The circle of Apollonius is named after the ancient geometrician Apollonius of Perga. This beautiful geometric construct can be helpful when solving some general problems of geometry and mathematical physics, optics, and electricity. Here we discuss two of its applications: localizing an object in space and calculating electric fields. First, we pose an entertaining localization problem to trigger students' interest in the subject. Analyzing this problem, we introduce the circle of Apollonius and show that this geometric technique helps solve the problem in an elegant and intuitive manner. Then we switch to seemingly unrelated problems of calculating the electric fields. We show that the zero equipotential line for two unlike charges is the Apollonius circle for these two charges and use this discovery to find the electric field of a charge positioned near a grounded conductive sphere. Finally, we pose some questions for further examination.

Topics
Mathematical physics, Educational aids, Educational assessment
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Eric W. Weisstein, “Apollonius circle,” from Math World—A Wolfram Web Resource; http://mathworld.wolfram.com/ApolloniusCircle.html.
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V.V. Batygin and I.N. Toptygin, Problems in Electrodynamics, 2nd ed. (Academic Press, London, 1978), Problem 118.
5.
David J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, Upper Saddle River, NJ, 1998), p. 121.
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© 2008 American Association of Physics Teachers.
2008
American Association of Physics Teachers
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