What shape would produce the greatest electric field?

[Helmholtz]

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.

$$\vec E = \frac{\rho}{4 \pi \epsilon_0} \iiint \frac{\hat r}{r^2}dx' dy' dz'$$

 

[UltrafastPED]

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.

 

[A.T.]

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

 

[UltrafastPED]

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.

 

[A.T.]

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?

 

[Helmholtz]

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.