Page 186 - Engineering Electromagnetics, 8th Edition

168 ENGINEERING ELECTROMAGNETICS

In order to find the capacitance between a conducting cone with its vertex sepa-
rated from a conducting plane by an infinitesimal insulating gap and its axis normal
to the plane, we first find the field strength:

−1 ∂V V 0
E =−∇V = a θ =− a θ
r ∂θ α
r sin θ ln tan
2
The surface charge density on the cone is then

− V 0
ρ S =
r sin α ln tan α
2

producing a total charge Q,

2π
− V 0 ∞ r sin α dφ dr
Q = r
sin α ln tan α 0 0
2

−2π 0 V 0 ∞
= dr
ln tan α 0
2
This leads to an infinite value of charge and capacitance, and it becomes necessary to
consider a cone of finite size. Our answer will now be only an approximation because
the theoretical equipotential surface is θ = α,a conical surface extending from r = 0
to r =∞, whereas our physical conical surface extends only from r = 0 to, say,
r = r 1 . The approximate capacitance is

2π r 1
C ˙= (43)
ln cot α
2

If we desire a more accurate answer, we may make an estimate of the capacitance
of the base of the cone to the zero-potential plane and add this amount to our answer.
Fringing, or nonuniform, fields in this region have been neglected and introduce an
additional source of error.

D6.6. Find |E| at P(3, 1, 2) in rectangular coordinates for the field of: (a)
two coaxial conducting cylinders, V = 50 V at ρ = 2m, and V = 20 V
at ρ = 3m;(b)two radial conducting planes, V = 50 V at φ = 10 , and
◦
◦
V = 20Vat φ = 30 .
Ans. 23.4 V/m; 27.2 V/m