What shape would produce the greatest electric field?

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Discussion Overview

The discussion centers on the question of what shape would produce the greatest electric field at a given point in space, considering an incompressible material with a constant charge density. Participants explore the implications of charge distribution and geometry on electric field strength, touching on concepts from electrostatics and potential analogies with gravitational fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that the largest electric field concentration occurs at pointed ends and is minimized by flat or spherical surfaces, referencing practical applications in high voltage devices.
  • Another participant questions whether the initial claim applies to conducting materials, noting that for a constant charge density, the scenario may resemble maximizing gravitational fields with constant mass density objects.
  • A participant mentions that brush discharge from insulators increases with angularity, indicating that sharp tips can generate discharges more readily than smooth surfaces.
  • Concerns are raised about the assumption of constant charge density in the context of potential charge movement during discharge events.
  • One participant expresses appreciation for a referenced paper, indicating that it helped clarify their understanding of the topic.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the applicability of certain principles to non-conducting materials with constant charge density. There is no consensus on the optimal shape for maximizing electric field strength, as various viewpoints and analogies are presented.

Contextual Notes

Some assumptions regarding charge movement and the nature of the materials discussed remain unresolved. The relationship between electric field strength and geometry is complex and may depend on additional factors not fully explored in the discussion.

Who May Find This Useful

This discussion may be of interest to those studying electrostatics, materials science, or anyone exploring the effects of geometry on electric fields in theoretical or practical contexts.

Helmholtz
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Suppose you are given an incompressible material with a constant charge density. What shape would create the largest electric field at a given point in space? These seems like a calculus of variation problem, but I am wondering if there might be any clever trick.

$$\vec E = \frac{\rho}{4 \pi \epsilon_0} \iiint \frac{\hat r}{r^2}dx' dy' dz'$$
 
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The largest field concentration occurs at the pointy end ... and is minimized by flat or spherical surfaces.
Thus inside any high voltage device a very sharp tip is used for emission, and mirror polished smooth surfaces are used wherever emission is not desired. Knowing the answer, you can now look for clever tricks.
 
UltrafastPED said:
The largest field concentration occurs at the pointy end ... and is minimized by flat or spherical surfaces.
Isn't this for conducting materials, where the charges can move? For "constant charge density" the problem seems similar to this problem about maximizing the gravitational field with a constant mass density object:

http://pages.physics.cornell.edu/~aalemi/random/planet.pdf
 
Last edited:
A.T. said:
Isn't this for conducting materials, where the charges can move? For "constant charge density" the problem seems similar to this problem about maximizing the gravitational field with a constant mass density object:

http://pages.physics.cornell.edu/~aalemi/random/planet.pdf

Brush discharge from an insulator increases with angularity ... a sharp tip (eg, a crack or edge) will generate a discharge long before a nice smooth surface. You can see this in action if you have a Van de Graaff generator handy.
 
UltrafastPED said:
Brush discharge from an insulator...
If there can be a discharge, it means that charges can move. So how can you be sure there is constant charge density, as the OP states?
 
Thank you for the reference paper, I found that to be a great help. I believe I now understand what the answer is, as derived in the paper.
 

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