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.
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Part I - Finding the potential everywhere in space
Students should be assigned to work in groups of three and given the following instructions using the visual of a hula hoop or other large ring:
Prompt: "This is a ring with radius \(R\) and total charge \(Q\). Find a formula for the electrostatic potential \(V\) due to this ring that is valid everywhere in space".
Part II (Optional) - Power series expansion along an axis
With the charged ring in the \(x,y\)-plane, student groups are asked to make the power series expansion for either near or far from the plane on the \(z\) axis or near or far from the \(z\) axis in the \(x,y\)-plane. The instructor may choose to have the whole class do one particular case or have different groups do different cases, in a Compare and Contrast strategy (Compare and Contrast Activities).
Part I - Finding the potential everywhere in space
Part II (Optional) - Power series expansion along an axis
The Electrostatic Potential Due to a Ring of Charge
Find the electrostatic potential everywhere in space due to a charged ring with radius \(R\) and total charge \(Q\).
Evaluate your expression for the special case that \(\vec{r}\) is on the \(z\)-axis.
- Find a series expansion for the electrostatic potential at these special locations:
- Near the center of the ring, in the plane of the ring;
- Near the center of the ring, on the axis of the ring;
- Far from the ring on the axis of symmetry;
- Far from the ring, in the plane of the ring.