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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.

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DETAILS

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.
References
[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. www.benthamscience.com/open/toapj/articles/V002/32TOAPJ.pdf.
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