Electrostatic force between two arbitrary shapes

[yiorgos]

I am looking some help on applying the well known formula of the electrostatic force between two parallel capacitor plates on charged surfaces which have non usual shapes.
My approach is to break apart the total surface into segments with common shapes
and calculating the force between each two of them. Although this approach neglects the mutual interaction among one segment with all the others I think it would result in an acceptable approximation. My question is if the total force is given by arithmetically adding all the single forces or it is more complicated.

My second and more important for me question is how to calculate the force between two surfaces when the shape of one or both of them is that of a cone. I already know that at sharp surfaces the charge concentration is increased but I can't find any formula for calculating it.

Third and last, for two given surfaces, which are the shapes resulting in the greatest electrostatic force between them?

Thanks in advance for your time

 

[Shooting Star]

I am looking some help on applying the well known formula of the electrostatic force between two parallel capacitor plates on charged surfaces which have non usual shapes.
My approach is to break apart the total surface into segments with common shapes
and calculating the force between each two of them. Although this approach neglects the mutual interaction among one segment with all the others I think it would result in an acceptable approximation. My question is if the total force is given by arithmetically adding all the single forces or it is more complicated.

This is not really an answer, just a note of caution. Forces are vectors, so you can't add them arithmetically. Break them into the three Cartesian components, and add those arithmetically.

 

[yiorgos]

Thanks for your reply.
You are right, I had in mind the parallel plate case where all forces are parallel.

But still I can't figure out how to manipulate the case of the cone (two cones).
Any suggestions on this?

 

[Shooting Star]

I will really have to think about that one. If anything comes to my mind, I'll let you know. The vertex of the cone is not smooth...

 

[yiorgos]

That's why the charge density is increased as we get closer to the edge.
If you can quantify the above density then you can relatively easy calculate
the force between them.
But as I said I can't find any formula of calculating that.