Activity: Acting Out Flux

Static Fields 2023 (5 years)
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.
  • group Small Group Activity schedule 5 min. build Rulers and Meter Sticks, one for each of a group of students,Hula Hoop
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    flux electrostatics vector fields
What students learn The geometric definition of the flux of a vector field \(\vec{F}\) through a surface:
  • Only the vectors whose tails are in the surface contribute to the flux.
  • Only the components of the vectors perpendicular to the surface contribute to the flux.
  • The surface needs an orientation so that the vectors that make a positive contribution to the flux can be distinguished from the vectors that make a negative contribution.

Instructor's Guide

Introduction

  • If you haven't yet done so, this is the time to introduce the vector surface element \(d\vec{A}= \hat{n}\; dA\). Point out that the only unique direction related to a surface (in three dimensions) is the normal to the surface.
  • Start with the [[swbq:emsw:vfswflux|Recall Definition of Flux]] SWBQ. This SWBQ is an opportunity for the instructor to find out what the students already know about the flux so as to focus the succeeding discussion appropriately.

Activity Prompt: Give a group of rulers (of different sizes) to a small group of students and ask them to form a vector field.

  • Hold the hoop with some of the rulers pointing through it and lead a whole class discussion about how you would measure the flux through the hoop.
  • Specify that the students hands are the points at which the field is being evaluated, and the length of the ruler is the magnitude of the field at that point.
  • See [[activities:reflections:adopters:vffluxconcept|reflections]] and [[whitepapers:narratives:fluxshort|narrative]] for examples from specific classes.

Student Conversations

  • Evaluating the field: Because of students' soundbite understanding of the flux that it involves the field “pointing through” the surface, students may not realize that only the points (i.e.  hands) lying on the surface (i.e. in the plane of the loop) contribute to the flux. Emphasize this by holding the hula hoop with only the rulers pointing through it, but with the hands not on the “surface” of the hoop.
  • Only perpendicular component: Have one of the rulers lie in the plane of the hoop or at an angle to the plane of the hoop and ask the students what the flux is.
  • Conceptualization of \(d\vec{A}\): The size of \(d\vec{A}\) in a field, \(d\vec{A}\) is small enough that the field appears constant over the surface of each “little chunk of \(d\vec{A}\).”
  • Sign of the flux: Discuss the need to specify a “direction” for the area. Emphasize that the dot product only picks up the component of the vector field perpendicular to the surface. Since the differential area element is a vector quantity, the flux can be positive or negative.
  • Static vs. Moving Flux: Since this is a course in electroSTATICS, it is important to avoid examples with time dependence and language like the amount of stuff that “gets through” or ”flows through” the surface. “Points through” is more helpful language.
  • computer Visualizing Flux through a Cube

    computer Computer Simulation

    30 min.

    Visualizing Flux through a Cube
    Static Fields 2023 (6 years) 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.
  • assignment Divergence through a Prism

    assignment Homework

    Divergence through a Prism
    Static Fields 2023 (6 years)

    Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).

    1. Calculate the divergence of \(\vec F\).
    2. In which direction does the vector field \(\vec F\) point on the plane \(z=x\)? What is the value of \(\vec F\cdot \hat n\) on this plane where \(\hat n\) is the unit normal to the plane?
    3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

  • group Flux through a Cone

    group Small Group Activity

    30 min.

    Flux through a Cone
    Static Fields 2021 (4 years)

    Integration Sequence

    Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
  • group Visualization of Divergence

    group Small Group Activity

    30 min.

    Visualization of Divergence
    Vector Calculus II 23 (12 years) 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.
  • assignment Flux through a Plane

    assignment Homework

    Flux through a Plane
    Static Fields 2023 (4 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
  • assignment Divergence

    assignment Homework

    Divergence
    Static Fields 2023 (6 years)

    Shown above is a two-dimensional vector field.

    Determine whether the divergence at point A and at point C is positive, negative, or zero.

  • accessibility_new Acting Out Current Density

    accessibility_new Kinesthetic

    10 min.

    Acting Out Current Density
    Static Fields 2023 (6 years)

    Steady current current density magnetic field idealization

    Integration Sequence

    Ring Cycle Sequence

    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.
  • assignment Flux through a Paraboloid

    assignment Homework

    Flux through a Paraboloid
    Static Fields 2021 (5 years)

    Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.

  • assignment Gravitational Field and Mass

    assignment Homework

    Gravitational Field and Mass
    Static Fields 2023 (5 years)

    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 description 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. Briefly 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. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

  • assignment Gauss's Law for a Rod inside a Cube

    assignment Homework

    Gauss's Law for a Rod inside a Cube
    Static Fields 2023 (4 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(x,y\)-plane. The charge density \(\lambda_0\) is constant. Find the total flux of the electric field through a closed cubical surface with sides of length \(3L\) centered at the origin.
Author Information
Corinne Manogue, Liz Gire
Learning Outcomes