Spherical Shell Step Functions

  • step function charge density
    • 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}

    • 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 Total Current, Square Cross-Section

      assignment Homework

      Total Current, Square Cross-Section

      Integration Sequence

      Static Fields 2022 (5 years)
      1. Current \(I\) flows down a wire with square cross-section. The length of the square side is \(L\). If the current is uniformly distributed over the entire area, find the current density .
      2. If the current is uniformly distributed over the outer surface only, find the current density .
    • assignment Volume Charge Density, Version 2

      assignment Homework

      Volume Charge Density, Version 2
      charge density delta function Static Fields 2022 (5 years)

      You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge \(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).

      1. Sketch the charge distribution.
      2. Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.

    • assignment Volume Charge Density

      assignment Homework

      Volume Charge Density
      Static Fields 2022 (5 years)

      Sketch the volume charge density: \begin{equation} \rho (x,y,z)=c\,\delta (x-3) \end{equation}

    • accessibility_new Acting Out Charge Densities

      accessibility_new Kinesthetic

      10 min.

      Acting Out Charge Densities
      Static Fields 2022 (5 years)

      density charge density mass density linear density uniform idealization

      Ring Cycle Sequence

      Integration Sequence

      Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.
    • assignment Theta Parameters

      assignment Homework

      Theta Parameters
      Static Fields 2022 (5 years)

      The function \(\theta(x)\) (the Heaviside or unit step function) is a defined as: \begin{equation} \theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases} \end{equation} This function is discontinuous at \(x=0\) and is generally taken to have a value of \(\theta(0)=1/2\).

      Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{align} y &= \theta (x)\\ y &= 2+\theta (x)\\ y &= \theta(2+x)\\ y &= 2\theta (x)\\ y &= \theta (2x) \end{align}

    • assignment Total Current, Circular Cross Section

      assignment Homework

      Total Current, Circular Cross Section

      Integration Sequence

      Static Fields 2022 (4 years)

      A current \(I\) flows down a cylindrical wire of radius \(R\).

      1. If it is uniformly distributed over the surface, give a formula for the surface current density \(\vec K\).
      2. If it is distributed in such a way that the volume current density, \(|\vec J|\), is inversely proportional to the distance from the axis, give a formula for \(\vec J\).

    • assignment Mass of a Slab

      assignment Homework

      Mass of a Slab
      Static Fields 2022 (5 years)

      Determine the total mass of each of the slabs below.

      1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
      2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
      3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
      4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
      5. What are the dimensions of \(A\)?
      6. Write several sentences comparing your answers to the different cases above.

    • assignment Cube Charge

      assignment Homework

      Cube Charge
      charge density

      Integration Sequence

      Static Fields 2022 (5 years)
      1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
      2. In a new physical situation: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.
  • Static Fields 2022 (5 years)

    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://paradigms.oregonstate.eduhttps://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.