatics vector fields

    Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.
  • assignment Gravitational Field and Mass

    assignment Homework

    Gravitational Field and Mass
    AIMS Maxwell AIMS 21

    The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}

    This problem explores the consequences of the divergence theorem for this shell.

    1. Using the given value of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
    2. Discuss the physical meaning of the divergence in this particular example.
    3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\). ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
    4. Discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

  • assignment Electric Field and Charge

    assignment Homework

    Electric Field and Charge
    divergence charge density Maxwell's equations electric field AIMS Maxwell AIMS 21 Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
    1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
    2. Find a formula for the charge density that creates this electric field.
    3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
  • assignment_ind Vector Differential--Rectangular

    assignment_ind Small White Board Question

    10 min.

    Vector Differential--Rectangular
    AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021

    vector differential rectangular coordinates math

    Integration Sequence

    In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

    This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

  • assignment_ind Partial Derivatives from a Contour Map

    assignment_ind Small White Board Question

    10 min.

    Partial Derivatives from a Contour Map
    AIMS Maxwell AIMS 21 Students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
  • assignment_ind Dot Product Review

    assignment_ind Small White Board Question

    10 min.

    Dot Product Review
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    dot product math

    This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
  • group Electrostatic Potential Due to a Pair of Charges (without Series)

    group Small Group Activity

    30 min.

    Electrostatic Potential Due to a Pair of Charges (without Series)
    AIMS Maxwell AIMS 21 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). Students then evaluate the limiting cases of the potential on the axes of symmetry.

Instructor's Guide

Introduction

As described here, this activity has been done as a whole class activity where students answer a series of small whiteboard questions (SWBQ) as the instructor makes cuts in a pumpkin to construct a volume element in spherical coordinates. This activity can be done as a small group activity by giving pieces of pineapple slices and pumpkins, together with appropriate children's pumpkin cutting tools, while students are doing Scalar Surface and Volume Elements or Vector Surface and Volume Elements.

If done as an instructor led whole class activity, the following provides a structure for the activity done in spherical coordinates using a pumpkin.

  • Let's define a set of coordinates \(\hat{x}\) and \(\hat{y}\) (\(\hat{z}\) is through the pumpkin stem). I'm going to start by drawing a line around the equator to help my drawing.
    • SWBQ: What is the circumference of this circle? Answer: \(l=2\pi r\)
  • Now I'm going to pick a vector \(\vec{r}\), which we will examine. When drawing figures (or pumpkins), I recommend that you never pick an angle that is close to \(90^{\circ}\) or \(45^{\circ}\), since that makes it way easier to distinguish between \(\theta\) and \(\frac{\pi}{2}-\theta\).
  • Draw a longitude line.
    • SWBQ: What is the arc length from \(\vec{r}\) to the stem? Answer: \(l=r\theta\)
    • SWBQ: What is the distance along the equator from \(\hat{x}\)? Answer: \(r\phi\)
  • Now let's consider an infinitesimal volume of pumpkin. Pro tip: you always want to draw infinitesimal quantities as medium-sized, otherwise your drawing will be illegible, and won't actually help you. In this case, I'll pick a \(d\theta\) and a \(d\phi\).
    • SWBQ: What is this small distance along the equator? Answer: \(dl=rd\phi\)
    • SWBQ: What is this small distance along the longitude? Answer: \(dl=rd\theta\)
  • Draw a latitude circle at \(\vec{r}\).
    • SWBQ: What is the circumference of this circle? Answer: \(l=2\pi s = 2\pi r\sin{\theta}\)
  • Cut out the pumpkin chunk.
    • SWBQ: What is the width of my pumpkin chunk? Answer: \(dl=sd\phi=r\sin{\theta}d\phi\)
  • So when we put these distances together (including the thickness of the chunk dr), we find that the volume of our pumpkin chunk is \(d\tau=dr(rd\theta)(r\sin{\theta}d\phi)=r^2\sin{\theta}drd\phi d\theta\)


Author Information
David Roundy and Corinne Manogue
Learning Outcomes