Linear Quadrupole (w/ series)

• This problem is used in the following sequences

  1. << Electrostatic Potential Due to a Pair of Charges (with Series) | Power Series Sequence (E&M) | Electrostatic Potential Due to a Ring of Charge >>

• assignment Linear Quadrupole (w/o series)

  assignment Homework

  Linear Quadrupole (w/o series)

  Static Fields 2023 (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 \(\vec{r}\) 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?

• 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 2023 (4 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.
• group Equipotential Surfaces

  group Small Group Activity

  120 min.

  Equipotential Surfaces

  E&M Quadrupole Scalar Fields

  Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
• computer Using Technology to Visualize Potentials

  computer Mathematica Activity

  30 min.

  Using Technology to Visualize Potentials

  Static Fields 2023 (6 years)

  electrostatic potential visualization

  Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.
• keyboard Electrostatic potential of four point charges

  keyboard Computational Activity

  120 min.

  Electrostatic potential of four point charges

  Computational Physics Lab II 2023 (2 years)

  electrostatic potential python

  Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
• assignment Differential Form of Gauss's Law

  assignment Homework

  Differential Form of Gauss's Law

  Static Fields 2023 (6 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 from a Rod

  assignment Homework

  Electric Field from a Rod

  Static Fields 2023 (5 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 Gauss's Law for a Rod inside a Cube

  assignment Homework

  Gauss's Law for a Rod inside a Cube

  Static Fields 2023 (4 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 Electric Field and Charge

  assignment Homework

  Electric Field and Charge

  divergence charge density Maxwell's equations electric field Static Fields 2023 (4 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 Electrostatic Potential Due to a Ring of Charge

  group Small Group Activity

  30 min.

  Electrostatic Potential Due to a Ring of Charge

  Static Fields 2023 (8 years)

  electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula

  Power Series Sequence (E&M)

  Warm-Up

  Ring Cycle Sequence

  Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.

  In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• Static Fields 2023 (6 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.

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