Divergence through a Prism

    • assignment Divergence

      assignment Homework

      Divergence
      Static Fields 2022 (5 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.

    • group Visualization of Divergence

      group Small Group Activity

      30 min.

      Visualization of Divergence
      Vector Calculus II 2022 (8 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 Gravitational Field and Mass

      assignment Homework

      Gravitational Field and Mass
      Static Fields 2022 (4 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 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 (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 Curl

      assignment Homework

      Curl
      Static Fields 2022 (5 years)

      Shown above is a two-dimensional cross-section of a vector field. All the parallel cross-sections of this field look exactly the same. Determine the direction of the curl at points A, B, and C.

    • assignment Divergence Practice including Curvilinear Coordinates

      assignment Homework

      Divergence Practice including Curvilinear Coordinates

      Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.

      1. \begin{equation} \hat{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z} \end{equation}
      2. \begin{equation} \hat{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z} \end{equation}
      3. \begin{equation} \hat{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z} \end{equation}
      4. \begin{equation} \hat{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z} \end{equation}
      5. \begin{equation} \hat{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z} \end{equation}
      6. \begin{equation} \hat{K} = s^2\,\hat{s} \end{equation}
      7. \begin{equation} \hat{L} = r^3\,\hat{\phi} \end{equation}

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

      assignment Homework

      Gauss's Law for a Rod inside a Cube
      Static Fields 2022 (3 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.
    • 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 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 \(0\le x\le3\).
  • Static Fields 2022 (5 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.)

  • Media & Figures
    • figures/vfprism.png