9772

Electrostatic Fields Using Conformal Mapping

A conformal mapping produces a complex function of a complex variable, , so that the analytical function maps the complex plane into the complex plane. This technique is useful for calculating two-dimensional electric fields: the curve in the plane where either or is constant corresponds to either an equipotential line or electric flux. This Demonstration shows 10 examples of electrostatic fields often encountered in high voltage applications. The electric field is shown in the - plane (or the plane, where ). The electrodes correspond to either or , where (). The 10 examples are:
• concentric circles:
• ellipses:
• hyperbolas:
• parabolas:
• bipolar circles:
• Cassinian ovals:
• elliptical pairs:
• blade to plate shape:
• Maxwell curves:
• square edge using a function derived by the Schwarz–Christoffel method
Three options give slightly different boundary conditions or electrode potentials or . The parameters are shown on the right. The calculated electric fields are shown by color, normalized to the average field or , where is the smallest distance between two electrodes. If you select option 1, the local field is high in the vicinity of sharp electrode edges. When selecting option 2 or 3, the values are reduced owing to blunted edge conditions. The white lines indicate the flux line and the dashed lines are the equipotential lines for constant or constant . Those two families of curves are orthogonal.

SNAPSHOTS

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

DETAILS

Snapshot 1: field of knife edge to knife edge using
Snapshot 2: field of parallel plate capacitor edge: Maxwell curves using
Snapshot 3: field of square edge to plane using a function derived using the Schwarz–Christoffel transformation
When a curve from a constant represents an equipotential line, the electric field can be calculated from . Therefore, its magnitude is given by
.
The calculation is done for a limited number of and values to save time. Please be patient, particularly for and .
References
[1] H. Prinz, Hochspannungsfelder, München: R. Oldenbourg Verlag, 1969.
[2] P. Moon and D. E. Spencer, Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions, 2nd ed., Cleveland: John T. Zubal, 2003.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+