Electric Fields for Pairs of Cylinders or Spheres

Electric fields for either a pair of parallel cylinders or a pair of spheres (a sphere gap) are calculated and plotted. The radii of the two cylinders or spheres are assumed to be same.
For a pair of parallel cylinders, the electric field is equivalent to that of parallel line charges with a separation distance , where is the gap length and is the common cylinder radius.
For a pair of spheres (sphere gap), the electric field can be calculated analytically using bispherical coordinates. However, it is far simpler to use the image method, which is applied here.
In both cases, the gap length and radius are selected as the configuration parameters. You can set the voltages of the conductors and using the sliders.
An asymmetric field appears in the sphere gap case if the applied voltage is not symmetrical (i.e. ). However, in the cylinder system, the field is always symmetrical since the potentials of the cylinders extend to infinity.


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


Snapshot 1: a pair of cylinders, symmetrical field observed for voltage difference
Snapshot 2: a sphere gap, symmetrical field observed for
Snapshot 3: a sphere gap, asymmetrical field observed for (the usual case for high voltage testing, in which one sphere is grounded)
In both cases, the electric field can be calculated from the potential function by .
The energy density is obtainable by .
The potential at any point is expressed as follows:
Pair of Parallel Cylinders
Assuming the line charges are , separated by the length , then , where , are the distances to the line charges. The value of is determined by per-unit-length capacitance .
Pair of Spheres (sphere gap)
Denoting the image charges of order , at shifted positions, , where , are the distances to the image charges. An upper limit of is found satisfactory in all these examples.
[1] C. R. Paul, Analysis of Multiconductor Transmission Lines, New York: John Wiley & Sons, 1994.
[2] P. T. Metzer and J. E. Lane, "Electric Potential Due to a System of Conducting Spheres," The Open Applied Physics Journal, 2, 2009 pp. 32–48. http://benthamopen.com/ABSTRACT/TOAPJ-2-32.
    • 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+