Page 192 - Engineering Electromagnetics, 8th Edition

174 ENGINEERING ELECTROMAGNETICS

y = 5 mm. Calculate the capacitance per square meter of surface area
if (a)region 3 is air; (b) r3 = r1 ;(c) r3 = r2 ;(d)region 3 is silver.
6.8 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 is given by
2
2
(ρ) = 0 (1 + ρ /a ). Find (a) E;(b) D;(c) Q;(d) C.
6.9 Two coaxial conducting cylinders of radius 2 cm and 4 cm have a length
of 1 m. The region between the cylinders contains a layer of dielectric from
ρ = c to ρ = d with r = 4. Find the capacitance if (a) c = 2 cm, d = 3 cm;
(b) d = 4 cm, and the volume of the dielectric is the same as in part (a).
6.10 A coaxial cable has conductor dimensions of a = 1.0mmand b = 2.7 mm.
The inner conductor is supported by dielectric spacers ( r = 5) in the
form of washers with a hole radius of 1 mm and an outer radius of 2.7 mm,
and with a thickness of 3.0 mm. The spacers are located every 2 cm down
the cable. (a)By what factor do the spacers increase the capacitance per
unit length? (b)If 100 V is maintained across the cable, find E at all points.
6.11 Two conducting spherical shells have radii a = 3cmand b = 6 cm. The
interior is a perfect dielectric for which r = 8. (a) Find C.(b)A portion of
the dielectric is now removed so that r = 1.0, 0 <φ <π/2, and r = 8,
π/2 <φ < 2π.Again find C.
6.12 (a) Determine the capacitance of an isolated conducting sphere of radius a in
free space (consider an outer conductor existing at r →∞). (b) The sphere is
to be covered with a dielectric layer of thickness d and dielectric contant r .If
r = 3, find d in terms of a such that the capacitance is twice that of part (a).

6.13 With reference to Figure 6.5, let b = 6m, h = 15 m, and the conductor
potential be 250 V. Take = 0 . Find values for K 1 , ρ L , a, and C.
6.14 Two #16 copper conductors (1.29 mm diameter) are parallel with a separation
d between axes. Determine d so that the capacitance between wires in air
is 30 pF/m.
6.15 A 2-cm-diameter conductor is suspended in air with its axis 5 cm from a
conducting plane. Let the potential of the cylinder be 100 V and that of the
plane be 0 V. (a) Find the surface charge density on the cylinder at a point
nearest the plane. (b) Plane at a point nearest the cylinder; (c) find
the capacitance per unit length.
6.16 Consider an arrangement of two isolated conducting
surfaces of any shape that form a capacitor. Use the definitions of capacitance
(Eq. (2) in this chapter) and resistance (Eq. (14) in Chapter 5) to show
that when the region between the conductors is filled with either conductive
material (conductivity σ)ora perfect dielectric (permittivity ), the resulting