Page 126 - Engineering Electromagnetics, 8th Edition

108 ENGINEERING ELECTROMAGNETICS

same r and φ coordinates. Under what conditions does the answer agree with
Eq. (33), for the potential at θ a ?
4.29 A dipole having a moment p = 3a x − 5a y + 10a z nC · mis located at
Q(1, 2, −4) in free space. Find V at P(2, 3, 4).
4.30 A dipole for which p = 10 0 a z C · mis located at the origin. What is the
equation of the surface on which E z = 0but E = 0?
4.31 A potential field in free space is expressed as V = 20/(xyz)V. (a) Find the
total energy stored within the cube 1 < x, y, z < 2. (b) What value would be
obtained by assuming a uniform energy density equal to the value at the
center of the cube?
4.32 (a) Using Eq. (35), find the energy stored in the dipole field in the region
r > a. (b)Why can we not let a approach zero as a limit?
4.33 A copper sphere of radius 4 cm carries a uniformly distributed total charge
of 5 µCin free space. (a) Use Gauss’s law to find D external to the sphere.
(b) Calculate the total energy stored in the electrostatic field. (c) Use W E =
2
Q /(2C)to calculate the capacitance of the isolated sphere.
3
4.34 A sphere of radius a contains volume charge of uniform density ρ 0 C/m .
Find the total stored energy by applying (a) Eq. (42); (b) Eq. (44).
4.35 Four 0.8 nC point charges are located in free space at the corners of a square
4cmona side. (a) Find the total potential energy stored. (b)A fifth 0.8 nC
charge is installed at the center of the square. Again find the total stored
energy.
4.36 Surface charge of uniform density ρ s lies on a spherical shell of radius b,
centered at the origin in free space. (a) Find the absolute potential
everywhere, with zero reference at infinity. (b) Find the stored energy in the
sphere by considering the charge density and the potential in a
two-dimensional version of Eq. (42). (c) Find the stored energy in the electric
field and show that the results of parts (b) and (c) are identical.