Activity: Flux through a Cone

Static Fields 2022 (4 years)
Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
What students learn
  • Calculational fluency with flux;
  • Finding the component of a field perpendicular to a surface;
  • Finding the differential area element of a surface by taking the cross product of two vector differentials in the surface, \(d\vec{A}=d\vec{r}_1\times d\vec{r}_2\).
Flux through a Cone
Find the flux through a cone of height \(H\) and radius \(R\) due to the vector field \(\vec{F} = C\,z\,\hat{z}\).

Instructor's Guide

Introduction

This activity is part of a sequence on flux Flux Sequence, which we strongly recommend. If you don't have time, the minimum introduction is a short lecture introducing the concept of flux (as the amount of a vector field perpendicular to a surface) and how to calculate it: \[ \Phi = \int_S\, \vec{F}\, \cdot \,d\vec{A}\]

Prompt: Find the flux through a cone of height \(H\) and radius \(R\) due to the vector field \(\vec{F} = C\,z\,\hat{z}\).

This prompt is open-ended in that it doesn't specify either the location of the cone or whether or not the circular top of the cone is to be considered part of the surface. We like to leave it open-ended, see what students do, and when students question the open-endedness, give a mini-”sermon” on the ill-posedness of most real world problems. If you are short of time, or otherwise want to avoid these questions, you should use a more explicit prompt.

If you choose the point of the cone at the origin (and allow it to open upward, like an icecream cone), then the problem can easily be solved in spherical coordinates as well as the obvious cylindrical coordinates.

Student Conversations

  • Choice of coordinates - some groups will choose cylindrical and others will choose spherical coordinates and it can be done with either. Occasionally, a group will attempt to use Cartesian coordinates, but they usually realize quickly that this is not a good choice.
  • Into or Out of the Cone? - the sign of the flux depends on whether we choose to calculate the flux up through the cone or down through the cone. Students need to be aware of this choice. The answers differ by a minus sign.
  • Evaluating the field on the surface - in this case, the vector field varies along the surface of the cone so students must integrate over the surface rather than finding a constant flux times a surface area.
  • Finding the differential surface element - students can find the differential surface element by taking the cross product of two \(d\vec{r}\) vectors lying on the cone. Several issues arise, such as:
    • writing down the \(d\vec{r}\)'s using the "use what you know" strategy;
    • choosing the direction of the area element (i.e. the order of the vectors in the cross product);
    • making sure that the \(d\vec{r}\) they choose actually lies on the cone.
  • Limits of integration - some students have difficulty determining the limits of integration.
  • Only two parameters - students will often forget to change all of the variables in the integrand into the two variables that are being integrated over.

Wrap-up

We do a brief summary of the main points to wrap up the activity.

  • This is also good place to talk about the affordances of different choices for coordinates (e.g. ask a group that solved it in cylindrical and one that solved it in spherical to compare).
  • It is important to reinforce the method of constructing the \(d\vec{A}\) vector by taking the cross product \(d\vec{r_1} \times d\vec{r_2}\).

  • assignment Cone Surface

    assignment Homework

    Cone Surface
    Static Fields 2022 (5 years)

    • Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
    • Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

  • assignment Icecream Mass

    assignment Homework

    Icecream Mass
    Static Fields 2022 (5 years)

    Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

  • keyboard Electric field for a waffle cone of charge

    keyboard Computational Activity

    120 min.

    Electric field for a waffle cone of charge
    Computational Physics Lab II 2022

    electric field cone

    Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
  • assignment Flux through a Plane

    assignment Homework

    Flux through a Plane
    Static Fields 2022 (3 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 Find Area/Volume from $d\vec{r}$

    assignment Homework

    Find Area/Volume from \(d\vec{r}\)
    Static Fields 2022 (4 years)

    Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

    1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
    2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
    3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

  • group Vector Surface and Volume Elements

    group Small Group Activity

    30 min.

    Vector Surface and Volume Elements
    Static Fields 2022 (3 years)

    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 vector approach to find directed surface and volume elements.

  • assignment Flux through a Paraboloid

    assignment Homework

    Flux through a Paraboloid
    Static Fields 2022 (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 Electric Field and Charge

    assignment Homework

    Electric Field and Charge
    divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) 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.
  • group Acting Out Flux

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    5 min.

    Acting Out Flux
    Static Fields 2022 (3 years)

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    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.
  • computer Visualizing Flux through a Cube

    computer Computer Simulation

    30 min.

    Visualizing Flux through a Cube
    Static Fields 2022 (5 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.

Author Information
Corinne Manogue, Liz Gire
Learning Outcomes