Memorize $d\vec{r}$

    • 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 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 Flux through a Cone

      group Small Group Activity

      30 min.

      Flux through a Cone
      Static Fields 2022 (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 Vector Differential--Curvilinear

      group Small Group Activity

      30 min.

      Vector Differential--Curvilinear
      Vector Calculus II 2022 (8 years)

      vector calculus coordinate systems curvilinear coordinates

      Integration Sequence

      In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

      Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

    • 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).

    • group Scalar Surface and Volume Elements

      group Small Group Activity

      30 min.

      Scalar Surface and Volume Elements
      Static Fields 2022 (6 years)

      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.

    • groups Pineapples and Pumpkins

      groups Whole Class Activity

      10 min.

      Pineapples and Pumpkins
      Static Fields 2022 (5 years)

      Integration Sequence

      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.

    • 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 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 Total Charge

      assignment Homework

      Total Charge
      charge density curvilinear coordinates

      Integration Sequence

      Static Fields 2022 (5 years)

      For each case below, find the total charge.

      1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
      2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}

  • Static Fields 2022 (2 years)

    Write \(\vec{dr}\) in rectangular, cylindrical, and spherical coordinates.

    1. Rectangular: \begin{equation} \vec{dr}= \end{equation}
    2. Cylindrical: \begin{equation} \vec{dr}= \end{equation}
    3. Spherical: \begin{equation} \vec{dr}= \end{equation}