The electrostatic potential of a uniformly charged ring

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When faced with mathematical methods, undergraduate students have difficulty in grasping the reality of various approaches and special functions. It is only when they take a more specialized course such as classical electromagnetism that they finally see the connection. A problem that we believe illustrates very well the depth and variety of methods to be employed in a realistic physics situation is the calculation of the off-axis electrostatic potential created by a single uniformly charged ring. By using two methods to solve this problem, we pedagogically show how certain special functions appear and how some interesting connections can be uncovered. Results derived by using conventional direct integration techniques (in terms of complete elliptic integrals of the first kind) are compared with solutions obtained by using a different method that starts with the integrals of Bessel functions. Comparison of the respective results uncovers an interesting transformation formula between the complete elliptic integrals of the first kind with different arguments. © 2009 IOP Publishing Ltd.

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