Electrostatic Orbits

This Demonstration shows the orbit of two charged spheres attracted to each other by electrostatic forces. When the spheres are far apart, the electrostatic force between them can be found using Coulomb's law for point charges. When the spheres are closer to each other, however, the actual force becomes much larger than Coulomb's law predictions. This difference is caused by large polarization of the charge on each sphere (the charge distributions become asymmetric) as their separation becomes relatively small. By adjusting their initial separation, charge ratio, and size ratio and then giving them some angular momentum, it is possible to create and sustain a binary orbiting system. This Demonstration initially sets these factors to create a stable, circular orbit for each size ratio. By increasing or decreasing the other values, the orbits will become "unstable," which can be seen by an early collision, a wobbly orbital pattern, or an escaping orbit with a hyperbolic trajectory.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Electrostatic orbits generally occur in nature only on a microscopic scale (i.e. the electron orbiting the nucleus). This is because larger versions of the orbits require much more electrostatic energy than is usually available. Even so, a macroscopic version of electrostatic orbits is instructive. This Demonstration shows a simulation of electrostatic orbits that complements theory and experiments that are already available in the literature [1, 2, 3].
For this simulation, Newton's second law , ) is used to numerically solve for the positions of each sphere using Mathematica's built-in function NDSolve. For the electrostatic force between the two spheres, unpublished data [4] is used. The force data was accurately calculated (for several different sphere sizes) at some specific distances using the method of image charges [5]. A fitting function is then used to express the force as a function of the distance. The adjustable parameters that determine the motion of the spheres are the charge ratio of the spheres (with one charge set equal to 1), the size ratio (with one sphere's radius set to 1), the initial center-to-center separation, and the initial angular momentum. One sphere is held fixed, so only the mass of the orbiting sphere is varied. Since the angular momentum is adjustable, the mass of the orbiting sphere can be set equal to 1. The variation of mass is accounted for through the value of the angular momentum [1].
[1] S. Banerjee, B. Taylor, and A. Banerjee, "On the Stability of Electrostatic Orbits," American Journal of Physics, 77(5), 2009 pp. 396–400. doi:10.1119/1.3100774.
[2]. S. Banerjee, K. W. Andring, D. L. Campbell, J. A. Janeski, D. A. Keedy, S. P. Quinn, and B. K. Hoffmeister, "Orbital Motion of Electrically Charged Spheres in Microgravity," The Physics Teacher, 46(8), 2008 pp. 460–464. doi:10.1119/1.2999060.
[3] B. K. Hoffmeister, D. A. Meyer, B. M. Atkins, G. A. Franks, J. T. Fuchs, L. Li, C. W. Sliger, and J. E. Thompson, "Orbital Dynamics of Two Electrically Charged Conducting Spheres," America Journal of Physics, 78(10), 2010 pp. 1002–1006. doi:10.1119/1.3456117.
[4] S. Banerjee and E. Nelsen, data to be published.
[5] J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1, New York: Dover, 1954 pp. 281–283.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+