Electric field of uniformly charged tetrahedron

How to calculate electric field of arbitrary tetrahedron with uniform charge density at arbitrary point? I got:

• 4 tetrahedron vertices
• total charge in Coulombs
• point at which I want to compute E field vector

The point can be outside or inside the tetrahedron. The charge density is uniform across tetrahedron volume (not surface!).

EDIT

This is not a homework. I'm writing a simulation program to simulate electric fields around and inside an object defined by its 3D mesh. The field to calculate is of all the protons in an object. It is assumed they are uniformly spread across entire volume. Imagine protons field only as if you removed all valence electrons from the metal, so each atom would have +1e charge. I was thinking about subdividing the mesh to smaller chunks (tetrahedrons) and adding each tetrahedron fields at particular point.

EDIT 2

I came up with a solution for the inner field part if outside field equation is known. The idea is to check if the point at which we want compute the field is inside the tetrahedron. If it's not then we compute field regularly, if it's inside then we choose a plane containing one tetrahedron's edge (or any two vertices) and that point. This plane will split the tetrahedron into two smaller ones. We then compute and add fields of both of them. In this case point can be considered to be outside of each smaller tetrahedron, or lying on their surfaces.

• This sounds like a nasty problem. The locations of the verticies are arbitrary? The observation point is arbitrary? It might help a little to know why you want to calculate this. – garyp Apr 28 '17 at 22:12
• The answer I'm thinking of involves a triple integral. With integration variables inside a square root. And the bounds aren't that great either. I don't fancy solving it right now, sorry. – John Dvorak Apr 28 '17 at 22:15
• For outside, you can do multipole expansion. If you keep sufficient order (octopole and beyond), you can get good approximation, even surprisingly close to the charge. – AHusain Apr 28 '17 at 22:24