Electrostatic Fields Using Conformal Mapping

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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:
Contributed by: Y. Shibuya (January 2013)
Open content licensed under CC BY-NC-SA
Snapshots
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.
Permanent Citation