Ym. Zhang et Ah. Zemanian, A SIMPLE FORMULA FOR THE CONCENTRATION OF CHARGE ON A 3-DIMENSIONAL CORNER OF A CONDUCTOR, IEEE transactions on microwave theory and techniques, 44(6), 1996, pp. 975-979
A major problem in the computation of capacitance coefficients for mic
rowave transmission and VLSI interconnection systems is caused by the
singularities in the electric field at the corners and edges of conduc
tors. For edges, a solution is given by the Duncan correction, which i
s based on a two-dimensional (2-D) polar expansion of the field. No su
ch exact expansion exists for corners. Recent research by Beagles and
Whiteman has yielded an asymptotic expansion for the electric field In
the vicinity of a rectangular three-dimensional conductive corner, an
d this is used to derive a simple formula for the charge Q (in coulomb
s) concentrated at any such corner. The formula is Q = 1.307 epsilon d
(V-c - V-s), where epsilon is the dielectric permittivity (in farads p
er meter) of the medium surrounding the conductive corner, d is the le
ngth (in meters) of one Side of a cubic region situated on the conduct
or adjacent to the corner, V-c is the electric potential (in volts) of
the conductor, and V-s is the electric potential at a point in the me
dium displaced from the corner's apex along a line through the cube's
diagonal and at a distance equal to that diagonal. Q is the charge on
the cube's three surfaces lying along the conductor's surfaces. Such a
configuration is convenient for a finite-difference computation of ca
pacitance.