Activity: Electrostatic Potential Due to a Ring of Charge

Static Fields 2023 (8 years)

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.

What students learn
  • The electrostatic potential \(V\) from a distribution of charges can be found, via the superposition principle, by adding up the contribution from many small chunks of charge;
  • For round problems, the superposition should be performed as an integral over round coordinates;
  • The analytical and geometric meaning of the distance formula \(\vert\vec{r} - \vec{r}^{\prime}\vert\);
  • How to calculate linear charge density from a total charge and a distance;
  • How to use power series expansions to approximate integrals.

The Electrostatic Potential Due to a Ring of Charge
  1. Use the superposition principle for the electrostatic potential due to a continuous charge distribution: \begin{align} V(\vec{r})=\frac{1}{4\pi \epsilon_0} \int \frac{\rho'(\vec{r}^{\,\prime})}{\left| \vec{r}-\vec{r}'\right|}\, d\tau', \end{align} to find the electrostatic potential everywhere in space due to a uniformly charged ring with radius \(R\) and total charge \(Q\).

    Check with a teaching team member before moving on to subsequent parts below.

  2. Evaluate your expression for the special case of the potential on the \(z\)-axis.
  3. Evaluate your expression for the special case of the potential on the \(x\)-axis.
  4. Find a series expansion for the electrostatic potential in these special regions:
    1. Near the center of the ring, in the plane of the ring;
    2. Near the center of the ring, on the axis of the ring;
    3. Far from the ring on the axis of symmetry;
    4. Far from the ring, in the plane of the ring.

  • 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 2023 (8 years)

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

    Power Series Sequence (E&M)

    Ring Cycle Sequence

    Students work in small groups 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.

  • 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 Magnetic Vector Potential Due to a Spinning Charged Ring

    group Small Group Activity

    30 min.

    Magnetic Vector Potential Due to a Spinning Charged Ring
    Static Fields 2023 (6 years)

    compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

    Power Series Sequence (E&M)

    Ring Cycle Sequence

    Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

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

  • group Magnetic Field Due to a Spinning Ring of Charge

    group Small Group Activity

    30 min.

    Magnetic Field Due to a Spinning Ring of Charge
    Static Fields 2023 (7 years)

    magnetic fields current Biot-Savart law vector field symmetry

    Power Series Sequence (E&M)

    Ring Cycle Sequence

    Students work in small groups to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

    In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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 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 (with Series)

    group Small Group Activity

    60 min.

    Electrostatic Potential Due to a Pair of Charges (with Series)
    Static Fields 2023 (6 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.
  • assignment Linear Quadrupole (w/ series)

    assignment Homework

    Linear Quadrupole (w/ series)

    Power Series Sequence (E&M)

    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?

  • format_list_numbered Ring Cycle Sequence

    format_list_numbered Sequence

    Ring Cycle Sequence
    Students calculate electrostatic fields (\(V\), \(\vec{E}\)) and magnetostatic fields (\(\vec{A}\), \(\vec{B}\)) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.
  • assignment_ind Electrostatic Potential Due to a Point Charge

    assignment_ind Small White Board Question

    10 min.

    Electrostatic Potential Due to a Point Charge
    Static Fields 2023 (2 years)

    Warm-Up

    Ring Cycle Sequence

  • assignment Potential vs. Potential Energy

    assignment Homework

    Potential vs. Potential Energy
    Static Fields 2023 (6 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?


Author Information
Corinne Manogue, Leonard Cerny
Learning Outcomes