9716

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 .

SNAPSHOTS

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

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.
.
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.
    • 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+