Page 124 - Engineering Electromagnetics, 8th Edition

106 ENGINEERING ELECTROMAGNETICS

and a point charge of 2 µCbe present at P(2, 0, 0). If V = 0at M(0, 0, 5),
find V at N(1, 2, 3).
2 2
4.12 In spherical coordinates, E = 2r/(r + a ) a r V/m. Find the potential at any
2
point, using the reference (a)V = 0at infinity; (b) V = 0at r = 0;
(c)V = 100Vat r = a.
4.13 Three identical point charges of 4 pC each are located at the corners of an
equilateral triangle 0.5 mm on a side in free space. How much work must be
done to move one charge to a point equidistant from the other two and on the
line joining them?
4.14 Given the electric field E = (y + 1)a x + (x − 1)a y + 2a z find the potential
difference between the points (a)(2, −2, −1) and (0, 0, 0); (b)(3, 2, −1) and
(−2, −3, 4).
4.15 Two uniform line charges, 8 nC/m each, are located at x = 1, z = 2, and at
x =−1, y = 2in free space. If the potential at the origin is 100 V, find V at
P(4, 1, 3).
4.16 A spherically symmetric charge distribution in free space (with 0 < r < ∞)
2
is known to have a potential function V (r) = V 0 a /r , where V 0 and a are
2
constants. (a) Find the electric field intensity. (b) Find the volume charge
density. (c) Find the charge contained inside radius a.(d) Find the total
energy stored in the charge (or equivalently, in its electric field).
4.17 Uniform surface charge densities of 6 and 2 nC/m are present at ρ = 2 and
2
6 cm, respectively, in free space. Assume V = 0at ρ = 4 cm, and calculate
V at (a) ρ = 5 cm; (b) ρ = 7 cm.
2
2
4.18 Find the potential at the origin produced by a line charge ρ L = kx/(x + a )
extending along the x axis from x = a to +∞, where a > 0. Assume a zero
reference at infinity.
4.19 The annular surface 1 cm <ρ < 3 cm, z = 0, carries the nonuniform surface
2
charge density ρ s = 5ρ nC/m . Find V at P(0, 0, 2 cm) if V = 0at infinity.
4.20 In a certain medium, the electric potential is given by

ρ 0 −ax
V (x) = 1 − e
a 0
where ρ 0 and a are constants. (a) Find the electric field intensity, E.(b) Find
the potential difference between the points x = d and x = 0. (c)Ifthe
ax
medium permittivity is given by (x) = 0 e , find the electric flux density,
D, and the volume charge density, ρ v ,in the region. (d) Find the stored
energy in the region (0 < x < d), (0 < y < 1), (0 < z < 1).
2
2
2 3
2
4.21 Let V = 2xy z + 3 ln(x + 2y + 3z )V in free space. Evaluate each of the
following quantities at P(3, 2, −1) (a) V ;(b) |V |;(c) E;(d) |E|;(e) a N ;
( f ) D.