Sphere in Cylindrical 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}

    • 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 \(0\le x\le3\).
    • 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 Basic Calculus: Practice Exercises

      assignment Homework

      Basic Calculus: Practice Exercises
      Static Fields 2022 (3 years) Determine the following derivatives and evaluate the following integrals.
      1. \(\frac{d}{du}\left(u^2\sin u\right)\)
      2. \(\frac{d}{dz}\left(\ln(z^2+1)\right)\)
      3. \(\displaystyle\int v\cos(v^2)\,dv\)
      4. \(\displaystyle\int v\cos v\,dv\)
    • assignment Icecream Mass

      assignment Homework

      Icecream Mass
      Static Fields 2022 (4 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).

    • 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 Scalar Surface and Volume Elements

      group Small Group Activity

      30 min.

      Scalar Surface and Volume Elements
      Static Fields 2022 (4 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 (4 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.

    • assignment Helix

      assignment Homework

      Helix

      Integration Sequence

      Static Fields 2022 (4 years)

      A helix with 17 turns has height \(H\) and radius \(R\). Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix. At the bottom of the helix the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the top of the helix, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the helix?

    • assignment Electric Field of a Finite Line

      assignment Homework

      Electric Field of a Finite Line

      Consider the finite line with a uniform charge density from class.

      1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
      2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

  • Static Fields 2022 (3 years) Find the surface area of a sphere using cylindrical coordinates.