Gauss's Law for a Rod inside a Cube

    • assignment Mass Density

      assignment Homework

      Mass Density
      Static Fields 2022 (3 years) Consider a rod of length \(L\) lying on the \(z\)-axis. Find an algebraic expression for the mass density of the rod if the mass density at \(z=0\) is \(\lambda_0\) and at \(z=L\) is \(7\lambda_0\) and you know that the mass density increases
      • linearly;
      • like the square of the distance along the rod;
      • exponentially.
    • assignment Electric Field from a Rod

      assignment Homework

      Electric Field from a Rod
      Static Fields 2022 (4 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(xy\)-plane. The charge density \(\lambda\) is constant. Find the electric field at the point \((0,0,2L)\).
    • assignment Differential Form of Gauss's Law

      assignment Homework

      Differential Form of Gauss's Law
      Static Fields 2022 (5 years)

      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.

    • computer Visualizing Flux through a Cube

      computer Computer Simulation

      30 min.

      Visualizing Flux through a Cube
      Static Fields 2022 (5 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.
    • assignment Line Sources Using Coulomb's Law

      assignment Homework

      Line Sources Using Coulomb's Law
      Static Fields 2022 (5 years)
      1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
      2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.
    • 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 Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient

      Gradient Sequence

      Static Fields 2022 (5 years)
      1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

    • group Electrostatic Potential Due to a Pair of Charges (without Series)

      group Small Group Activity

      30 min.

      Electrostatic Potential Due to a Pair of Charges (without Series)
      Static Fields 2022 (3 years) Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.
    • assignment Linear Quadrupole (w/o series)

      assignment Homework

      Linear Quadrupole (w/o series)
      Static Fields 2022 (3 years) Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
      1. Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.

      2. A series of charges arranged in this way is called a linear quadrupole. Why?

    • assignment Linear Quadrupole (w/ series)

      assignment Homework

      Linear Quadrupole (w/ series)

      Power Series Sequence (E&M)

      Static Fields 2022 (5 years)

      Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

      1. Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r'}\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert} \] Electrostatic potentials satisfy the superposition principle.
      2. Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

  • 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.