Page 125 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 107

4.22 A line charge of infinite length lies along the z axis and carries a uniform
linear charge density of ρ C/m. A perfectly conducting cylindrical shell,
whose axis is the z axis, surrounds the line charge. The cylinder (of radius b),
is at ground potential. Under these conditions, the potential function inside
the cylinder (ρ< b)isgivenby
ρ
V (ρ) = k − ln(ρ)
2π 0
where k is a constant. (a) Find k in terms of given or known parameters.
(b) Find the electric field strength, E, for ρ< b.(c) Find the electric field
strength, E, for ρ> b.(d) Find the stored energy in the electric field per unit
length in the z direction within the volume defined by ρ> a, where a < b.
4.23 It is known that the potential is given as V = 80ρ 0.6 V. Assuming free space
conditions, find. (a) E;(b) the volume charge density at ρ = 0.5m;(c) the
total charge lying within the closed surface ρ = 0.6, 0 < z < 1.
4.24 A certain spherically symmetric charge configuration in free space produces
an electric field given in spherical coordinates by
(ρ 0 r )/(100 0 ) a r V/m (r ≤ 10)
2
E(r) =
2
(100ρ 0 )/( 0 r ) a r V/m (r ≥ 10)
where ρ 0 is a constant. (a) Find the charge density as a function of position.
(b) Find the absolute potential as a function of position in the two regions,
r ≤ 10 and r ≥ 10. (c) Check your result of part b by using the gradient.
(d) Find the stored energy in the charge by an integral of the form of Eq. (43).
(e) Find the stored energy in the field by an integral of the form of Eq. (45).

4.25 Within the cylinder ρ = 2, 0 < z < 1, the potential is given by V = 100 +
50ρ + 150ρ sin φV. (a) Find V, E, D, and ρ ν at P(1, 60 , 0.5) in free space.
◦
(b)How much charge lies within the cylinder?
4.26 Let us assume that we have a very thin, square, imperfectly conducting plate
2mona side, located in the plane z = 0 with one corner at the origin such
that it lies entirely within the first quadrant. The potential at any point in
the plate is given as V =−e −x sin y. (a)An electron enters the plate at
x = 0, y = π/3 with zero initial velocity; in what direction is its initial
movement? (b) Because of collisions with the particles in the plate, the
electron achieves a relatively low velocity and little acceleration (the work
that the field does on it is converted largely into heat). The electron therefore
moves approximately along a streamline. Where does it leave the plate and in
what direction is it moving at the time?
4.27 Two point charges, 1 nC at (0, 0, 0.1) and −1nCat(0, 0, −0.1), are in free
space. (a) Calculate V at P(0.3, 0, 0.4). (b) Calculate |E| at P.(c)Now treat
the two charges as a dipole at the origin and find V at P.
4.28 Use the electric field intensity of the dipole [Section 4.7, Eq. (35)] to find the
difference in potential between points at θ a and θ b , each point having the