Electric Field of a Finite Line

    • 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 Line Sources Using Coulomb's Law

      assignment Homework

      Line Sources Using Coulomb's Law
      Static Fields 2022 (4 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 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.
    • assignment Differential Form of Gauss's Law

      assignment Homework

      Differential Form of Gauss's Law
      Static Fields 2022 (4 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.

    • 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.
    • 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.

    • 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 Gravitational Field and Mass

      assignment Homework

      Gravitational Field and Mass
      Static Fields 2022 (3 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 Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient
      Static Fields 2022 (4 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}

    • assignment Linear Quadrupole (w/ series)

      assignment Homework

      Linear Quadrupole (w/ series)

      Power Series Sequence (E&M)

      Static Fields 2022 (4 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.

  • 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!)

  • Media & Figures
    • figures/efiniteline.svg