Page 89 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 71

5. Thomas, G. B., Jr., and R. L. Finney. (see Suggested References for Chapter 1). The
divergence theorem is developed and illustrated from several different points of view on
pp. 976–980.

CHAPTER 3 PROBLEMS
3.1 Suppose that the Faraday concentric sphere experiment is performed in free
space using a central charge at the origin, Q 1 , and with hemispheres of radius
a.A second charge Q 2 (this time a point charge) is located at distance R
from Q 1 , where R >> a.(a) What is the force on the point charge before the
hemispheres are assembled around Q 1 ?(b) What is the force on the point
charge after the hemispheres are assembled but before they are discharged?
(c) What is the force on the point charge after the hemispheres are assembled
and after they are discharged? (d) Qualitatively, describe what happens as Q 2
is moved toward the sphere assembly to the extent that the condition R >> a
is no longer valid.
2
3.2 An electric field in free space is E = (5z / 0 ) ˆa z V/m. Find the total charge
contained within a cube, centered at the origin, of 4-m side length, in which
all sides are parallel to coordinate axes (and therefore each side intersects an
axis at ±2).
3.3 The cylindrical surface ρ = 8cm contains the surface charge density, ρ S =
2
5e −20|z| nC/m .(a) What is the total amount of charge present? (b)How
much electric flux leaves the surface ρ = 8cm,1cm < z < 5 cm,
30 <φ < 90 ?
◦
◦
3.4 An electric field in free space is E = (5z / 0 ) ˆa z V/m. Find the total charge
3
contained within a sphere of 3-m radius, centered at the origin.
2
2
2
3.5 Let D = 4xya x + 2(x + z )a y + 4yza z nC/m and evaluate surface integrals
to find the total charge enclosed in the rectangular parallelepiped 0 < x < 2,
0 < y < 3, 0 < z < 5m.
3.6 In free space, a volume charge of constant density ρ ν = ρ 0 exists within the
region −∞ < x < ∞, −∞ < y < ∞, and −d/2 < z < d/2. Find D and E
everywhere.
3.7 Volume charge density is located in free space as ρ ν = 2e −1000r nC/m for
3
0 < r < 1 mm, and ρ ν = 0 elsewhere. (a) Find the total charge enclosed by
the spherical surface r = 1 mm. (b)By using Gauss’s law, calculate the value
of D r on the surface r = 1 mm.
3.8 Use Gauss’s law in integral form to show that an inverse distance field in
spherical coordinates, D = Aa r /r, where A is a constant, requires every
spherical shell of1m thickness to contain 4πA coulombs of charge. Does
this indicate a continuous charge distribution? If so, find the charge density
variation with r.