Electric Field and Charge

  • divergence charge density Maxwell's equations electric field
    • assignment Gravitational Field and Mass

      assignment Homework

      Gravitational Field and Mass
      AIMS Maxwell AIMS 21

      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 value 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. 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. Discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

    • assignment Divergence

      assignment Homework

      Divergence
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      Shown above is a two-dimensional vector field.

      Determine whether the divergence at point A and at point C is positive, negative, or zero.

    • assignment Differential Form of Gauss's Law

      assignment Homework

      Differential Form of Gauss's Law
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

    • group Visualization of Divergence

      group Small Group Activity

      30 min.

      Visualization of Divergence
      AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021 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.
    • group Total Charge

      group Small Group Activity

      30 min.

      Total Charge
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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.
    • assignment Total Charge

      assignment Homework

      Total Charge
      charge density curvilinear coordinates

      Integration Sequence

      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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}

    • assignment Spherical Shell Step Functions

      assignment Homework

      Spherical Shell Step Functions
      step function charge density AIMS Maxwell AIMS 21 Static Fields Winter 2021

      One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. If you need to review this, see the following link in the math-physics book: https://books.physics.oregonstate.edu/GMM/step.html

      Consider a spherical shell with charge density \(\rho (\vec{r})=\alpha3e^{(k r)^3}\) between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else. Use step functions to write this charge density as a single function valid everywhere in space.

    • assignment Divergence through a Prism

      assignment Homework

      Divergence through a Prism
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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.

    • 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 Magnetic Field and Current

      assignment Homework

      Magnetic Field and Current
      AIMS Maxwell AIMS 21 Consider the magnetic field \[ \vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases} \]
      1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
      2. Find a formula for the current density that creates this magnetic field.
      3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
  • AIMS Maxwell AIMS 21 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.