Electric field enhancement near a surface bump

[somearfar]

Homework Statement

Well I got a question regarding the electric field. So let's say an electric field E exists between two parallel plates. There is a hemispherical bump on one of the plates with radius of r, so how to calculate the electric field enhancement?

Thanks guys.

 

[rude man]

If h is the distance between plates, then far away from the hump, V = Eh. At the tip of the hump, the average electric field has to be V/(h-r).

Away from the tip, you have to determine the path of least distance s between the spot on the hump and the flat plate. The boundary conditions are that the E field must be perpendicular to the hump surface at the hump surface, and to and at the bottom plate. This could be a problem in the calculus of variations. Once you have determined this distance the average E field is again V/s.

 

[mingda]

You have to expand the potential using the orthogonal special functions under your symmetry, at each region, and match the boundary condition for dielectric displacements. Actually there is a paper about this problem: http://www.sciencedirect.com/science/article/pii/003810988190034X