Page 63 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 45

2.8 A crude device for measuring charge consists of two small insulating spheres
of radius a, one of which is fixed in position. The other is movable along the
x axis and is subject to a restraining force kx, where k is a spring constant.
The uncharged spheres are centered at x = 0 and x = d, the latter fixed. If
the spheres are given equal and opposite charges of Q/C, obtain the
expression by which Q may be found as a function of x. Determine the
maximum charge that can be measured in terms of 0 , k, and d, and state
the separation of the spheres then. What happens if a larger charge is applied?
2.9 A 100-nC point charge is located at A(−1, 1, 3) in free space. (a) Find the
locus of all points P(x, y, z)at which E x = 500 V/m. (b) Find y 1 if
P(−2, y 1 , 3) lies on that locus.
2.10 A charge of −1nCis located at the origin in free space. What charge must be
located at (2, 0, 0) to cause E x to be zero at (3, 1, 1)?

2.11 A charge Q 0 located at the origin in free space produces a field for which
E z = 1 kV/m at point P(−2, 1, −1). (a) Find Q 0 . Find E at M(1, 6, 5) in
(b) rectangular coordinates; (c)cylindrical coordinates; (d) spherical
coordinates.
2.12 Electrons are in random motion in a fixed region in space. During any 1 µs
interval, the probability of finding an electron in a subregion of volume
2
10 −15 m is 0.27. What volume charge density, appropriate for such time
durations, should be assigned to that subregion?
3
2.13 A uniform volume charge density of 0.2 µC/m is present throughout the
spherical shell extending from r = 3cmto r = 5 cm. If ρ ν = 0 elsewhere,
find (a) the total charge present throughout the shell, and (b) r 1 if half the
total charge is located in the region 3 cm < r < r 1 .
2.14 The electron beam in a certain cathode ray tube possesses cylindrical
2
−8
symmetry, and the charge density is represented by ρ v =−0.1/(ρ + 10 )
3
pC/m for 0 <ρ < 3 × 10 −4 m, and ρ v = 0 for ρ> 3 × 10 −4 m. (a) Find
the total charge per meter along the length of the beam; (b)if the electron
7
velocity is 5 × 10 m/s, and with one ampere defined as 1C/s, find the beam
current.
2.15 A spherical volume having a 2-µm radius contains a uniform volume charge
15
3
density of 10 C/m . (a) What total charge is enclosed in the spherical
volume? (b)Now assume that a large region contains one of these little
spheres at every corner of a cubical grid 3 mm on a side and that there is no
charge between the spheres. What is the average volume charge density
throughout this large region?
3
ρ 0 rcosθ C/m ,
2.16 Within a region of free space, charge density is given as ρ ν = a
where ρ 0 and a are constants. Find the total charge lying within (a) the
sphere, r ≤ a;(b) the cone, r ≤ a,0 ≤ θ ≤ 0.1π;(c) the region, r ≤ a,
0 ≤ θ ≤ 0.1π,0 ≤ φ ≤ 0.2π.