Page 191 - Engineering Electromagnetics, 8th Edition

CHAPTER 6 Capacitance 173

3. Dekker, A. J. See references for Chapter 5.
4. Hayt, W. H. Jr., and J. E. Kemmerly. Engineering Circuit Analysis. 5th ed. New York:
McGraw-Hill, 1993.
5. Collin, R. E., and R. E. Plonsey. Principles and Applications of Electromagnetic Fields.
New York: McGraw-Hill, 1961. Provides an excellent treatment of methods of solving
Laplace’s and Poisson’s equations.
6. Smythe, W. R. Static and Dynamic Electricity. 3rd ed. Taylor and Francis, 1989. An
advanced treatment of potential theory is given in Chapter 4.

CHAPTER 6 PROBLEMS

6.1 Consider a coaxial capacitor having inner radius a, outer radius b, unit
length, and filled with a material with dielectric constant, r . Compare this to
a parallel-plate capacitor having plate width w, plate separation d, filled with
the same dielectric, and having unit length. Express the ratio b/a in terms of
the ratio d/w, such that the two structures will store the same energy for a
given applied voltage.
6.2 Let S = 100 mm , d = 3 mm, and r = 12 for a parallel-plate capacitor.
2
(a) Calculate the capacitance. (b) After connecting a 6-V battery across the
capacitor, calculate E, D, Q, and the total stored electrostatic energy.
(c)With the source still connected, the dielectric is carefully withdrawn
from between the plates. With the dielectric gone, recalculate E, D, Q, and
the energy stored in the capacitor. (d)If the charge and energy found in
part (c) are less than the values found in part (b) (which you should have
discovered), what became of the missing charge and energy?
6.3 Capacitors tend to be more expensive as their capacitance and
maximum voltage V max increase. The voltage V max is limited by the field
strength at which the dielectric breaks down, E BD . Which of these dielectrics
will give the largest CV max product for equal plate areas? (a) Air: r = 1,
E BD = 3 MV/m. (b) Barium titanate: r = 1200, E BD = 3 MV/m. (c) Silicon
dioxide: r = 3.78, E BD = 16 MV/m. (d) Polyethylene: r = 2.26, E BD =
4.7 MV/m.
6.4 An air-filled parallel-plate capacitor with plate separation d and plate
area A is connected to a battery that applies a voltage V 0 between
plates. With the battery left connected, the plates are moved apart to a
distance of 10d. Determine by what factor each of the following
quantities changes: (a) V 0 ;(b) C;(c) E;(d) D;(e) Q;( f ) ρ S ;(g) W E .
6.5 A parallel-plate capacitor is filled with a nonuniform dielectric characterized
6 2
by r = 2 + 2 × 10 x , where x is the distance from one plate in meters.
2
If S = 0.02 m and d = 1 mm, find C.
6.6 Repeat Problem 6.4, assuming the battery is disconnected before the plate
separation is increased.
6.7 Let r1 = 2.5 for 0 < y < 1 mm, r2 = 4 for 1 < y < 3 mm, and r3 for
3 < y < 5mm (region 3). Conducting surfaces are present at y = 0 and