Activity: Scalar Surface and Volume Elements

Static Fields 2022 (4 years)

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.

What students learn
  • How to find area, and volume elements in curvilinear coordinates using geometric methods.

Instructor's Guide

Notes:

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements. Therefore, this version of the activity does not require knowledge of the \(d\vec{r}\) vector, cross products, and dot products which can make this activity more accessible to students earlier in a course on electricity and magnetism.

This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins.

Introduction

In a previous activity, Vector Differential--Curvilinear, students are asked to find the line element, \(d\ell\), along each side of an “infinitesimal box” in cylindrical and spherical coordinates. Using the \(d\ell\), they are now asked to construct the area (\(dA\)) and volume (\(dV\)) elements in each coordinate system. This prepares students to integrate scalar-valued functions in curvilinear coordinates.

Scalar Surface and Volume Elements

Find the formulas for the differential surface \(dA\) and volume \(d\tau\) elements (little chopped pieces of the surface and/or volume) for a plane, for a finite cylinder (including the top and bottom), and for a hemisphere. Make sure to draw an appropriate figure.

  • 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 Gravitational Field and Mass

    assignment Homework

    Gravitational Field and Mass
    Static Fields 2022 (3 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 Cone Surface

    assignment Homework

    Cone Surface
    Static Fields 2022 (4 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.

  • group Total Charge

    group Small Group Activity

    30 min.

    Total Charge
    Static Fields 2022 (4 years)

    charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

    Integration Sequence

    In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
  • group Acting Out Flux

    group Small Group Activity

    5 min.

    Acting Out Flux
    Static Fields 2022 (3 years)

    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 Sphere in Cylindrical Coordinates

    assignment Homework

    Sphere in Cylindrical Coordinates
    Static Fields 2022 (3 years) Find the surface area of a sphere using cylindrical coordinates.
  • 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.
  • assignment Flux through a Paraboloid

    assignment Homework

    Flux through a Paraboloid
    Static Fields 2022 (4 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 Current from a Spinning Cylinder

    assignment Homework

    Current from a Spinning Cylinder
    A solid cylinder with radius \(R\) and height \(H\) has its base on the \(x,y\)-plane and is symmetric around the \(z\)-axis. There is a fixed volume charge density on the cylinder \(\rho=\alpha z\). If the cylinder is spinning with period \(T\):
    1. Find the volume current density.
    2. Find the total current.

Learning Outcomes