Page 193 - Engineering Electromagnetics, 8th Edition

CHAPTER 6 Capacitance 175

resistance and capacitance of the structures are related through the simple
formula RC = /σ. What basic properties must be true about both the
dielectric and the conducting medium for this condition to hold for certain?
6.17 Construct a curvilinear-square map for a coaxial capacitor of 3 cm inner
radius and 8 cm outer radius. These dimensions are suitable for the drawing.
(a) Use your sketch to calculate the capacitance per meter length, assuming
r = 1. (b) Calculate an exact value for the capacitance per unit length.

6.18 Construct a curvilinear-square map of the potential field about two
parallel circular cylinders, each of 2.5 cm radius, separated by a center-
to-center distance of 13 cm. These dimensions are suitable for the actual
sketch if symmetry is considered. As a check, compute the capacitance
per meter both from your sketch and from the exact formula. Assume r = 1.
6.19 Construct a curvilinear-square map of the potential field between two
parallel circular cylinders, one of 4 cm radius inside another of 8 cm radius.
The two axes are displaced by 2.5 cm. These dimensions are suitable for
the drawing. As a check on the accuracy, compute the capacitance per meter
from the sketch and from the exact expression:
2π
C =
2
cosh −1 [(a + b − D )/(2ab)]
2
2
where a and b are the conductor radii and D is the axis separation.
6.20 A solid conducting cylinder of 4 cm radius is centered within a rectangular
conducting cylinder with a 12 cm by 20 cm cross section. (a) Make a full-size
sketch of one quadrant of this configuration and construct a curvilinear-square
map for its interior. (b) Assume = 0 and estimate C per meter length.
6.21 The inner conductor of the transmission line shown in Figure 6.13 has a
square cross section 2a × 2a, whereas the outer square is 4a × 5a. The axes
are displaced as shown. (a) Construct a good-sized drawing of this
transmission line, say with a = 2.5 cm, and then prepare a curvilinear-square
plot of the electrostatic field between the conductors. (b) Use the map to
calculate the capacitance per meter length if = 1.6 0 .(c)How would your
result to part (b) change if a = 0.6 cm?
6.22 Two conducting plates, each 3 × 6 cm, and three slabs of dielectric, each
1 × 3 × 6 cm, and having dielectric constants of 1, 2, and 3, are assembled
into a capacitor with d = 3 cm. Determine the two values of capacitance
obtained by the two possible methods of assembling the capacitor.
6.23 Atwo-wire transmission line consists of two parallel perfectly conducting
cylinders, each having a radius of 0.2 mm, separated by a center-to-center
distance of 2 mm. The medium surrounding the wires has r = 3 and σ =
1.5 mS/m. A 100-V battery is connected between the wires. (a) Calculate
the magnitude of the charge per meter length on each wire. (b) Using
the result of Problem 6.16, find the battery current.