## Activity: Electric Field Due to a Ring of Charge

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.

• This activity is used in the following sequences
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;
• Media
The Electrostatic Field Due to a Ring of Charge
• Find the electric field 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 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 of the ring;
3. Far from the ring on the axis of symmetry;
4. Far from the ring, in the plane of the ring;

## Instructor's Guide

### Introduction

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 electric field $\vec{E}$ due to this ring that is valid everywhere in space".

### Student Conversations

This activity is part of a sequence (the Ring Cycle Sequence) of four electrostatics activities involving a ring of charge: $V$, $\vec{E}$, $\vec{A}$, $\vec{B}$. They are arranged so that the mathematical complexity of the problems increases in a natural way. If you are doing this activity as a standalone, please see the Student Conversations section of the previous activity (Electrostatic Potential Due to a Ring of Charge) for further advice.

Part I - Finding the electric field everywhere in space

The new idea in the electric field case is that the numerator is a vector. The basis vectors in cylindrical or spherical coordinates differ from point to point in space. Therefore, you CANNOT subtract two vectors that "live" at different points if they are expanded in curvilinear coordinate basis vectors. The numerator in this case must be expanded in rectangular basis vectors (so you can subtract) and components written in curvilinear coordinates (so that you can integrate)

$\vec{r}-\vec{r}^{\prime}=(s\cos{\phi}-R\cos{\phi^{\prime}})\hat{x}+(s\sin{\phi}-R\sin{\phi^{\prime}})\hat{y}+(z)\hat{z}$

Part II (Optional) - Power series expansion along an axis

This part will go much the same as for the potential case.

### Wrap-up

• Show a graph of the value of the electric field.
• Compare to the electrostatic potential.

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