Activity: 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.
  • group Small Group Activity schedule 30 min. build Medium whiteboards/markers/erasers, balls to represent the charges, voltmeter, coordinate axes on the ceiling description Student handout (PDF)
What students learn
  • The superposition principle for the electrostatic potential;
  • How to calculate the distance formula \(\frac{1}{|\vec{r} - \vec{r}'|}\) for a simple specific geometric situation;
Electrostatic Potential from Two Charges
  • Find a formula for the electrostatic potential \(V(\vec{r})\) that is valid everywhere in space for the following two physical situations:
    • Two charges \(+Q\) and \(+Q\) placed on a line at \(z'=D\) and \(z''=-D\).
    • Two charges \(+Q\) and \(-Q\) placed on a line at \(z'=D\) and \(z''=-D\), respectively.
  • Simplify your formulas in the limiting cases of:
    • the \(x\)-axis
    • the \(z\)-axis
  • Discuss the relationship between the symmetries of the physical situations and the symmetries of the functions in these limiting cases.

Instructor's Guide

Introduction

It may help to do the The Distance Formula (Star Trek) activity before this one. There is an alternative version of this activity Electrostatic Potential Due to a Pair of Charges (with Series) in which students find series expansions of the potential along the axes of symmetry.

Students typically know the iconic formula for the electrostatic potential of a point charge \(V=\frac{kq}{r}\). We begin this activity with a short lecture/discussion that generalizes this formula in a coordinate independent way to the situation where the source is moved away from the origin to the point \(\vec{r}^{\prime}\), \(V(\vec{r})=\frac{kq}{|\vec{r} - \vec{r}^{\prime}|}\). (A nice warm-up (SWBQ) to lead off the discussion:

Introductory SWBQ Prompt: “Write down the electrostatic potential everywhere in space due to a point charge that is not at the origin.” The lecture should also review the superposition principle. \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0}\sum_{i}\frac{q_i}{|\vec{r} - \vec{r_i}|} \]

This general, coordinate-independent formula should be left on the board for students to consult as they do this activity.

Student Conversations

  • Students create an expression such as \[V(x,y,z) = \frac{Q}{ 4\pi\epsilon_0} {\left(\frac{1}{|z - D|} + \frac{1}{|z + D|}\right)}\] Each axis and charge distribution has a slightly different formula. A few groups may have trouble coordinatizing \(|\vec{r} - \vec{r}^{\prime}|\) into an expression in rectangular coordinates, but because the coordinate system is set up for them, most students are successful with this part fairly quickly.
  • Many students are likely to treat this as a two-dimensional case from the start, ignoring the \(y\) axis entirely. Look for expressions like \[V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^N {q_i\over\sqrt{x ^2 + (z - z_i)^2}}. \] Prompt them to consider the larger, 3-dimensional picture.
  • Most students will leave off the absolute value signs when evaluating the potential on the \(z\)-axis. If they do this, their formulas will not be correct for negative values of \(z\). The subtlety here is that \[\sqrt{a^2}=\vert a\vert\] not \[\sqrt{a^2}=a\] in contexts like this when \(\sqrt{a^2}\) is a distance and therefore necessarily positive and when \(a\) itself might be either positive or negative.
  • 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 (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.

  • computer Visualizing Flux through a Cube

    computer Computer Simulation

    30 min.

    Visualizing Flux through a Cube
    Static Fields 2022 (4 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.
  • group Electrostatic Potential Due to a Pair of Charges (with Series)

    group Small Group Activity

    60 min.

    Electrostatic Potential Due to a Pair of Charges (with Series)
    Static Fields 2022 (4 years)

    electrostatic potential multipole charge symmetry scalar field superposition coulomb's Law

    Power Series Sequence (E&M)

    Ring Cycle Sequence

    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). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
  • 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 2022 (6 years)

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

    Power Series Sequence (E&M)

    Ring Cycle Sequence

    Students work in groups of three 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.

  • assignment Potential vs. Potential Energy

    assignment Homework

    Potential vs. Potential Energy
    Static Fields 2022 (4 years)

    In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

    1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
    2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
    3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

  • 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}

  • group Electric Field Due to a Ring of Charge

    group Small Group Activity

    30 min.

    Electric Field Due to a Ring of Charge
    Static Fields 2022 (6 years)

    coulomb's law electric field charge ring symmetry integral power series superposition

    Power Series Sequence (E&M)

    Ring Cycle Sequence

    Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

    In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.

  • 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 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 \((0,0,2L)\).

Learning Outcomes