Page 196 - Engineering Electromagnetics, 8th Edition

178 ENGINEERING ELECTROMAGNETICS

Figure 6.14 See Problem 6.39.

Determine a general rule on the allowed directions in which may vary
with respect to the local electric field.
6.37 Coaxial conducting cylinders are located at ρ = 0.5cmand ρ = 1.2 cm.
The region between the cylinders is filled with a homogeneous perfect
dielectric. If the inner cylinder is at 100 V and the outer at 0 V, find
(a) the location of the 20 V equipotential surface; (b) E ρ max ;(c) r if the
charge per meter length on the inner cylinder is 20 nC/m.
6.38 Repeat Problem 6.37, but with the dielectric only partially filling
the volume, within 0 <φ <π, and with free space in the remaining volume.

6.39 The two conducting planes illustrated in Figure 6.14 are
defined by 0.001 <ρ < 0.120 m, 0 < z < 0.1m, φ = 0.179 and 0.188 rad.
The medium surrounding the planes is air. For Region 1, 0.179 <φ < 0.188;
neglect fringing and find (a) V (φ); (b) E(ρ); (c) D(ρ); (d) ρ s on the upper
surface of the lower plane; (e) Q on the upper surface of the lower plane.
( f ) Repeat parts (a) through (c) for Region 2 by letting the location of
the upper plane be φ = .188 − 2π, and then find ρ s and Q on the lower
surface of the lower plane. (g) Find the total charge on the lower plane and
the capacitance between the planes.

6.40 A parallel-plate capacitor is made using two circular plates
of radius a, with the bottom plate on the xy plane, centered at the origin.
The top plate is located at z = d, with its center on the z axis. Potential V 0
is on the top plate; the bottom plate is grounded. Dielectric having radially
dependent permittivity fills the region between plates. The permittivity
2
2
is given by (ρ) = 0 (1 + ρ /a ). Find (a)V (z); (b) E;(c) Q;(d) C.
This is a reprise of Problem 6.8, but it starts with Laplace’s equation.
6.41 Concentric conducting spheres are located at r = 5mmand r = 20 mm.
The region between the spheres is filled with a perfect dielectric. If
the inner sphere is at 100 V and the outer sphere is at0V(a) Find the