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Electrostatic Potential Due to a Ring

Keywords: Small Group Activity, Compare and Contrast, Computer Visualization, Maple, Mathematica, Electrostatic Potential and Potential Energy, Charge, Taylor and Power Series, Scalar Field, Superposition, Symmetry

Highlights of the activity

This small group activity is designed to help students wrestle with the concepts of charge density and electrostatic potential, as well as provide practice using power series approximations.
Students are asked to create an integral expression for the electrostatic potential everywhere in space caused by a ring with total charge Q and radius R and to develop the power series expansion for the potential near the center or far from the ring.
The wrap-up discussion focuses on how to create an appropriate expression for an electrostatic potential.

Reasons to spend class time on the activity

The first concept students need to understand is linear charge density. Students must grapple with the underlying concept of charge density, but also understand how this linear density relates to the “chopping and adding” aspect of integration. Students frequently leave math classes understanding integration as “the area under a curve”. This activity pushes students to transform their understanding of integration to focus on “chopping and adding”.

This activity also gives students the opportunity to use curvilinear coordinates and then realize that they cannot successfully integrate without transforming them into rectangular coordinates. Understanding that $|\Vec{r} - \Vec r'|$ cannot be integrated by simply using “$r$” in curvilinear coordinates is an important realization.

The final component is that students need to recognize an elliptic integral and what to do when they run into one. Most commonly students have never seen such “unsolvable” integrals in their calculus classes.