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:
Let
represent the spheroidal protrusion of height
and radius
. The parameters are selected as
and
. The imposed electric field in the
direction
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
(from
) 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
.
[1] J. A. Stratton,
Electromagnetic Theory, 1st ed., New York: McGraw–Hill, 1941.
[2] P. Moon and D. E. Spencer,
Field Theory Handbook: Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed., Cleveland: John T. Zubal, 2003.