Page 197 - Engineering Electromagnetics, 8th Edition

CHAPTER 6 Capacitance 179

location of the 20 V equipotential surface. (b) Find E r,max .(c) Find r if
2
the surface charge density on the inner sphere is 1.0 µC/m .
6.42 The hemisphere 0 < r < a, 0 <θ <π/2, is composed of homogeneous
conducting material of conductivity σ. The flat side of the hemisphere
rests on a perfectly conducting plane. Now, the material within the
conical region 0 <θ <α, 0 < r < a is drilled out and replaced with
material that is perfectly conducting. An air gap is maintained between the
r = 0 tip of this new material and the plane. What resistance
is measured between the two perfect conductors? Neglect fringing fields.
6.43 Two coaxial conducting cones have their vertices at the origin and the z axis
as their axis. Cone A has the point A(1, 0, 2) on its surface, while cone B
has the point B(0, 3, 2) on its surface. Let V A = 100 V and V B = 20 V. Find
(a) α for each cone; (b) V at P(1, 1, 1).
6.44 A potential field in free space is given as V = 100 ln tan(θ/2) + 50 V.
(a) Find the maximum value of |E θ | on the surface θ = 40 ◦
for 0.1 < r < 0.8m,60 <φ < 90 .(b) Describe the surface V = 80 V.
◦
◦
6.45 In free space, let ρ ν = 200 0 /r 2.4 .(a) Use Poisson’s equation to
2
find V (r)ifitis assumed that r E r → 0 when r → 0, and also that V → 0
as r →∞.(b)Now find V (r)by using Gauss’s law and a line integral.
6.46 By appropriate solution of Laplace’s and Poisson’s equations, determine
the absolute potential at the center of a sphere of radius a, containing
uniform volume charge of density ρ 0 . Assume permittivity 0 everywhere.
Hint: What must be true about the potential and the electric
field at r = 0 and at r = a?