Student handout: Electrostatic Potential Due to a Ring of Charge

Static Fields 2024 (12 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.

Author Information
Corinne Manogue, Leonard Cerny
Keywords
electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula
Learning Outcomes