Snapshot 1: protrusion with a moderate aspect ratio (
Snapshot 2: protrusion with a small aspect ratio (
Snapshot 3: protrusion with a large aspect ratio (
Prolate spherical coordinates
are related to Cartesian coordinates by:
represent the spheroidal protrusion of height
. The parameters are selected as
. The imposed electric field in the
is related to the potential by
, which is given in prolate spherical coordinates as
, with the boundary conditions
. The following differential equation is derived from Laplace's equation.
It is evident that
) is a solution of this equation. Using this particular solution, it is possible to fix
to fit the boundary conditions, and to obtain the formulation for
. Eventually, the potential for the present problem takes the following form:
The electric field can be obtained from
. The calculation again uses the prolate spherical coordinates
, but the results are converted to Cartesian
coordinates. The maximum field occurs at the protrusion tip
 J. A. Stratton, Electromagnetic Theory
, 1st ed., New York: McGraw–Hill, 1941.
 P. Moon and D. E. Spencer, Field Theory Handbook: Including Coordinate Systems, Differential Equations, and Their Solutions
, 2nd ed., Cleveland: John T. Zubal, 2003.