Spheroidal Protrusion in a Uniform Electric Field

This Demonstration shows how the electric field is produced by a spheroidal protrusion. The main feature is the enhancement of the field around the tip. The electric field is shown by coloring and arrows, and the corresponding equipotential lines are drawn as white dashed lines. You can vary the height of protrusion and the aspect ratio , where is the radius of the protrusion. The indicated maximum electric field is the value at the tip of the spheroid. All field values are normalized to the applied uniform field .

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DETAILS

Snapshot 1: protrusion with a moderate aspect ratio ()
Snapshot 2: protrusion with a small aspect ratio ()
Snapshot 3: protrusion with a large aspect ratio ()
Prolate spherical coordinates are related to Cartesian coordinates by:
, , .
Let represent the spheroidal protrusion of height and radius . The parameters are selected as and . The imposed electric field in the direction is related to the potential by , which is given in prolate spherical coordinates as , with the boundary conditions , . The following differential equation is derived from Laplace's equation.
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It is evident that (from ) is a solution of this equation. Using this particular solution, it is possible to fix to fit the boundary conditions, and to obtain the formulation for . Eventually, the potential for the present problem takes the following form:
.
The electric field can be obtained from . The calculation again uses the prolate spherical coordinates , but the results are converted to Cartesian - coordinates. The maximum field occurs at the protrusion tip .
References
[1] J. A. Stratton, Electromagnetic Theory, 1st ed., New York: McGraw–Hill, 1941.
[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.
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