## Student handout: Electric Field Due to a Ring of Charge

Static Fields 2021

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.

What students learn
• to perform a electric field calculation using Coulomb's Law;
• to decide which form of Coulomb's Law to use, depending on the dimensions of the charge density;
• how to find charge density from total charge $Q$ and the geometry of the problem, radius $R$;
• to write the distance formula $\vec{r}-\vec{r'}$ in both the numerator and denominator of Coulomb's Law in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
The Electrostatic Field Due to a Ring of Charge
1. 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 the electric field everywhere in space due to a charged ring with radius $R$ and total charge $Q$.
2. Evaluate your expression for the special case that $\vec{r}$ is on the $z$-axis.
3. Evaluate your expression for the special case that $\vec{r}$ is on the $x$-axis.
4. Find a series expansion for the electric field at these special locations:
1. Near the center of the ring, in the plane of the ring;
2. Near the center of the ring, on the axis perpendicular to the plane of the ring;
3. Far from the ring on the axis perpendicular to the plane of the ring;
4. Far from the ring, in the plane of the ring;

Author Information
Corinne Manogue, Leonard Cerny
Keywords
coulomb's law electric field charge ring symmetry integral power series superposition
Learning Outcomes